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G = C5×C23.24D4order 320 = 26·5

Direct product of C5 and C23.24D4

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Aliases: C5×C23.24D4, C4○D41C20, (C22×C40)⋊8C2, (C22×C8)⋊4C10, D4.5(C2×C20), C4.53(D4×C10), Q8.5(C2×C20), C20.460(C2×D4), (C2×C20).518D4, C4.3(C22×C20), D4⋊C420C10, C23.23(C5×D4), C42⋊C22C10, Q8⋊C420C10, C22.43(D4×C10), C10.114(C4○D8), (C2×C40).358C22, C20.207(C22×C4), (C2×C20).892C23, (C22×C10).127D4, C20.163(C22⋊C4), (D4×C10).287C22, (Q8×C10).251C22, (C22×C20).583C22, C2.1(C5×C4○D8), (C5×C4○D4)⋊13C4, C4⋊C4.38(C2×C10), (C2×C8).61(C2×C10), (C2×C4).49(C2×C20), (C2×C4○D4).4C10, (C5×D4).41(C2×C4), (C2×C4).122(C5×D4), C4.32(C5×C22⋊C4), (C5×Q8).44(C2×C4), (C5×D4⋊C4)⋊43C2, (C2×C20).443(C2×C4), (C5×Q8⋊C4)⋊43C2, (C10×C4○D4).18C2, (C2×D4).45(C2×C10), (C2×C10).619(C2×D4), C2.19(C10×C22⋊C4), (C2×Q8).36(C2×C10), C22.4(C5×C22⋊C4), (C5×C42⋊C2)⋊23C2, (C5×C4⋊C4).359C22, C10.148(C2×C22⋊C4), (C2×C4).67(C22×C10), (C2×C10).95(C22⋊C4), (C22×C4).112(C2×C10), SmallGroup(320,917)

Series: Derived Chief Lower central Upper central

C1C4 — C5×C23.24D4
C1C2C22C2×C4C2×C20C5×C4⋊C4C5×D4⋊C4 — C5×C23.24D4
C1C2C4 — C5×C23.24D4
C1C2×C20C22×C20 — C5×C23.24D4

Generators and relations for C5×C23.24D4
 G = < a,b,c,d,e,f | a5=b2=c2=d2=1, e4=d, f2=c, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, fbf-1=bd=db, be=eb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=cde3 >

Subgroups: 258 in 158 conjugacy classes, 82 normal (30 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×2], C4 [×2], C4 [×4], C22, C22 [×2], C22 [×6], C5, C8 [×2], C2×C4 [×2], C2×C4 [×4], C2×C4 [×7], D4 [×2], D4 [×5], Q8 [×2], Q8, C23, C23, C10, C10 [×2], C10 [×4], C42, C22⋊C4, C4⋊C4 [×2], C2×C8 [×2], C2×C8 [×2], C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4 [×4], C4○D4 [×2], C20 [×2], C20 [×2], C20 [×4], C2×C10, C2×C10 [×2], C2×C10 [×6], D4⋊C4 [×2], Q8⋊C4 [×2], C42⋊C2, C22×C8, C2×C4○D4, C40 [×2], C2×C20 [×2], C2×C20 [×4], C2×C20 [×7], C5×D4 [×2], C5×D4 [×5], C5×Q8 [×2], C5×Q8, C22×C10, C22×C10, C23.24D4, C4×C20, C5×C22⋊C4, C5×C4⋊C4 [×2], C2×C40 [×2], C2×C40 [×2], C22×C20, C22×C20, D4×C10, D4×C10, Q8×C10, C5×C4○D4 [×4], C5×C4○D4 [×2], C5×D4⋊C4 [×2], C5×Q8⋊C4 [×2], C5×C42⋊C2, C22×C40, C10×C4○D4, C5×C23.24D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C5, C2×C4 [×6], D4 [×4], C23, C10 [×7], C22⋊C4 [×4], C22×C4, C2×D4 [×2], C20 [×4], C2×C10 [×7], C2×C22⋊C4, C4○D8 [×2], C2×C20 [×6], C5×D4 [×4], C22×C10, C23.24D4, C5×C22⋊C4 [×4], C22×C20, D4×C10 [×2], C10×C22⋊C4, C5×C4○D8 [×2], C5×C23.24D4

Smallest permutation representation of C5×C23.24D4
On 160 points
Generators in S160
(1 98 23 90 15)(2 99 24 91 16)(3 100 17 92 9)(4 101 18 93 10)(5 102 19 94 11)(6 103 20 95 12)(7 104 21 96 13)(8 97 22 89 14)(25 41 116 33 108)(26 42 117 34 109)(27 43 118 35 110)(28 44 119 36 111)(29 45 120 37 112)(30 46 113 38 105)(31 47 114 39 106)(32 48 115 40 107)(49 124 140 57 132)(50 125 141 58 133)(51 126 142 59 134)(52 127 143 60 135)(53 128 144 61 136)(54 121 137 62 129)(55 122 138 63 130)(56 123 139 64 131)(65 82 156 73 148)(66 83 157 74 149)(67 84 158 75 150)(68 85 159 76 151)(69 86 160 77 152)(70 87 153 78 145)(71 88 154 79 146)(72 81 155 80 147)
(1 70)(2 71)(3 72)(4 65)(5 66)(6 67)(7 68)(8 69)(9 147)(10 148)(11 149)(12 150)(13 151)(14 152)(15 145)(16 146)(17 155)(18 156)(19 157)(20 158)(21 159)(22 160)(23 153)(24 154)(25 127)(26 128)(27 121)(28 122)(29 123)(30 124)(31 125)(32 126)(33 135)(34 136)(35 129)(36 130)(37 131)(38 132)(39 133)(40 134)(41 143)(42 144)(43 137)(44 138)(45 139)(46 140)(47 141)(48 142)(49 105)(50 106)(51 107)(52 108)(53 109)(54 110)(55 111)(56 112)(57 113)(58 114)(59 115)(60 116)(61 117)(62 118)(63 119)(64 120)(73 93)(74 94)(75 95)(76 96)(77 89)(78 90)(79 91)(80 92)(81 100)(82 101)(83 102)(84 103)(85 104)(86 97)(87 98)(88 99)
(1 127)(2 128)(3 121)(4 122)(5 123)(6 124)(7 125)(8 126)(9 54)(10 55)(11 56)(12 49)(13 50)(14 51)(15 52)(16 53)(17 62)(18 63)(19 64)(20 57)(21 58)(22 59)(23 60)(24 61)(25 70)(26 71)(27 72)(28 65)(29 66)(30 67)(31 68)(32 69)(33 78)(34 79)(35 80)(36 73)(37 74)(38 75)(39 76)(40 77)(41 87)(42 88)(43 81)(44 82)(45 83)(46 84)(47 85)(48 86)(89 134)(90 135)(91 136)(92 129)(93 130)(94 131)(95 132)(96 133)(97 142)(98 143)(99 144)(100 137)(101 138)(102 139)(103 140)(104 141)(105 150)(106 151)(107 152)(108 145)(109 146)(110 147)(111 148)(112 149)(113 158)(114 159)(115 160)(116 153)(117 154)(118 155)(119 156)(120 157)
(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)(25 29)(26 30)(27 31)(28 32)(33 37)(34 38)(35 39)(36 40)(41 45)(42 46)(43 47)(44 48)(49 53)(50 54)(51 55)(52 56)(57 61)(58 62)(59 63)(60 64)(65 69)(66 70)(67 71)(68 72)(73 77)(74 78)(75 79)(76 80)(81 85)(82 86)(83 87)(84 88)(89 93)(90 94)(91 95)(92 96)(97 101)(98 102)(99 103)(100 104)(105 109)(106 110)(107 111)(108 112)(113 117)(114 118)(115 119)(116 120)(121 125)(122 126)(123 127)(124 128)(129 133)(130 134)(131 135)(132 136)(137 141)(138 142)(139 143)(140 144)(145 149)(146 150)(147 151)(148 152)(153 157)(154 158)(155 159)(156 160)
(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 8 127 126)(2 125 128 7)(3 6 121 124)(4 123 122 5)(9 12 54 49)(10 56 55 11)(13 16 50 53)(14 52 51 15)(17 20 62 57)(18 64 63 19)(21 24 58 61)(22 60 59 23)(25 28 70 65)(26 72 71 27)(29 32 66 69)(30 68 67 31)(33 36 78 73)(34 80 79 35)(37 40 74 77)(38 76 75 39)(41 44 87 82)(42 81 88 43)(45 48 83 86)(46 85 84 47)(89 135 134 90)(91 133 136 96)(92 95 129 132)(93 131 130 94)(97 143 142 98)(99 141 144 104)(100 103 137 140)(101 139 138 102)(105 151 150 106)(107 149 152 112)(108 111 145 148)(109 147 146 110)(113 159 158 114)(115 157 160 120)(116 119 153 156)(117 155 154 118)

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

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

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

140 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E4F4G···4L5A5B5C5D8A···8H10A···10L10M···10T10U···10AB20A···20P20Q···20X20Y···20AV40A···40AF
order122222224444444···455558···810···1010···1010···1020···2020···2020···2040···40
size111122441111224···411112···21···12···24···41···12···24···42···2

140 irreducible representations

dim11111111111111222222
type++++++++
imageC1C2C2C2C2C2C4C5C10C10C10C10C10C20D4D4C4○D8C5×D4C5×D4C5×C4○D8
kernelC5×C23.24D4C5×D4⋊C4C5×Q8⋊C4C5×C42⋊C2C22×C40C10×C4○D4C5×C4○D4C23.24D4D4⋊C4Q8⋊C4C42⋊C2C22×C8C2×C4○D4C4○D4C2×C20C22×C10C10C2×C4C23C2
# reps12211184884443231812432

Matrix representation of C5×C23.24D4 in GL4(𝔽41) generated by

1000
01600
0010
0001
,
1000
04000
0009
00320
,
40000
0100
0010
0001
,
1000
0100
00400
00040
,
9000
04000
002912
002929
,
9000
0100
002912
001212
G:=sub<GL(4,GF(41))| [1,0,0,0,0,16,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,40,0,0,0,0,0,32,0,0,9,0],[40,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,40,0,0,0,0,40],[9,0,0,0,0,40,0,0,0,0,29,29,0,0,12,29],[9,0,0,0,0,1,0,0,0,0,29,12,0,0,12,12] >;

C5×C23.24D4 in GAP, Magma, Sage, TeX

C_5\times C_2^3._{24}D_4
% in TeX

G:=Group("C5xC2^3.24D4");
// GroupNames label

G:=SmallGroup(320,917);
// by ID

G=gap.SmallGroup(320,917);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,560,589,856,7004,3511,172]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^5=b^2=c^2=d^2=1,e^4=d,f^2=c,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,f*b*f^-1=b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=c*d*e^3>;
// generators/relations

׿
×
𝔽