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G = C24.8D10order 320 = 26·5

8th non-split extension by C24 of D10 acting via D10/C5=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C24.8D10, C2.5(D4×Dic5), C10.117(C4×D4), C22⋊C45Dic5, C22.99(D4×D5), C2.5(D10⋊D4), C10.30(C4⋊D4), C23.7(C2×Dic5), (C2×Dic5).151D4, (C22×C4).307D10, C10.32(C4.4D4), C22.52(C4○D20), (C23×C10).34C22, C57(C24.C22), C23.281(C22×D5), C10.10C4231C2, C10.63(C42⋊C2), C10.14(C422C2), C2.6(D10.12D4), C22.47(D42D5), (C22×C20).343C22, (C22×C10).326C23, C2.6(Dic5.5D4), C2.7(C23.D10), C22.40(C22×Dic5), C10.31(C22.D4), C2.8(C23.21D10), (C22×Dic5).208C22, (C2×C4×Dic5)⋊23C2, (C5×C22⋊C4)⋊14C4, (C2×C4⋊Dic5)⋊11C2, (C2×C20).333(C2×C4), (C2×C10).320(C2×D4), (C2×C22⋊C4).14D5, (C2×C4).16(C2×Dic5), (C2×C10).79(C4○D4), (C10×C22⋊C4).19C2, (C2×C23.D5).13C2, (C2×C10).280(C22×C4), (C22×C10).119(C2×C4), SmallGroup(320,578)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C24.8D10
C1C5C10C2×C10C22×C10C22×Dic5C2×C4×Dic5 — C24.8D10
C5C2×C10 — C24.8D10
C1C23C2×C22⋊C4

Generators and relations for C24.8D10
 G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e10=c, f2=cb=bc, ab=ba, ac=ca, eae-1=ad=da, faf-1=acd, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e9 >

Subgroups: 590 in 190 conjugacy classes, 75 normal (51 characteristic)
C1, C2 [×7], C2 [×2], C4 [×10], C22 [×7], C22 [×10], C5, C2×C4 [×4], C2×C4 [×18], C23, C23 [×2], C23 [×6], C10 [×7], C10 [×2], C42 [×2], C22⋊C4 [×4], C22⋊C4 [×4], C4⋊C4 [×2], C22×C4 [×2], C22×C4 [×4], C24, Dic5 [×6], C20 [×4], C2×C10 [×7], C2×C10 [×10], C2.C42 [×2], C2×C42, C2×C22⋊C4, C2×C22⋊C4 [×2], C2×C4⋊C4, C2×Dic5 [×4], C2×Dic5 [×10], C2×C20 [×4], C2×C20 [×4], C22×C10, C22×C10 [×2], C22×C10 [×6], C24.C22, C4×Dic5 [×2], C4⋊Dic5 [×2], C23.D5 [×4], C5×C22⋊C4 [×4], C22×Dic5 [×4], C22×C20 [×2], C23×C10, C10.10C42 [×2], C2×C4×Dic5, C2×C4⋊Dic5, C2×C23.D5 [×2], C10×C22⋊C4, C24.8D10
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, D5, C22×C4, C2×D4 [×2], C4○D4 [×4], Dic5 [×4], D10 [×3], C42⋊C2, C4×D4 [×2], C4⋊D4, C22.D4, C4.4D4, C422C2, C2×Dic5 [×6], C22×D5, C24.C22, C4○D20 [×2], D4×D5 [×2], D42D5 [×2], C22×Dic5, C23.D10, D10.12D4, D10⋊D4, Dic5.5D4, C23.21D10, D4×Dic5 [×2], C24.8D10

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

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

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

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

68 conjugacy classes

class 1 2A···2G2H2I4A4B4C4D4E4F4G···4N4O4P4Q4R5A5B10A···10N10O···10V20A···20P
order12···2224444444···444445510···1010···1020···20
size11···14422224410···1020202020222···24···44···4

68 irreducible representations

dim1111111222222244
type++++++++-+++-
imageC1C2C2C2C2C2C4D4D5C4○D4Dic5D10D10C4○D20D4×D5D42D5
kernelC24.8D10C10.10C42C2×C4×Dic5C2×C4⋊Dic5C2×C23.D5C10×C22⋊C4C5×C22⋊C4C2×Dic5C2×C22⋊C4C2×C10C22⋊C4C22×C4C24C22C22C22
# reps12112184288421644

Matrix representation of C24.8D10 in GL6(𝔽41)

100000
0400000
001000
000100
000010
00003140
,
100000
010000
0040000
0004000
0000400
0000040
,
4000000
0400000
001000
000100
0000400
0000040
,
4000000
0400000
001000
000100
000010
000001
,
010000
4000000
000700
0035600
0000210
0000839
,
3200000
0320000
0021300
0032000
00002638
00002015

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,31,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[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],[0,40,0,0,0,0,1,0,0,0,0,0,0,0,0,35,0,0,0,0,7,6,0,0,0,0,0,0,21,8,0,0,0,0,0,39],[32,0,0,0,0,0,0,32,0,0,0,0,0,0,21,3,0,0,0,0,3,20,0,0,0,0,0,0,26,20,0,0,0,0,38,15] >;

C24.8D10 in GAP, Magma, Sage, TeX

C_2^4._8D_{10}
% in TeX

G:=Group("C2^4.8D10");
// GroupNames label

G:=SmallGroup(320,578);
// by ID

G=gap.SmallGroup(320,578);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,120,422,387,12550]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^10=c,f^2=c*b=b*c,a*b=b*a,a*c=c*a,e*a*e^-1=a*d=d*a,f*a*f^-1=a*c*d,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