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## G = C5×C22.34C24order 320 = 26·5

### Direct product of C5 and C22.34C24

direct product, metabelian, nilpotent (class 2), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C5×C22.34C24
 Chief series C1 — C2 — C22 — C2×C10 — C22×C10 — C5×C22⋊C4 — C5×C4⋊D4 — C5×C22.34C24
 Lower central C1 — C22 — C5×C22.34C24
 Upper central C1 — C2×C10 — C5×C22.34C24

Generators and relations for C5×C22.34C24
G = < a,b,c,d,e,f,g | a5=b2=c2=d2=f2=1, e2=c, g2=b, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, ede-1=gdg-1=bd=db, fef=be=eb, bf=fb, bg=gb, fdf=cd=dc, ce=ec, cf=fc, cg=gc, eg=ge, fg=gf >

Subgroups: 402 in 240 conjugacy classes, 146 normal (26 characteristic)
C1, C2, C2 [×2], C2 [×5], C4 [×2], C4 [×9], C22, C22 [×15], C5, C2×C4 [×2], C2×C4 [×8], C2×C4 [×6], D4 [×12], C23, C23 [×4], C10, C10 [×2], C10 [×5], C42 [×2], C22⋊C4 [×10], C4⋊C4 [×8], C22×C4, C22×C4 [×4], C2×D4 [×10], C20 [×2], C20 [×9], C2×C10, C2×C10 [×15], C42⋊C2, C4×D4 [×2], C4⋊D4 [×6], C22.D4 [×4], C42.C2, C41D4, C2×C20 [×2], C2×C20 [×8], C2×C20 [×6], C5×D4 [×12], C22×C10, C22×C10 [×4], C22.34C24, C4×C20 [×2], C5×C22⋊C4 [×10], C5×C4⋊C4 [×8], C22×C20, C22×C20 [×4], D4×C10 [×10], C5×C42⋊C2, D4×C20 [×2], C5×C4⋊D4 [×6], C5×C22.D4 [×4], C5×C42.C2, C5×C41D4, C5×C22.34C24
Quotients: C1, C2 [×15], C22 [×35], C5, C23 [×15], C10 [×15], C4○D4 [×2], C24, C2×C10 [×35], C2×C4○D4, 2+ 1+4 [×2], C22×C10 [×15], C22.34C24, C5×C4○D4 [×2], C23×C10, C10×C4○D4, C5×2+ 1+4 [×2], C5×C22.34C24

Smallest permutation representation of C5×C22.34C24
On 160 points
Generators in S160
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)(21 22 23 24 25)(26 27 28 29 30)(31 32 33 34 35)(36 37 38 39 40)(41 42 43 44 45)(46 47 48 49 50)(51 52 53 54 55)(56 57 58 59 60)(61 62 63 64 65)(66 67 68 69 70)(71 72 73 74 75)(76 77 78 79 80)(81 82 83 84 85)(86 87 88 89 90)(91 92 93 94 95)(96 97 98 99 100)(101 102 103 104 105)(106 107 108 109 110)(111 112 113 114 115)(116 117 118 119 120)(121 122 123 124 125)(126 127 128 129 130)(131 132 133 134 135)(136 137 138 139 140)(141 142 143 144 145)(146 147 148 149 150)(151 152 153 154 155)(156 157 158 159 160)
(1 35)(2 31)(3 32)(4 33)(5 34)(6 160)(7 156)(8 157)(9 158)(10 159)(11 17)(12 18)(13 19)(14 20)(15 16)(21 27)(22 28)(23 29)(24 30)(25 26)(36 55)(37 51)(38 52)(39 53)(40 54)(41 47)(42 48)(43 49)(44 50)(45 46)(56 75)(57 71)(58 72)(59 73)(60 74)(61 67)(62 68)(63 69)(64 70)(65 66)(76 95)(77 91)(78 92)(79 93)(80 94)(81 87)(82 88)(83 89)(84 90)(85 86)(96 115)(97 111)(98 112)(99 113)(100 114)(101 107)(102 108)(103 109)(104 110)(105 106)(116 135)(117 131)(118 132)(119 133)(120 134)(121 127)(122 128)(123 129)(124 130)(125 126)(136 155)(137 151)(138 152)(139 153)(140 154)(141 147)(142 148)(143 149)(144 150)(145 146)
(1 21)(2 22)(3 23)(4 24)(5 25)(6 20)(7 16)(8 17)(9 18)(10 19)(11 157)(12 158)(13 159)(14 160)(15 156)(26 34)(27 35)(28 31)(29 32)(30 33)(36 41)(37 42)(38 43)(39 44)(40 45)(46 54)(47 55)(48 51)(49 52)(50 53)(56 61)(57 62)(58 63)(59 64)(60 65)(66 74)(67 75)(68 71)(69 72)(70 73)(76 81)(77 82)(78 83)(79 84)(80 85)(86 94)(87 95)(88 91)(89 92)(90 93)(96 101)(97 102)(98 103)(99 104)(100 105)(106 114)(107 115)(108 111)(109 112)(110 113)(116 121)(117 122)(118 123)(119 124)(120 125)(126 134)(127 135)(128 131)(129 132)(130 133)(136 141)(137 142)(138 143)(139 144)(140 145)(146 154)(147 155)(148 151)(149 152)(150 153)
(1 156)(2 157)(3 158)(4 159)(5 160)(6 34)(7 35)(8 31)(9 32)(10 33)(11 22)(12 23)(13 24)(14 25)(15 21)(16 27)(17 28)(18 29)(19 30)(20 26)(36 136)(37 137)(38 138)(39 139)(40 140)(41 141)(42 142)(43 143)(44 144)(45 145)(46 146)(47 147)(48 148)(49 149)(50 150)(51 151)(52 152)(53 153)(54 154)(55 155)(56 127)(57 128)(58 129)(59 130)(60 126)(61 135)(62 131)(63 132)(64 133)(65 134)(66 120)(67 116)(68 117)(69 118)(70 119)(71 122)(72 123)(73 124)(74 125)(75 121)(76 107)(77 108)(78 109)(79 110)(80 106)(81 115)(82 111)(83 112)(84 113)(85 114)(86 100)(87 96)(88 97)(89 98)(90 99)(91 102)(92 103)(93 104)(94 105)(95 101)
(1 101 21 96)(2 102 22 97)(3 103 23 98)(4 104 24 99)(5 105 25 100)(6 94 20 86)(7 95 16 87)(8 91 17 88)(9 92 18 89)(10 93 19 90)(11 82 157 77)(12 83 158 78)(13 84 159 79)(14 85 160 80)(15 81 156 76)(26 114 34 106)(27 115 35 107)(28 111 31 108)(29 112 32 109)(30 113 33 110)(36 121 41 116)(37 122 42 117)(38 123 43 118)(39 124 44 119)(40 125 45 120)(46 134 54 126)(47 135 55 127)(48 131 51 128)(49 132 52 129)(50 133 53 130)(56 141 61 136)(57 142 62 137)(58 143 63 138)(59 144 64 139)(60 145 65 140)(66 154 74 146)(67 155 75 147)(68 151 71 148)(69 152 72 149)(70 153 73 150)
(1 56)(2 57)(3 58)(4 59)(5 60)(6 120)(7 116)(8 117)(9 118)(10 119)(11 128)(12 129)(13 130)(14 126)(15 127)(16 121)(17 122)(18 123)(19 124)(20 125)(21 61)(22 62)(23 63)(24 64)(25 65)(26 66)(27 67)(28 68)(29 69)(30 70)(31 71)(32 72)(33 73)(34 74)(35 75)(36 76)(37 77)(38 78)(39 79)(40 80)(41 81)(42 82)(43 83)(44 84)(45 85)(46 86)(47 87)(48 88)(49 89)(50 90)(51 91)(52 92)(53 93)(54 94)(55 95)(96 155)(97 151)(98 152)(99 153)(100 154)(101 147)(102 148)(103 149)(104 150)(105 146)(106 145)(107 141)(108 142)(109 143)(110 144)(111 137)(112 138)(113 139)(114 140)(115 136)(131 157)(132 158)(133 159)(134 160)(135 156)
(1 55 35 36)(2 51 31 37)(3 52 32 38)(4 53 33 39)(5 54 34 40)(6 154 160 140)(7 155 156 136)(8 151 157 137)(9 152 158 138)(10 153 159 139)(11 142 17 148)(12 143 18 149)(13 144 19 150)(14 145 20 146)(15 141 16 147)(21 47 27 41)(22 48 28 42)(23 49 29 43)(24 50 30 44)(25 46 26 45)(56 95 75 76)(57 91 71 77)(58 92 72 78)(59 93 73 79)(60 94 74 80)(61 87 67 81)(62 88 68 82)(63 89 69 83)(64 90 70 84)(65 86 66 85)(96 135 115 116)(97 131 111 117)(98 132 112 118)(99 133 113 119)(100 134 114 120)(101 127 107 121)(102 128 108 122)(103 129 109 123)(104 130 110 124)(105 126 106 125)

G:=sub<Sym(160)| (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20)(21,22,23,24,25)(26,27,28,29,30)(31,32,33,34,35)(36,37,38,39,40)(41,42,43,44,45)(46,47,48,49,50)(51,52,53,54,55)(56,57,58,59,60)(61,62,63,64,65)(66,67,68,69,70)(71,72,73,74,75)(76,77,78,79,80)(81,82,83,84,85)(86,87,88,89,90)(91,92,93,94,95)(96,97,98,99,100)(101,102,103,104,105)(106,107,108,109,110)(111,112,113,114,115)(116,117,118,119,120)(121,122,123,124,125)(126,127,128,129,130)(131,132,133,134,135)(136,137,138,139,140)(141,142,143,144,145)(146,147,148,149,150)(151,152,153,154,155)(156,157,158,159,160), (1,35)(2,31)(3,32)(4,33)(5,34)(6,160)(7,156)(8,157)(9,158)(10,159)(11,17)(12,18)(13,19)(14,20)(15,16)(21,27)(22,28)(23,29)(24,30)(25,26)(36,55)(37,51)(38,52)(39,53)(40,54)(41,47)(42,48)(43,49)(44,50)(45,46)(56,75)(57,71)(58,72)(59,73)(60,74)(61,67)(62,68)(63,69)(64,70)(65,66)(76,95)(77,91)(78,92)(79,93)(80,94)(81,87)(82,88)(83,89)(84,90)(85,86)(96,115)(97,111)(98,112)(99,113)(100,114)(101,107)(102,108)(103,109)(104,110)(105,106)(116,135)(117,131)(118,132)(119,133)(120,134)(121,127)(122,128)(123,129)(124,130)(125,126)(136,155)(137,151)(138,152)(139,153)(140,154)(141,147)(142,148)(143,149)(144,150)(145,146), (1,21)(2,22)(3,23)(4,24)(5,25)(6,20)(7,16)(8,17)(9,18)(10,19)(11,157)(12,158)(13,159)(14,160)(15,156)(26,34)(27,35)(28,31)(29,32)(30,33)(36,41)(37,42)(38,43)(39,44)(40,45)(46,54)(47,55)(48,51)(49,52)(50,53)(56,61)(57,62)(58,63)(59,64)(60,65)(66,74)(67,75)(68,71)(69,72)(70,73)(76,81)(77,82)(78,83)(79,84)(80,85)(86,94)(87,95)(88,91)(89,92)(90,93)(96,101)(97,102)(98,103)(99,104)(100,105)(106,114)(107,115)(108,111)(109,112)(110,113)(116,121)(117,122)(118,123)(119,124)(120,125)(126,134)(127,135)(128,131)(129,132)(130,133)(136,141)(137,142)(138,143)(139,144)(140,145)(146,154)(147,155)(148,151)(149,152)(150,153), (1,156)(2,157)(3,158)(4,159)(5,160)(6,34)(7,35)(8,31)(9,32)(10,33)(11,22)(12,23)(13,24)(14,25)(15,21)(16,27)(17,28)(18,29)(19,30)(20,26)(36,136)(37,137)(38,138)(39,139)(40,140)(41,141)(42,142)(43,143)(44,144)(45,145)(46,146)(47,147)(48,148)(49,149)(50,150)(51,151)(52,152)(53,153)(54,154)(55,155)(56,127)(57,128)(58,129)(59,130)(60,126)(61,135)(62,131)(63,132)(64,133)(65,134)(66,120)(67,116)(68,117)(69,118)(70,119)(71,122)(72,123)(73,124)(74,125)(75,121)(76,107)(77,108)(78,109)(79,110)(80,106)(81,115)(82,111)(83,112)(84,113)(85,114)(86,100)(87,96)(88,97)(89,98)(90,99)(91,102)(92,103)(93,104)(94,105)(95,101), (1,101,21,96)(2,102,22,97)(3,103,23,98)(4,104,24,99)(5,105,25,100)(6,94,20,86)(7,95,16,87)(8,91,17,88)(9,92,18,89)(10,93,19,90)(11,82,157,77)(12,83,158,78)(13,84,159,79)(14,85,160,80)(15,81,156,76)(26,114,34,106)(27,115,35,107)(28,111,31,108)(29,112,32,109)(30,113,33,110)(36,121,41,116)(37,122,42,117)(38,123,43,118)(39,124,44,119)(40,125,45,120)(46,134,54,126)(47,135,55,127)(48,131,51,128)(49,132,52,129)(50,133,53,130)(56,141,61,136)(57,142,62,137)(58,143,63,138)(59,144,64,139)(60,145,65,140)(66,154,74,146)(67,155,75,147)(68,151,71,148)(69,152,72,149)(70,153,73,150), (1,56)(2,57)(3,58)(4,59)(5,60)(6,120)(7,116)(8,117)(9,118)(10,119)(11,128)(12,129)(13,130)(14,126)(15,127)(16,121)(17,122)(18,123)(19,124)(20,125)(21,61)(22,62)(23,63)(24,64)(25,65)(26,66)(27,67)(28,68)(29,69)(30,70)(31,71)(32,72)(33,73)(34,74)(35,75)(36,76)(37,77)(38,78)(39,79)(40,80)(41,81)(42,82)(43,83)(44,84)(45,85)(46,86)(47,87)(48,88)(49,89)(50,90)(51,91)(52,92)(53,93)(54,94)(55,95)(96,155)(97,151)(98,152)(99,153)(100,154)(101,147)(102,148)(103,149)(104,150)(105,146)(106,145)(107,141)(108,142)(109,143)(110,144)(111,137)(112,138)(113,139)(114,140)(115,136)(131,157)(132,158)(133,159)(134,160)(135,156), (1,55,35,36)(2,51,31,37)(3,52,32,38)(4,53,33,39)(5,54,34,40)(6,154,160,140)(7,155,156,136)(8,151,157,137)(9,152,158,138)(10,153,159,139)(11,142,17,148)(12,143,18,149)(13,144,19,150)(14,145,20,146)(15,141,16,147)(21,47,27,41)(22,48,28,42)(23,49,29,43)(24,50,30,44)(25,46,26,45)(56,95,75,76)(57,91,71,77)(58,92,72,78)(59,93,73,79)(60,94,74,80)(61,87,67,81)(62,88,68,82)(63,89,69,83)(64,90,70,84)(65,86,66,85)(96,135,115,116)(97,131,111,117)(98,132,112,118)(99,133,113,119)(100,134,114,120)(101,127,107,121)(102,128,108,122)(103,129,109,123)(104,130,110,124)(105,126,106,125)>;

G:=Group( (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20)(21,22,23,24,25)(26,27,28,29,30)(31,32,33,34,35)(36,37,38,39,40)(41,42,43,44,45)(46,47,48,49,50)(51,52,53,54,55)(56,57,58,59,60)(61,62,63,64,65)(66,67,68,69,70)(71,72,73,74,75)(76,77,78,79,80)(81,82,83,84,85)(86,87,88,89,90)(91,92,93,94,95)(96,97,98,99,100)(101,102,103,104,105)(106,107,108,109,110)(111,112,113,114,115)(116,117,118,119,120)(121,122,123,124,125)(126,127,128,129,130)(131,132,133,134,135)(136,137,138,139,140)(141,142,143,144,145)(146,147,148,149,150)(151,152,153,154,155)(156,157,158,159,160), (1,35)(2,31)(3,32)(4,33)(5,34)(6,160)(7,156)(8,157)(9,158)(10,159)(11,17)(12,18)(13,19)(14,20)(15,16)(21,27)(22,28)(23,29)(24,30)(25,26)(36,55)(37,51)(38,52)(39,53)(40,54)(41,47)(42,48)(43,49)(44,50)(45,46)(56,75)(57,71)(58,72)(59,73)(60,74)(61,67)(62,68)(63,69)(64,70)(65,66)(76,95)(77,91)(78,92)(79,93)(80,94)(81,87)(82,88)(83,89)(84,90)(85,86)(96,115)(97,111)(98,112)(99,113)(100,114)(101,107)(102,108)(103,109)(104,110)(105,106)(116,135)(117,131)(118,132)(119,133)(120,134)(121,127)(122,128)(123,129)(124,130)(125,126)(136,155)(137,151)(138,152)(139,153)(140,154)(141,147)(142,148)(143,149)(144,150)(145,146), (1,21)(2,22)(3,23)(4,24)(5,25)(6,20)(7,16)(8,17)(9,18)(10,19)(11,157)(12,158)(13,159)(14,160)(15,156)(26,34)(27,35)(28,31)(29,32)(30,33)(36,41)(37,42)(38,43)(39,44)(40,45)(46,54)(47,55)(48,51)(49,52)(50,53)(56,61)(57,62)(58,63)(59,64)(60,65)(66,74)(67,75)(68,71)(69,72)(70,73)(76,81)(77,82)(78,83)(79,84)(80,85)(86,94)(87,95)(88,91)(89,92)(90,93)(96,101)(97,102)(98,103)(99,104)(100,105)(106,114)(107,115)(108,111)(109,112)(110,113)(116,121)(117,122)(118,123)(119,124)(120,125)(126,134)(127,135)(128,131)(129,132)(130,133)(136,141)(137,142)(138,143)(139,144)(140,145)(146,154)(147,155)(148,151)(149,152)(150,153), (1,156)(2,157)(3,158)(4,159)(5,160)(6,34)(7,35)(8,31)(9,32)(10,33)(11,22)(12,23)(13,24)(14,25)(15,21)(16,27)(17,28)(18,29)(19,30)(20,26)(36,136)(37,137)(38,138)(39,139)(40,140)(41,141)(42,142)(43,143)(44,144)(45,145)(46,146)(47,147)(48,148)(49,149)(50,150)(51,151)(52,152)(53,153)(54,154)(55,155)(56,127)(57,128)(58,129)(59,130)(60,126)(61,135)(62,131)(63,132)(64,133)(65,134)(66,120)(67,116)(68,117)(69,118)(70,119)(71,122)(72,123)(73,124)(74,125)(75,121)(76,107)(77,108)(78,109)(79,110)(80,106)(81,115)(82,111)(83,112)(84,113)(85,114)(86,100)(87,96)(88,97)(89,98)(90,99)(91,102)(92,103)(93,104)(94,105)(95,101), (1,101,21,96)(2,102,22,97)(3,103,23,98)(4,104,24,99)(5,105,25,100)(6,94,20,86)(7,95,16,87)(8,91,17,88)(9,92,18,89)(10,93,19,90)(11,82,157,77)(12,83,158,78)(13,84,159,79)(14,85,160,80)(15,81,156,76)(26,114,34,106)(27,115,35,107)(28,111,31,108)(29,112,32,109)(30,113,33,110)(36,121,41,116)(37,122,42,117)(38,123,43,118)(39,124,44,119)(40,125,45,120)(46,134,54,126)(47,135,55,127)(48,131,51,128)(49,132,52,129)(50,133,53,130)(56,141,61,136)(57,142,62,137)(58,143,63,138)(59,144,64,139)(60,145,65,140)(66,154,74,146)(67,155,75,147)(68,151,71,148)(69,152,72,149)(70,153,73,150), (1,56)(2,57)(3,58)(4,59)(5,60)(6,120)(7,116)(8,117)(9,118)(10,119)(11,128)(12,129)(13,130)(14,126)(15,127)(16,121)(17,122)(18,123)(19,124)(20,125)(21,61)(22,62)(23,63)(24,64)(25,65)(26,66)(27,67)(28,68)(29,69)(30,70)(31,71)(32,72)(33,73)(34,74)(35,75)(36,76)(37,77)(38,78)(39,79)(40,80)(41,81)(42,82)(43,83)(44,84)(45,85)(46,86)(47,87)(48,88)(49,89)(50,90)(51,91)(52,92)(53,93)(54,94)(55,95)(96,155)(97,151)(98,152)(99,153)(100,154)(101,147)(102,148)(103,149)(104,150)(105,146)(106,145)(107,141)(108,142)(109,143)(110,144)(111,137)(112,138)(113,139)(114,140)(115,136)(131,157)(132,158)(133,159)(134,160)(135,156), (1,55,35,36)(2,51,31,37)(3,52,32,38)(4,53,33,39)(5,54,34,40)(6,154,160,140)(7,155,156,136)(8,151,157,137)(9,152,158,138)(10,153,159,139)(11,142,17,148)(12,143,18,149)(13,144,19,150)(14,145,20,146)(15,141,16,147)(21,47,27,41)(22,48,28,42)(23,49,29,43)(24,50,30,44)(25,46,26,45)(56,95,75,76)(57,91,71,77)(58,92,72,78)(59,93,73,79)(60,94,74,80)(61,87,67,81)(62,88,68,82)(63,89,69,83)(64,90,70,84)(65,86,66,85)(96,135,115,116)(97,131,111,117)(98,132,112,118)(99,133,113,119)(100,134,114,120)(101,127,107,121)(102,128,108,122)(103,129,109,123)(104,130,110,124)(105,126,106,125) );

G=PermutationGroup([(1,2,3,4,5),(6,7,8,9,10),(11,12,13,14,15),(16,17,18,19,20),(21,22,23,24,25),(26,27,28,29,30),(31,32,33,34,35),(36,37,38,39,40),(41,42,43,44,45),(46,47,48,49,50),(51,52,53,54,55),(56,57,58,59,60),(61,62,63,64,65),(66,67,68,69,70),(71,72,73,74,75),(76,77,78,79,80),(81,82,83,84,85),(86,87,88,89,90),(91,92,93,94,95),(96,97,98,99,100),(101,102,103,104,105),(106,107,108,109,110),(111,112,113,114,115),(116,117,118,119,120),(121,122,123,124,125),(126,127,128,129,130),(131,132,133,134,135),(136,137,138,139,140),(141,142,143,144,145),(146,147,148,149,150),(151,152,153,154,155),(156,157,158,159,160)], [(1,35),(2,31),(3,32),(4,33),(5,34),(6,160),(7,156),(8,157),(9,158),(10,159),(11,17),(12,18),(13,19),(14,20),(15,16),(21,27),(22,28),(23,29),(24,30),(25,26),(36,55),(37,51),(38,52),(39,53),(40,54),(41,47),(42,48),(43,49),(44,50),(45,46),(56,75),(57,71),(58,72),(59,73),(60,74),(61,67),(62,68),(63,69),(64,70),(65,66),(76,95),(77,91),(78,92),(79,93),(80,94),(81,87),(82,88),(83,89),(84,90),(85,86),(96,115),(97,111),(98,112),(99,113),(100,114),(101,107),(102,108),(103,109),(104,110),(105,106),(116,135),(117,131),(118,132),(119,133),(120,134),(121,127),(122,128),(123,129),(124,130),(125,126),(136,155),(137,151),(138,152),(139,153),(140,154),(141,147),(142,148),(143,149),(144,150),(145,146)], [(1,21),(2,22),(3,23),(4,24),(5,25),(6,20),(7,16),(8,17),(9,18),(10,19),(11,157),(12,158),(13,159),(14,160),(15,156),(26,34),(27,35),(28,31),(29,32),(30,33),(36,41),(37,42),(38,43),(39,44),(40,45),(46,54),(47,55),(48,51),(49,52),(50,53),(56,61),(57,62),(58,63),(59,64),(60,65),(66,74),(67,75),(68,71),(69,72),(70,73),(76,81),(77,82),(78,83),(79,84),(80,85),(86,94),(87,95),(88,91),(89,92),(90,93),(96,101),(97,102),(98,103),(99,104),(100,105),(106,114),(107,115),(108,111),(109,112),(110,113),(116,121),(117,122),(118,123),(119,124),(120,125),(126,134),(127,135),(128,131),(129,132),(130,133),(136,141),(137,142),(138,143),(139,144),(140,145),(146,154),(147,155),(148,151),(149,152),(150,153)], [(1,156),(2,157),(3,158),(4,159),(5,160),(6,34),(7,35),(8,31),(9,32),(10,33),(11,22),(12,23),(13,24),(14,25),(15,21),(16,27),(17,28),(18,29),(19,30),(20,26),(36,136),(37,137),(38,138),(39,139),(40,140),(41,141),(42,142),(43,143),(44,144),(45,145),(46,146),(47,147),(48,148),(49,149),(50,150),(51,151),(52,152),(53,153),(54,154),(55,155),(56,127),(57,128),(58,129),(59,130),(60,126),(61,135),(62,131),(63,132),(64,133),(65,134),(66,120),(67,116),(68,117),(69,118),(70,119),(71,122),(72,123),(73,124),(74,125),(75,121),(76,107),(77,108),(78,109),(79,110),(80,106),(81,115),(82,111),(83,112),(84,113),(85,114),(86,100),(87,96),(88,97),(89,98),(90,99),(91,102),(92,103),(93,104),(94,105),(95,101)], [(1,101,21,96),(2,102,22,97),(3,103,23,98),(4,104,24,99),(5,105,25,100),(6,94,20,86),(7,95,16,87),(8,91,17,88),(9,92,18,89),(10,93,19,90),(11,82,157,77),(12,83,158,78),(13,84,159,79),(14,85,160,80),(15,81,156,76),(26,114,34,106),(27,115,35,107),(28,111,31,108),(29,112,32,109),(30,113,33,110),(36,121,41,116),(37,122,42,117),(38,123,43,118),(39,124,44,119),(40,125,45,120),(46,134,54,126),(47,135,55,127),(48,131,51,128),(49,132,52,129),(50,133,53,130),(56,141,61,136),(57,142,62,137),(58,143,63,138),(59,144,64,139),(60,145,65,140),(66,154,74,146),(67,155,75,147),(68,151,71,148),(69,152,72,149),(70,153,73,150)], [(1,56),(2,57),(3,58),(4,59),(5,60),(6,120),(7,116),(8,117),(9,118),(10,119),(11,128),(12,129),(13,130),(14,126),(15,127),(16,121),(17,122),(18,123),(19,124),(20,125),(21,61),(22,62),(23,63),(24,64),(25,65),(26,66),(27,67),(28,68),(29,69),(30,70),(31,71),(32,72),(33,73),(34,74),(35,75),(36,76),(37,77),(38,78),(39,79),(40,80),(41,81),(42,82),(43,83),(44,84),(45,85),(46,86),(47,87),(48,88),(49,89),(50,90),(51,91),(52,92),(53,93),(54,94),(55,95),(96,155),(97,151),(98,152),(99,153),(100,154),(101,147),(102,148),(103,149),(104,150),(105,146),(106,145),(107,141),(108,142),(109,143),(110,144),(111,137),(112,138),(113,139),(114,140),(115,136),(131,157),(132,158),(133,159),(134,160),(135,156)], [(1,55,35,36),(2,51,31,37),(3,52,32,38),(4,53,33,39),(5,54,34,40),(6,154,160,140),(7,155,156,136),(8,151,157,137),(9,152,158,138),(10,153,159,139),(11,142,17,148),(12,143,18,149),(13,144,19,150),(14,145,20,146),(15,141,16,147),(21,47,27,41),(22,48,28,42),(23,49,29,43),(24,50,30,44),(25,46,26,45),(56,95,75,76),(57,91,71,77),(58,92,72,78),(59,93,73,79),(60,94,74,80),(61,87,67,81),(62,88,68,82),(63,89,69,83),(64,90,70,84),(65,86,66,85),(96,135,115,116),(97,131,111,117),(98,132,112,118),(99,133,113,119),(100,134,114,120),(101,127,107,121),(102,128,108,122),(103,129,109,123),(104,130,110,124),(105,126,106,125)])

110 conjugacy classes

 class 1 2A 2B 2C 2D ··· 2H 4A ··· 4F 4G ··· 4M 5A 5B 5C 5D 10A ··· 10L 10M ··· 10AF 20A ··· 20X 20Y ··· 20AZ order 1 2 2 2 2 ··· 2 4 ··· 4 4 ··· 4 5 5 5 5 10 ··· 10 10 ··· 10 20 ··· 20 20 ··· 20 size 1 1 1 1 4 ··· 4 2 ··· 2 4 ··· 4 1 1 1 1 1 ··· 1 4 ··· 4 2 ··· 2 4 ··· 4

110 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 4 4 type + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C5 C10 C10 C10 C10 C10 C10 C4○D4 C5×C4○D4 2+ 1+4 C5×2+ 1+4 kernel C5×C22.34C24 C5×C42⋊C2 D4×C20 C5×C4⋊D4 C5×C22.D4 C5×C42.C2 C5×C4⋊1D4 C22.34C24 C42⋊C2 C4×D4 C4⋊D4 C22.D4 C42.C2 C4⋊1D4 C20 C4 C10 C2 # reps 1 1 2 6 4 1 1 4 4 8 24 16 4 4 4 16 2 8

Matrix representation of C5×C22.34C24 in GL6(𝔽41)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 18 0 0 0 0 0 0 18 0 0 0 0 0 0 18 0 0 0 0 0 0 18
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40
,
 40 0 0 0 0 0 0 40 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 32 18 0 0 0 0 32 9 0 0 0 0 0 0 1 0 0 0 0 0 1 40 0 0 0 0 0 0 1 0 0 0 0 0 1 40
,
 32 0 0 0 0 0 0 32 0 0 0 0 0 0 0 0 1 39 0 0 0 0 1 40 0 0 40 2 0 0 0 0 40 1 0 0
,
 1 39 0 0 0 0 0 40 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 39 0 0 0 0 1 40 0 0 0 0 0 0 1 39 0 0 0 0 1 40

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,18,0,0,0,0,0,0,18,0,0,0,0,0,0,18,0,0,0,0,0,0,18],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[32,32,0,0,0,0,18,9,0,0,0,0,0,0,1,1,0,0,0,0,0,40,0,0,0,0,0,0,1,1,0,0,0,0,0,40],[32,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,40,40,0,0,0,0,2,1,0,0,1,1,0,0,0,0,39,40,0,0],[1,0,0,0,0,0,39,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,39,40,0,0,0,0,0,0,1,1,0,0,0,0,39,40] >;

C5×C22.34C24 in GAP, Magma, Sage, TeX

C_5\times C_2^2._{34}C_2^4
% in TeX

G:=Group("C5xC2^2.34C2^4");
// GroupNames label

G:=SmallGroup(320,1542);
// by ID

G=gap.SmallGroup(320,1542);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-5,-2,-2,1149,1128,3446,891,2467,304]);
// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^5=b^2=c^2=d^2=f^2=1,e^2=c,g^2=b,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,a*g=g*a,b*c=c*b,e*d*e^-1=g*d*g^-1=b*d=d*b,f*e*f=b*e=e*b,b*f=f*b,b*g=g*b,f*d*f=c*d=d*c,c*e=e*c,c*f=f*c,c*g=g*c,e*g=g*e,f*g=g*f>;
// generators/relations

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