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G = (C2×D4).D10order 320 = 26·5

36th non-split extension by C2×D4 of D10 acting via D10/C5=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C4⋊C4.56D10, C4⋊D4.4D5, (C2×D4).36D10, (C2×C20).261D4, C10.95(C4○D8), C10.D835C2, C20.55D49C2, D4⋊Dic514C2, C20.Q834C2, (C22×C10).81D4, C20.182(C4○D4), C4.92(D42D5), C10.89(C8⋊C22), (C2×C20).354C23, (D4×C10).52C22, (C22×C4).118D10, C23.22(C5⋊D4), C57(C23.19D4), C4⋊Dic5.336C22, C2.10(D4.D10), C2.14(D4.8D10), C23.21D1015C2, (C22×C20).158C22, C10.79(C22.D4), C2.13(C23.18D10), (C5×C4⋊D4).3C2, (C2×C10).485(C2×D4), (C2×C4).170(C5⋊D4), (C5×C4⋊C4).103C22, (C2×C4).454(C22×D5), C22.160(C2×C5⋊D4), (C2×C52C8).107C22, SmallGroup(320,662)

Series: Derived Chief Lower central Upper central

C1C2×C20 — (C2×D4).D10
C1C5C10C20C2×C20C4⋊Dic5C23.21D10 — (C2×D4).D10
C5C10C2×C20 — (C2×D4).D10
C1C22C22×C4C4⋊D4

Generators and relations for (C2×D4).D10
 G = < a,b,c,d,e | a2=b4=c2=1, d10=b2, e2=a, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe-1=b-1, bd=db, dcd-1=ab2c, ece-1=b-1c, ede-1=d9 >

Subgroups: 334 in 106 conjugacy classes, 39 normal (all characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×4], C22, C22 [×6], C5, C8 [×2], C2×C4 [×2], C2×C4 [×5], D4 [×4], C23, C23, C10 [×3], C10 [×2], C42, C22⋊C4 [×2], C4⋊C4, C4⋊C4 [×2], C2×C8 [×2], C22×C4, C2×D4, C2×D4, Dic5 [×2], C20 [×2], C20 [×2], C2×C10, C2×C10 [×6], C22⋊C8, D4⋊C4 [×2], C4.Q8, C2.D8, C42⋊C2, C4⋊D4, C52C8 [×2], C2×Dic5 [×2], C2×C20 [×2], C2×C20 [×3], C5×D4 [×4], C22×C10, C22×C10, C23.19D4, C2×C52C8 [×2], C4×Dic5, C4⋊Dic5 [×2], C23.D5, C5×C22⋊C4, C5×C4⋊C4, C22×C20, D4×C10, D4×C10, C10.D8, C20.Q8, C20.55D4, D4⋊Dic5 [×2], C23.21D10, C5×C4⋊D4, (C2×D4).D10
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, D5, C2×D4, C4○D4 [×2], D10 [×3], C22.D4, C4○D8, C8⋊C22, C5⋊D4 [×2], C22×D5, C23.19D4, D42D5 [×2], C2×C5⋊D4, D4.D10, C23.18D10, D4.8D10, (C2×D4).D10

Smallest permutation representation of (C2×D4).D10
On 160 points
Generators in S160
(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)(21 148)(22 149)(23 150)(24 151)(25 152)(26 153)(27 154)(28 155)(29 156)(30 157)(31 158)(32 159)(33 160)(34 141)(35 142)(36 143)(37 144)(38 145)(39 146)(40 147)(61 101)(62 102)(63 103)(64 104)(65 105)(66 106)(67 107)(68 108)(69 109)(70 110)(71 111)(72 112)(73 113)(74 114)(75 115)(76 116)(77 117)(78 118)(79 119)(80 120)(81 133)(82 134)(83 135)(84 136)(85 137)(86 138)(87 139)(88 140)(89 121)(90 122)(91 123)(92 124)(93 125)(94 126)(95 127)(96 128)(97 129)(98 130)(99 131)(100 132)
(1 106 11 116)(2 107 12 117)(3 108 13 118)(4 109 14 119)(5 110 15 120)(6 111 16 101)(7 112 17 102)(8 113 18 103)(9 114 19 104)(10 115 20 105)(21 96 31 86)(22 97 32 87)(23 98 33 88)(24 99 34 89)(25 100 35 90)(26 81 36 91)(27 82 37 92)(28 83 38 93)(29 84 39 94)(30 85 40 95)(41 75 51 65)(42 76 52 66)(43 77 53 67)(44 78 54 68)(45 79 55 69)(46 80 56 70)(47 61 57 71)(48 62 58 72)(49 63 59 73)(50 64 60 74)(121 151 131 141)(122 152 132 142)(123 153 133 143)(124 154 134 144)(125 155 135 145)(126 156 136 146)(127 157 137 147)(128 158 138 148)(129 159 139 149)(130 160 140 150)
(2 43)(4 45)(6 47)(8 49)(10 51)(12 53)(14 55)(16 57)(18 59)(20 41)(21 96)(22 139)(23 98)(24 121)(25 100)(26 123)(27 82)(28 125)(29 84)(30 127)(31 86)(32 129)(33 88)(34 131)(35 90)(36 133)(37 92)(38 135)(39 94)(40 137)(61 101)(62 72)(63 103)(64 74)(65 105)(66 76)(67 107)(68 78)(69 109)(70 80)(71 111)(73 113)(75 115)(77 117)(79 119)(81 143)(83 145)(85 147)(87 149)(89 151)(91 153)(93 155)(95 157)(97 159)(99 141)(102 112)(104 114)(106 116)(108 118)(110 120)(122 142)(124 144)(126 146)(128 148)(130 150)(132 152)(134 154)(136 156)(138 158)(140 160)
(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)
(1 140 52 88)(2 129 53 97)(3 138 54 86)(4 127 55 95)(5 136 56 84)(6 125 57 93)(7 134 58 82)(8 123 59 91)(9 132 60 100)(10 121 41 89)(11 130 42 98)(12 139 43 87)(13 128 44 96)(14 137 45 85)(15 126 46 94)(16 135 47 83)(17 124 48 92)(18 133 49 81)(19 122 50 90)(20 131 51 99)(21 118 148 78)(22 107 149 67)(23 116 150 76)(24 105 151 65)(25 114 152 74)(26 103 153 63)(27 112 154 72)(28 101 155 61)(29 110 156 70)(30 119 157 79)(31 108 158 68)(32 117 159 77)(33 106 160 66)(34 115 141 75)(35 104 142 64)(36 113 143 73)(37 102 144 62)(38 111 145 71)(39 120 146 80)(40 109 147 69)

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

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

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

47 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G4H4I5A5B8A8B8C8D10A···10F10G10H10I10J10K10L10M10N20A···20H20I20J20K20L
order12222244444444455888810···10101010101010101020···2020202020
size111148222282020202022202020202···2444488884···48888

47 irreducible representations

dim111111122222222224444
type++++++++++++++-
imageC1C2C2C2C2C2C2D4D4D5C4○D4D10D10D10C4○D8C5⋊D4C5⋊D4C8⋊C22D42D5D4.D10D4.8D10
kernel(C2×D4).D10C10.D8C20.Q8C20.55D4D4⋊Dic5C23.21D10C5×C4⋊D4C2×C20C22×C10C4⋊D4C20C4⋊C4C22×C4C2×D4C10C2×C4C23C10C4C2C2
# reps111121111242224441444

Matrix representation of (C2×D4).D10 in GL6(𝔽41)

4000000
0400000
001000
000100
0000400
0000040
,
1220000
26400000
001000
000100
0000400
0000040
,
100000
26400000
001000
000100
000010
00002640
,
900000
090000
00203500
00151400
0000122
0000040
,
020000
2000000
00363800
008500
0000327
000009

G:=sub<GL(6,GF(41))| [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],[1,26,0,0,0,0,22,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,26,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,26,0,0,0,0,0,40],[9,0,0,0,0,0,0,9,0,0,0,0,0,0,20,15,0,0,0,0,35,14,0,0,0,0,0,0,1,0,0,0,0,0,22,40],[0,20,0,0,0,0,2,0,0,0,0,0,0,0,36,8,0,0,0,0,38,5,0,0,0,0,0,0,32,0,0,0,0,0,7,9] >;

(C2×D4).D10 in GAP, Magma, Sage, TeX

(C_2\times D_4).D_{10}
% in TeX

G:=Group("(C2xD4).D10");
// GroupNames label

G:=SmallGroup(320,662);
// by ID

G=gap.SmallGroup(320,662);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,232,254,219,1123,297,136,12550]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^4=c^2=1,d^10=b^2,e^2=a,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=e*b*e^-1=b^-1,b*d=d*b,d*c*d^-1=a*b^2*c,e*c*e^-1=b^-1*c,e*d*e^-1=d^9>;
// generators/relations

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