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## G = C5×C22.26C24order 320 = 26·5

### Direct product of C5 and C22.26C24

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C5×C22.26C24
 Chief series C1 — C2 — C22 — C2×C10 — C22×C10 — D4×C10 — C5×C4⋊D4 — C5×C22.26C24
 Lower central C1 — C22 — C5×C22.26C24
 Upper central C1 — C2×C20 — C5×C22.26C24

Generators and relations for C5×C22.26C24
G = < a,b,c,d,e,f,g | a5=b2=c2=d2=e2=1, f2=c, g2=b, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, ede=bd=db, be=eb, bf=fb, bg=gb, fdf-1=cd=dc, ce=ec, cf=fc, cg=gc, dg=gd, ef=fe, eg=ge, fg=gf >

Subgroups: 466 in 310 conjugacy classes, 170 normal (26 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C5, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C10, C10, C10, C42, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, C20, C20, C2×C10, C2×C10, C2×C10, C2×C42, C4×D4, C4⋊D4, C4.4D4, C41D4, C4⋊Q8, C2×C4○D4, C2×C20, C2×C20, C2×C20, C5×D4, C5×Q8, C22×C10, C22×C10, C22.26C24, C4×C20, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C22×C20, C22×C20, D4×C10, Q8×C10, C5×C4○D4, C2×C4×C20, D4×C20, C5×C4⋊D4, C5×C4.4D4, C5×C41D4, C5×C4⋊Q8, C10×C4○D4, C5×C22.26C24
Quotients: C1, C2, C22, C5, D4, C23, C10, C2×D4, C4○D4, C24, C2×C10, C22×D4, C2×C4○D4, C5×D4, C22×C10, C22.26C24, D4×C10, C5×C4○D4, C23×C10, D4×C2×C10, C10×C4○D4, C5×C22.26C24

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

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

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

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

140 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 4A 4B 4C 4D 4E ··· 4N 4O 4P 4Q 4R 5A 5B 5C 5D 10A ··· 10L 10M ··· 10T 10U ··· 10AJ 20A ··· 20P 20Q ··· 20BD 20BE ··· 20BT order 1 2 2 2 2 2 2 2 2 2 4 4 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 4 4 1 1 1 1 2 ··· 2 4 4 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 2 2 2 2 type + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C5 C10 C10 C10 C10 C10 C10 C10 D4 C4○D4 C5×D4 C5×C4○D4 kernel C5×C22.26C24 C2×C4×C20 D4×C20 C5×C4⋊D4 C5×C4.4D4 C5×C4⋊1D4 C5×C4⋊Q8 C10×C4○D4 C22.26C24 C2×C42 C4×D4 C4⋊D4 C4.4D4 C4⋊1D4 C4⋊Q8 C2×C4○D4 C2×C20 C20 C2×C4 C4 # reps 1 1 4 4 2 1 1 2 4 4 16 16 8 4 4 8 4 8 16 32

Matrix representation of C5×C22.26C24 in GL4(𝔽41) generated by

 1 0 0 0 0 1 0 0 0 0 10 0 0 0 0 10
,
 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
,
 0 1 0 0 1 0 0 0 0 0 40 0 0 0 17 1
,
 0 9 0 0 32 0 0 0 0 0 24 39 0 0 21 17
,
 0 1 0 0 40 0 0 0 0 0 40 0 0 0 0 40
,
 32 0 0 0 0 32 0 0 0 0 9 0 0 0 0 9
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,10,0,0,0,0,10],[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],[0,1,0,0,1,0,0,0,0,0,40,17,0,0,0,1],[0,32,0,0,9,0,0,0,0,0,24,21,0,0,39,17],[0,40,0,0,1,0,0,0,0,0,40,0,0,0,0,40],[32,0,0,0,0,32,0,0,0,0,9,0,0,0,0,9] >;

C5×C22.26C24 in GAP, Magma, Sage, TeX

C_5\times C_2^2._{26}C_2^4
% in TeX

G:=Group("C5xC2^2.26C2^4");
// GroupNames label

G:=SmallGroup(320,1534);
// by ID

G=gap.SmallGroup(320,1534);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-5,-2,-2,1149,3446,856,304]);
// Polycyclic

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

׿
×
𝔽