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G = C24.12D10order 320 = 26·5

12nd non-split extension by C24 of D10 acting via D10/C5=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C24.12D10, (C2×C20)⋊20D4, C232(C4×D5), C10.94(C4×D4), (C2×Dic5)⋊16D4, C10.38C22≀C2, D103(C22⋊C4), C2.2(C202D4), C22.101(D4×D5), (C22×C4).31D10, C2.6(D10⋊D4), C2.3(C23⋊D10), C10.31(C4⋊D4), (C22×D5).125D4, C53(C23.23D4), C22.53(C4○D20), (C23×C10).39C22, C23.283(C22×D5), C10.10C4239C2, C2.27(Dic54D4), C2.7(D10.12D4), C22.48(D42D5), (C22×C20).344C22, (C22×C10).330C23, (C23×D5).100C22, C10.32(C22.D4), (C22×Dic5).43C22, (C2×C5⋊D4)⋊9C4, C2.9(C4×C5⋊D4), (C2×C22⋊C4)⋊3D5, (D5×C22×C4)⋊13C2, (C2×C4)⋊12(C5⋊D4), (C2×Dic5)⋊7(C2×C4), (C2×C23.D5)⋊3C2, C2.29(D5×C22⋊C4), (C2×D10⋊C4)⋊4C2, (C10×C22⋊C4)⋊22C2, C22.127(C2×C4×D5), (C22×C10)⋊13(C2×C4), (C2×C10).322(C2×D4), C10.69(C2×C22⋊C4), (C22×C5⋊D4).2C2, C22.51(C2×C5⋊D4), (C22×D5).78(C2×C4), (C2×C10).145(C4○D4), (C2×C10).210(C22×C4), SmallGroup(320,583)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C24.12D10
C1C5C10C2×C10C22×C10C23×D5C22×C5⋊D4 — C24.12D10
C5C2×C10 — C24.12D10
C1C23C2×C22⋊C4

Generators and relations for C24.12D10
 G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e10=f2=b, ab=ba, ac=ca, eae-1=ad=da, faf-1=acd, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e9 >

Subgroups: 1118 in 286 conjugacy classes, 77 normal (51 characteristic)
C1, C2 [×7], C2 [×6], C4 [×8], C22 [×7], C22 [×26], C5, C2×C4 [×2], C2×C4 [×24], D4 [×8], C23, C23 [×2], C23 [×16], D5 [×4], C10 [×7], C10 [×2], C22⋊C4 [×6], C22×C4 [×2], C22×C4 [×9], C2×D4 [×8], C24, C24, Dic5 [×5], C20 [×3], D10 [×4], D10 [×12], C2×C10 [×7], C2×C10 [×10], C2.C42 [×2], C2×C22⋊C4, C2×C22⋊C4 [×2], C23×C4, C22×D4, C4×D5 [×8], C2×Dic5 [×4], C2×Dic5 [×7], C5⋊D4 [×8], C2×C20 [×2], C2×C20 [×5], C22×D5 [×6], C22×D5 [×4], C22×C10, C22×C10 [×2], C22×C10 [×6], C23.23D4, D10⋊C4 [×2], C23.D5 [×2], C5×C22⋊C4 [×2], C2×C4×D5 [×6], C22×Dic5 [×3], C2×C5⋊D4 [×4], C2×C5⋊D4 [×4], C22×C20 [×2], C23×D5, C23×C10, C10.10C42 [×2], C2×D10⋊C4, C2×C23.D5, C10×C22⋊C4, D5×C22×C4, C22×C5⋊D4, C24.12D10
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×8], C23, D5, C22⋊C4 [×4], C22×C4, C2×D4 [×4], C4○D4 [×2], D10 [×3], C2×C22⋊C4, C4×D4 [×2], C22≀C2, C4⋊D4 [×2], C22.D4, C4×D5 [×2], C5⋊D4 [×2], C22×D5, C23.23D4, C2×C4×D5, C4○D20, D4×D5 [×3], D42D5, C2×C5⋊D4, D5×C22⋊C4, Dic54D4, D10.12D4, D10⋊D4, C4×C5⋊D4, C23⋊D10, C202D4, C24.12D10

Smallest permutation representation of C24.12D10
On 160 points
Generators in S160
(1 129)(2 23)(3 131)(4 25)(5 133)(6 27)(7 135)(8 29)(9 137)(10 31)(11 139)(12 33)(13 121)(14 35)(15 123)(16 37)(17 125)(18 39)(19 127)(20 21)(22 145)(24 147)(26 149)(28 151)(30 153)(32 155)(34 157)(36 159)(38 141)(40 143)(41 112)(42 90)(43 114)(44 92)(45 116)(46 94)(47 118)(48 96)(49 120)(50 98)(51 102)(52 100)(53 104)(54 82)(55 106)(56 84)(57 108)(58 86)(59 110)(60 88)(61 91)(62 115)(63 93)(64 117)(65 95)(66 119)(67 97)(68 101)(69 99)(70 103)(71 81)(72 105)(73 83)(74 107)(75 85)(76 109)(77 87)(78 111)(79 89)(80 113)(122 158)(124 160)(126 142)(128 144)(130 146)(132 148)(134 150)(136 152)(138 154)(140 156)
(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 17)(8 18)(9 19)(10 20)(21 31)(22 32)(23 33)(24 34)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)(41 51)(42 52)(43 53)(44 54)(45 55)(46 56)(47 57)(48 58)(49 59)(50 60)(61 71)(62 72)(63 73)(64 74)(65 75)(66 76)(67 77)(68 78)(69 79)(70 80)(81 91)(82 92)(83 93)(84 94)(85 95)(86 96)(87 97)(88 98)(89 99)(90 100)(101 111)(102 112)(103 113)(104 114)(105 115)(106 116)(107 117)(108 118)(109 119)(110 120)(121 131)(122 132)(123 133)(124 134)(125 135)(126 136)(127 137)(128 138)(129 139)(130 140)(141 151)(142 152)(143 153)(144 154)(145 155)(146 156)(147 157)(148 158)(149 159)(150 160)
(1 93)(2 94)(3 95)(4 96)(5 97)(6 98)(7 99)(8 100)(9 81)(10 82)(11 83)(12 84)(13 85)(14 86)(15 87)(16 88)(17 89)(18 90)(19 91)(20 92)(21 44)(22 45)(23 46)(24 47)(25 48)(26 49)(27 50)(28 51)(29 52)(30 53)(31 54)(32 55)(33 56)(34 57)(35 58)(36 59)(37 60)(38 41)(39 42)(40 43)(61 127)(62 128)(63 129)(64 130)(65 131)(66 132)(67 133)(68 134)(69 135)(70 136)(71 137)(72 138)(73 139)(74 140)(75 121)(76 122)(77 123)(78 124)(79 125)(80 126)(101 150)(102 151)(103 152)(104 153)(105 154)(106 155)(107 156)(108 157)(109 158)(110 159)(111 160)(112 141)(113 142)(114 143)(115 144)(116 145)(117 146)(118 147)(119 148)(120 149)
(1 145)(2 146)(3 147)(4 148)(5 149)(6 150)(7 151)(8 152)(9 153)(10 154)(11 155)(12 156)(13 157)(14 158)(15 159)(16 160)(17 141)(18 142)(19 143)(20 144)(21 128)(22 129)(23 130)(24 131)(25 132)(26 133)(27 134)(28 135)(29 136)(30 137)(31 138)(32 139)(33 140)(34 121)(35 122)(36 123)(37 124)(38 125)(39 126)(40 127)(41 79)(42 80)(43 61)(44 62)(45 63)(46 64)(47 65)(48 66)(49 67)(50 68)(51 69)(52 70)(53 71)(54 72)(55 73)(56 74)(57 75)(58 76)(59 77)(60 78)(81 104)(82 105)(83 106)(84 107)(85 108)(86 109)(87 110)(88 111)(89 112)(90 113)(91 114)(92 115)(93 116)(94 117)(95 118)(96 119)(97 120)(98 101)(99 102)(100 103)
(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 10 11 20)(2 19 12 9)(3 8 13 18)(4 17 14 7)(5 6 15 16)(21 45 31 55)(22 54 32 44)(23 43 33 53)(24 52 34 42)(25 41 35 51)(26 50 36 60)(27 59 37 49)(28 48 38 58)(29 57 39 47)(30 46 40 56)(61 140 71 130)(62 129 72 139)(63 138 73 128)(64 127 74 137)(65 136 75 126)(66 125 76 135)(67 134 77 124)(68 123 78 133)(69 132 79 122)(70 121 80 131)(81 94 91 84)(82 83 92 93)(85 90 95 100)(86 99 96 89)(87 88 97 98)(101 110 111 120)(102 119 112 109)(103 108 113 118)(104 117 114 107)(105 106 115 116)(141 158 151 148)(142 147 152 157)(143 156 153 146)(144 145 154 155)(149 150 159 160)

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

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

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

68 conjugacy classes

class 1 2A···2G2H2I2J2K2L2M4A4B4C4D4E4F4G4H4I4J4K4L4M4N5A5B10A···10N10O···10V20A···20P
order12···2222222444444444444445510···1010···1020···20
size11···144101010102222441010101020202020222···24···44···4

68 irreducible representations

dim11111111222222222244
type++++++++++++++-
imageC1C2C2C2C2C2C2C4D4D4D4D5C4○D4D10D10C5⋊D4C4×D5C4○D20D4×D5D42D5
kernelC24.12D10C10.10C42C2×D10⋊C4C2×C23.D5C10×C22⋊C4D5×C22×C4C22×C5⋊D4C2×C5⋊D4C2×Dic5C2×C20C22×D5C2×C22⋊C4C2×C10C22×C4C24C2×C4C23C22C22C22
# reps12111118224244288862

Matrix representation of C24.12D10 in GL6(𝔽41)

2360000
35180000
00183500
0062300
00003318
0000178
,
4000000
0400000
001000
000100
000010
000001
,
4000000
0400000
0040000
0004000
000010
000001
,
100000
010000
001000
000100
0000400
0000040
,
13130000
2890000
00353500
0064000
0000918
00003232
,
13130000
9280000
00353500
0040600
0000918
00003232

G:=sub<GL(6,GF(41))| [23,35,0,0,0,0,6,18,0,0,0,0,0,0,18,6,0,0,0,0,35,23,0,0,0,0,0,0,33,17,0,0,0,0,18,8],[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],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[13,28,0,0,0,0,13,9,0,0,0,0,0,0,35,6,0,0,0,0,35,40,0,0,0,0,0,0,9,32,0,0,0,0,18,32],[13,9,0,0,0,0,13,28,0,0,0,0,0,0,35,40,0,0,0,0,35,6,0,0,0,0,0,0,9,32,0,0,0,0,18,32] >;

C24.12D10 in GAP, Magma, Sage, TeX

C_2^4._{12}D_{10}
% in TeX

G:=Group("C2^4.12D10");
// GroupNames label

G:=SmallGroup(320,583);
// by ID

G=gap.SmallGroup(320,583);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,387,58,12550]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^10=f^2=b,a*b=b*a,a*c=c*a,e*a*e^-1=a*d=d*a,f*a*f^-1=a*c*d,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*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=e^9>;
// generators/relations

׿
×
𝔽