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## G = C5×C23.36C23order 320 = 26·5

### Direct product of C5 and C23.36C23

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C5×C23.36C23
 Chief series C1 — C2 — C22 — C2×C10 — C22×C10 — C5×C22⋊C4 — C5×C22.D4 — C5×C23.36C23
 Lower central C1 — C22 — C5×C23.36C23
 Upper central C1 — C2×C20 — C5×C23.36C23

Generators and relations for C5×C23.36C23
G = < a,b,c,d,e,f,g | a5=b2=c2=d2=e2=1, f2=d, g2=c, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, ebe=bd=db, bf=fb, bg=gb, cd=dc, fef-1=ce=ec, cf=fc, cg=gc, de=ed, df=fd, dg=gd, eg=ge, fg=gf >

Subgroups: 322 in 234 conjugacy classes, 154 normal (62 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C5, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C10, C10, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C20, C20, C2×C10, C2×C10, C2×C10, C2×C42, C42⋊C2, C4×D4, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C42.C2, C422C2, C2×C20, C2×C20, C2×C20, C5×D4, C5×Q8, C22×C10, C22×C10, C23.36C23, C4×C20, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C5×C4⋊C4, C22×C20, C22×C20, D4×C10, D4×C10, Q8×C10, C2×C4×C20, C5×C42⋊C2, D4×C20, D4×C20, Q8×C20, C5×C4⋊D4, C5×C22⋊Q8, C5×C22.D4, C5×C4.4D4, C5×C42.C2, C5×C422C2, C5×C23.36C23
Quotients: C1, C2, C22, C5, C23, C10, C4○D4, C24, C2×C10, C2×C4○D4, C22×C10, C23.36C23, C5×C4○D4, C23×C10, C10×C4○D4, C5×C23.36C23

Smallest permutation representation of C5×C23.36C23
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 56)(2 57)(3 58)(4 59)(5 60)(6 121)(7 122)(8 123)(9 124)(10 125)(11 126)(12 127)(13 128)(14 129)(15 130)(16 131)(17 132)(18 133)(19 134)(20 135)(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)(96 136)(97 137)(98 138)(99 139)(100 140)(101 141)(102 142)(103 143)(104 144)(105 145)(106 146)(107 147)(108 148)(109 149)(110 150)(111 151)(112 152)(113 153)(114 154)(115 155)(116 156)(117 157)(118 158)(119 159)(120 160)
(1 26)(2 27)(3 28)(4 29)(5 30)(6 16)(7 17)(8 18)(9 19)(10 20)(11 156)(12 157)(13 158)(14 159)(15 160)(21 31)(22 32)(23 33)(24 34)(25 35)(36 46)(37 47)(38 48)(39 49)(40 50)(41 51)(42 52)(43 53)(44 54)(45 55)(56 66)(57 67)(58 68)(59 69)(60 70)(61 71)(62 72)(63 73)(64 74)(65 75)(76 86)(77 87)(78 88)(79 89)(80 90)(81 91)(82 92)(83 93)(84 94)(85 95)(96 106)(97 107)(98 108)(99 109)(100 110)(101 111)(102 112)(103 113)(104 114)(105 115)(116 126)(117 127)(118 128)(119 129)(120 130)(121 131)(122 132)(123 133)(124 134)(125 135)(136 146)(137 147)(138 148)(139 149)(140 150)(141 151)(142 152)(143 153)(144 154)(145 155)
(1 21)(2 22)(3 23)(4 24)(5 25)(6 156)(7 157)(8 158)(9 159)(10 160)(11 16)(12 17)(13 18)(14 19)(15 20)(26 31)(27 32)(28 33)(29 34)(30 35)(36 41)(37 42)(38 43)(39 44)(40 45)(46 51)(47 52)(48 53)(49 54)(50 55)(56 61)(57 62)(58 63)(59 64)(60 65)(66 71)(67 72)(68 73)(69 74)(70 75)(76 81)(77 82)(78 83)(79 84)(80 85)(86 91)(87 92)(88 93)(89 94)(90 95)(96 101)(97 102)(98 103)(99 104)(100 105)(106 111)(107 112)(108 113)(109 114)(110 115)(116 121)(117 122)(118 123)(119 124)(120 125)(126 131)(127 132)(128 133)(129 134)(130 135)(136 141)(137 142)(138 143)(139 144)(140 145)(146 151)(147 152)(148 153)(149 154)(150 155)
(1 156)(2 157)(3 158)(4 159)(5 160)(6 21)(7 22)(8 23)(9 24)(10 25)(11 26)(12 27)(13 28)(14 29)(15 30)(16 31)(17 32)(18 33)(19 34)(20 35)(36 146)(37 147)(38 148)(39 149)(40 150)(41 151)(42 152)(43 153)(44 154)(45 155)(46 136)(47 137)(48 138)(49 139)(50 140)(51 141)(52 142)(53 143)(54 144)(55 145)(56 121)(57 122)(58 123)(59 124)(60 125)(61 116)(62 117)(63 118)(64 119)(65 120)(66 131)(67 132)(68 133)(69 134)(70 135)(71 126)(72 127)(73 128)(74 129)(75 130)(76 111)(77 112)(78 113)(79 114)(80 115)(81 106)(82 107)(83 108)(84 109)(85 110)(86 101)(87 102)(88 103)(89 104)(90 105)(91 96)(92 97)(93 98)(94 99)(95 100)
(1 101 21 96)(2 102 22 97)(3 103 23 98)(4 104 24 99)(5 105 25 100)(6 81 156 76)(7 82 157 77)(8 83 158 78)(9 84 159 79)(10 85 160 80)(11 86 16 91)(12 87 17 92)(13 88 18 93)(14 89 19 94)(15 90 20 95)(26 111 31 106)(27 112 32 107)(28 113 33 108)(29 114 34 109)(30 115 35 110)(36 121 41 116)(37 122 42 117)(38 123 43 118)(39 124 44 119)(40 125 45 120)(46 131 51 126)(47 132 52 127)(48 133 53 128)(49 134 54 129)(50 135 55 130)(56 141 61 136)(57 142 62 137)(58 143 63 138)(59 144 64 139)(60 145 65 140)(66 151 71 146)(67 152 72 147)(68 153 73 148)(69 154 74 149)(70 155 75 150)
(1 46 26 36)(2 47 27 37)(3 48 28 38)(4 49 29 39)(5 50 30 40)(6 141 16 151)(7 142 17 152)(8 143 18 153)(9 144 19 154)(10 145 20 155)(11 146 156 136)(12 147 157 137)(13 148 158 138)(14 149 159 139)(15 150 160 140)(21 51 31 41)(22 52 32 42)(23 53 33 43)(24 54 34 44)(25 55 35 45)(56 86 66 76)(57 87 67 77)(58 88 68 78)(59 89 69 79)(60 90 70 80)(61 91 71 81)(62 92 72 82)(63 93 73 83)(64 94 74 84)(65 95 75 85)(96 126 106 116)(97 127 107 117)(98 128 108 118)(99 129 109 119)(100 130 110 120)(101 131 111 121)(102 132 112 122)(103 133 113 123)(104 134 114 124)(105 135 115 125)

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

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

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

140 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E ··· 4N 4O ··· 4T 5A 5B 5C 5D 10A ··· 10L 10M ··· 10T 10U ··· 10AB 20A ··· 20P 20Q ··· 20BD 20BE ··· 20CB order 1 2 2 2 2 2 2 2 4 4 4 4 4 ··· 4 4 ··· 4 5 5 5 5 10 ··· 10 10 ··· 10 10 ··· 10 20 ··· 20 20 ··· 20 20 ··· 20 size 1 1 1 1 2 2 4 4 1 1 1 1 2 ··· 2 4 ··· 4 1 1 1 1 1 ··· 1 2 ··· 2 4 ··· 4 1 ··· 1 2 ··· 2 4 ··· 4

140 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 type + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C5 C10 C10 C10 C10 C10 C10 C10 C10 C10 C10 C4○D4 C4○D4 C5×C4○D4 C5×C4○D4 kernel C5×C23.36C23 C2×C4×C20 C5×C42⋊C2 D4×C20 Q8×C20 C5×C4⋊D4 C5×C22⋊Q8 C5×C22.D4 C5×C4.4D4 C5×C42.C2 C5×C42⋊2C2 C23.36C23 C2×C42 C42⋊C2 C4×D4 C4×Q8 C4⋊D4 C22⋊Q8 C22.D4 C4.4D4 C42.C2 C42⋊2C2 C20 C2×C10 C4 C22 # reps 1 1 2 3 1 1 1 2 1 1 2 4 4 8 12 4 4 4 8 4 4 8 8 4 32 16

Matrix representation of C5×C23.36C23 in GL4(𝔽41) generated by

 37 0 0 0 0 37 0 0 0 0 16 0 0 0 0 16
,
 1 39 0 0 0 40 0 0 0 0 1 0 0 0 0 1
,
 40 0 0 0 0 40 0 0 0 0 40 0 0 0 0 40
,
 40 0 0 0 0 40 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 1 40 0 0 0 0 14 23 0 0 4 27
,
 9 23 0 0 0 32 0 0 0 0 38 39 0 0 4 3
,
 32 0 0 0 0 32 0 0 0 0 9 0 0 0 0 9
G:=sub<GL(4,GF(41))| [37,0,0,0,0,37,0,0,0,0,16,0,0,0,0,16],[1,0,0,0,39,40,0,0,0,0,1,0,0,0,0,1],[40,0,0,0,0,40,0,0,0,0,40,0,0,0,0,40],[40,0,0,0,0,40,0,0,0,0,1,0,0,0,0,1],[1,1,0,0,0,40,0,0,0,0,14,4,0,0,23,27],[9,0,0,0,23,32,0,0,0,0,38,4,0,0,39,3],[32,0,0,0,0,32,0,0,0,0,9,0,0,0,0,9] >;

C5×C23.36C23 in GAP, Magma, Sage, TeX

C_5\times C_2^3._{36}C_2^3
% in TeX

G:=Group("C5xC2^3.36C2^3");
// GroupNames label

G:=SmallGroup(320,1531);
// by ID

G=gap.SmallGroup(320,1531);
# by ID

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

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

׿
×
𝔽