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G = C24.7D10order 320 = 26·5

7th non-split extension by C24 of D10 acting via D10/C5=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C24.7D10, C23.5Dic10, (C2×C20).50D4, C2.5(C20⋊D4), (C2×Dic5).63D4, (C22×C4).29D10, C22.239(D4×D5), (C22×C10).13Q8, C10.11(C41D4), C52(C23.4Q8), C10.57(C22⋊Q8), C2.8(C20.48D4), C22.96(C4○D20), (C23×C10).32C22, (C22×C20).58C22, C22.46(C2×Dic10), C23.368(C22×D5), C10.10C4229C2, C22.94(D42D5), (C22×C10).324C23, C2.20(D10.12D4), C2.6(C23.18D10), C10.29(C22.D4), (C22×Dic5).40C22, C2.22(Dic5.14D4), (C2×C4⋊Dic5)⋊9C2, (C2×C10).34(C2×Q8), (C2×C10.D4)⋊9C2, (C2×C10).318(C2×D4), (C2×C4).29(C5⋊D4), (C2×C22⋊C4).12D5, (C2×C10).78(C4○D4), (C10×C22⋊C4).14C2, C22.124(C2×C5⋊D4), (C2×C23.D5).11C2, SmallGroup(320,576)

Series: Derived Chief Lower central Upper central

C1C22×C10 — C24.7D10
C1C5C10C2×C10C22×C10C22×Dic5C2×C4⋊Dic5 — C24.7D10
C5C22×C10 — C24.7D10
C1C23C2×C22⋊C4

Generators and relations for C24.7D10
 G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e10=dc=cd, f2=c, ab=ba, ac=ca, eae-1=ad=da, faf-1=abd, bc=cb, bd=db, be=eb, bf=fb, ce=ec, cf=fc, de=ed, df=fd, fef-1=cde9 >

Subgroups: 614 in 186 conjugacy classes, 63 normal (27 characteristic)
C1, C2 [×3], C2 [×4], C2 [×2], C4 [×9], C22 [×3], C22 [×4], C22 [×10], C5, C2×C4 [×2], C2×C4 [×19], C23, C23 [×2], C23 [×6], C10 [×3], C10 [×4], C10 [×2], C22⋊C4 [×6], C4⋊C4 [×6], C22×C4 [×2], C22×C4 [×4], C24, Dic5 [×6], C20 [×3], C2×C10 [×3], C2×C10 [×4], C2×C10 [×10], C2.C42, C2×C22⋊C4, C2×C22⋊C4 [×2], C2×C4⋊C4 [×3], C2×Dic5 [×4], C2×Dic5 [×10], C2×C20 [×2], C2×C20 [×5], C22×C10, C22×C10 [×2], C22×C10 [×6], C23.4Q8, C10.D4 [×4], C4⋊Dic5 [×2], C23.D5 [×4], C5×C22⋊C4 [×2], C22×Dic5 [×4], C22×C20 [×2], C23×C10, C10.10C42, C2×C10.D4 [×2], C2×C4⋊Dic5, C2×C23.D5 [×2], C10×C22⋊C4, C24.7D10
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], Q8 [×2], C23, D5, C2×D4 [×3], C2×Q8, C4○D4 [×3], D10 [×3], C22⋊Q8 [×3], C22.D4 [×3], C41D4, Dic10 [×2], C5⋊D4 [×2], C22×D5, C23.4Q8, C2×Dic10, C4○D20, D4×D5 [×2], D42D5 [×2], C2×C5⋊D4, Dic5.14D4 [×2], D10.12D4 [×2], C20.48D4, C23.18D10, C20⋊D4, C24.7D10

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

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

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

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

62 conjugacy classes

class 1 2A···2G2H2I4A4B4C4D4E···4L5A5B10A···10N10O···10V20A···20P
order12···22244444···45510···1010···1020···20
size11···144444420···20222···24···44···4

62 irreducible representations

dim111111222222222244
type++++++++-+++-+-
imageC1C2C2C2C2C2D4D4Q8D5C4○D4D10D10C5⋊D4Dic10C4○D20D4×D5D42D5
kernelC24.7D10C10.10C42C2×C10.D4C2×C4⋊Dic5C2×C23.D5C10×C22⋊C4C2×Dic5C2×C20C22×C10C2×C22⋊C4C2×C10C22×C4C24C2×C4C23C22C22C22
# reps112121422264288844

Matrix representation of C24.7D10 in GL6(𝔽41)

100000
0400000
001000
000100
000010
0000140
,
4000000
0400000
001000
000100
0000400
0000040
,
100000
010000
0040000
0004000
000010
000001
,
4000000
0400000
001000
000100
000010
000001
,
010000
4000000
00391600
00252500
0000400
0000040
,
4000000
010000
0003200
0032000
0000402
000001

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,0,40],[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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[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],[0,40,0,0,0,0,1,0,0,0,0,0,0,0,39,25,0,0,0,0,16,25,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[40,0,0,0,0,0,0,1,0,0,0,0,0,0,0,32,0,0,0,0,32,0,0,0,0,0,0,0,40,0,0,0,0,0,2,1] >;

C24.7D10 in GAP, Magma, Sage, TeX

C_2^4._7D_{10}
% in TeX

G:=Group("C2^4.7D10");
// GroupNames label

G:=SmallGroup(320,576);
// by ID

G=gap.SmallGroup(320,576);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,701,344,254,387,12550]);
// Polycyclic

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

׿
×
𝔽