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G = C24.63D10order 320 = 26·5

3rd non-split extension by C24 of D10 acting via D10/C10=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C24.63D10, (C23×C4).5D5, (C22×C20)⋊21C4, (C23×C20).3C2, (C22×C4)⋊7Dic5, (C22×C10).192D4, (C22×C4).407D10, C54(C23.34D4), C23.82(C5⋊D4), C23.30(C2×Dic5), C22.62(C4○D20), (C23×C10).98C22, C23.302(C22×D5), C10.10C4223C2, C10.67(C42⋊C2), (C22×C20).483C22, (C22×C10).362C23, C22.19(C23.D5), C22.49(C22×Dic5), C10.68(C22.D4), C2.4(C23.23D10), (C22×Dic5).65C22, C2.11(C23.21D10), (C2×C20).454(C2×C4), C2.5(C2×C23.D5), (C2×C10).548(C2×D4), (C2×C4).66(C2×Dic5), C22.86(C2×C5⋊D4), (C2×C10).90(C4○D4), C10.110(C2×C22⋊C4), (C2×C23.D5).17C2, (C22×C10).203(C2×C4), (C2×C10).293(C22×C4), (C2×C10).173(C22⋊C4), SmallGroup(320,838)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C24.63D10
C1C5C10C2×C10C22×C10C22×Dic5C2×C23.D5 — C24.63D10
C5C2×C10 — C24.63D10
C1C23C23×C4

Generators and relations for C24.63D10
 G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e10=d, f2=bcd, ab=ba, ac=ca, faf-1=ad=da, ae=ea, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=ce9 >

Subgroups: 590 in 218 conjugacy classes, 87 normal (13 characteristic)
C1, C2, C2 [×6], C2 [×4], C4 [×8], C22, C22 [×10], C22 [×12], C5, C2×C4 [×4], C2×C4 [×24], C23, C23 [×6], C23 [×4], C10, C10 [×6], C10 [×4], C22⋊C4 [×4], C22×C4 [×6], C22×C4 [×8], C24, Dic5 [×4], C20 [×4], C2×C10, C2×C10 [×10], C2×C10 [×12], C2.C42 [×4], C2×C22⋊C4 [×2], C23×C4, C2×Dic5 [×12], C2×C20 [×4], C2×C20 [×12], C22×C10, C22×C10 [×6], C22×C10 [×4], C23.34D4, C23.D5 [×4], C22×Dic5 [×4], C22×C20 [×6], C22×C20 [×4], C23×C10, C10.10C42 [×4], C2×C23.D5 [×2], C23×C20, C24.63D10
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, D5, C22⋊C4 [×4], C22×C4, C2×D4 [×2], C4○D4 [×4], Dic5 [×4], D10 [×3], C2×C22⋊C4, C42⋊C2 [×2], C22.D4 [×4], C2×Dic5 [×6], C5⋊D4 [×4], C22×D5, C23.34D4, C23.D5 [×4], C4○D20 [×4], C22×Dic5, C2×C5⋊D4 [×2], C23.21D10 [×2], C23.23D10 [×4], C2×C23.D5, C24.63D10

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

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

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

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

92 conjugacy classes

class 1 2A···2G2H2I2J2K4A···4H4I···4P5A5B10A···10AD20A···20AF
order12···222224···44···45510···1020···20
size11···122222···220···20222···22···2

92 irreducible representations

dim1111122222222
type++++++-++
imageC1C2C2C2C4D4D5C4○D4Dic5D10D10C5⋊D4C4○D20
kernelC24.63D10C10.10C42C2×C23.D5C23×C20C22×C20C22×C10C23×C4C2×C10C22×C4C22×C4C24C23C22
# reps142184288421632

Matrix representation of C24.63D10 in GL5(𝔽41)

10000
040000
004000
000400
00001
,
400000
040000
004000
000400
000040
,
10000
040000
004000
000400
000040
,
10000
01000
00100
000400
000040
,
400000
018000
002500
000210
00002
,
90000
001600
018000
000039
000210

G:=sub<GL(5,GF(41))| [1,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,1],[40,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,40],[1,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,40],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,40,0,0,0,0,0,40],[40,0,0,0,0,0,18,0,0,0,0,0,25,0,0,0,0,0,21,0,0,0,0,0,2],[9,0,0,0,0,0,0,18,0,0,0,16,0,0,0,0,0,0,0,21,0,0,0,39,0] >;

C24.63D10 in GAP, Magma, Sage, TeX

C_2^4._{63}D_{10}
% in TeX

G:=Group("C2^4.63D10");
// GroupNames label

G:=SmallGroup(320,838);
// by ID

G=gap.SmallGroup(320,838);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,477,422,184,12550]);
// Polycyclic

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

׿
×
𝔽