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G = C20.(C2×D4)  order 320 = 26·5

125th non-split extension by C20 of C2×D4 acting via C2×D4/C22=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C4○D42Dic5, (C2×C20).501D4, C20.213(C2×D4), D4.6(C2×Dic5), Q8.6(C2×Dic5), (C2×D4).202D10, Q8⋊Dic544C2, D4⋊Dic544C2, (C2×Q8).171D10, C10.111(C4○D8), (C2×C20).480C23, C20.143(C22×C4), (C22×C10).112D4, (C22×C4).355D10, C23.44(C5⋊D4), C56(C23.24D4), C4.32(C23.D5), C4.14(C22×Dic5), C20.145(C22⋊C4), C2.7(D4.8D10), (D4×C10).243C22, C4⋊Dic5.355C22, (Q8×C10).206C22, C22.3(C23.D5), C23.21D1019C2, (C22×C20).206C22, (C5×C4○D4)⋊8C4, (C2×C4○D4).2D5, C4.95(C2×C5⋊D4), (C22×C52C8)⋊8C2, (C10×C4○D4).2C2, (C5×D4).37(C2×C4), (C5×Q8).39(C2×C4), (C2×C20).296(C2×C4), (C2×C10).566(C2×D4), (C2×C4).52(C2×Dic5), C22.97(C2×C5⋊D4), C2.18(C2×C23.D5), (C2×C4).281(C5⋊D4), C10.123(C2×C22⋊C4), (C2×C4).565(C22×D5), (C2×C10).89(C22⋊C4), (C2×C52C8).290C22, SmallGroup(320,860)

Series: Derived Chief Lower central Upper central

C1C20 — C20.(C2×D4)
C1C5C10C20C2×C20C4⋊Dic5C23.21D10 — C20.(C2×D4)
C5C10C20 — C20.(C2×D4)
C1C2×C4C22×C4C2×C4○D4

Generators and relations for C20.(C2×D4)
 G = < a,b,c,d | a20=c4=d2=1, b2=a10, ab=ba, cac-1=a-1, dad=a11, bc=cb, bd=db, dcd=a5c-1 >

Subgroups: 398 in 158 conjugacy classes, 71 normal (27 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×2], C4 [×2], C4 [×4], C22, C22 [×2], C22 [×6], C5, C8 [×2], C2×C4 [×2], C2×C4 [×4], C2×C4 [×7], D4 [×2], D4 [×5], Q8 [×2], Q8, C23, C23, C10, C10 [×2], C10 [×4], C42, C22⋊C4, C4⋊C4 [×2], C2×C8 [×4], C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4 [×4], C4○D4 [×2], Dic5 [×2], C20 [×2], C20 [×2], C20 [×2], C2×C10, C2×C10 [×2], C2×C10 [×6], D4⋊C4 [×2], Q8⋊C4 [×2], C42⋊C2, C22×C8, C2×C4○D4, C52C8 [×2], C2×Dic5 [×2], C2×C20 [×2], C2×C20 [×4], C2×C20 [×5], C5×D4 [×2], C5×D4 [×5], C5×Q8 [×2], C5×Q8, C22×C10, C22×C10, C23.24D4, C2×C52C8 [×2], C2×C52C8 [×2], C4×Dic5, C4⋊Dic5 [×2], C23.D5, C22×C20, C22×C20, D4×C10, D4×C10, Q8×C10, C5×C4○D4 [×4], C5×C4○D4 [×2], D4⋊Dic5 [×2], Q8⋊Dic5 [×2], C22×C52C8, C23.21D10, C10×C4○D4, C20.(C2×D4)
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, D5, C22⋊C4 [×4], C22×C4, C2×D4 [×2], Dic5 [×4], D10 [×3], C2×C22⋊C4, C4○D8 [×2], C2×Dic5 [×6], C5⋊D4 [×4], C22×D5, C23.24D4, C23.D5 [×4], C22×Dic5, C2×C5⋊D4 [×2], D4.8D10 [×2], C2×C23.D5, C20.(C2×D4)

Smallest permutation representation of C20.(C2×D4)
On 160 points
Generators in S160
(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)
(1 112 11 102)(2 113 12 103)(3 114 13 104)(4 115 14 105)(5 116 15 106)(6 117 16 107)(7 118 17 108)(8 119 18 109)(9 120 19 110)(10 101 20 111)(21 132 31 122)(22 133 32 123)(23 134 33 124)(24 135 34 125)(25 136 35 126)(26 137 36 127)(27 138 37 128)(28 139 38 129)(29 140 39 130)(30 121 40 131)(41 76 51 66)(42 77 52 67)(43 78 53 68)(44 79 54 69)(45 80 55 70)(46 61 56 71)(47 62 57 72)(48 63 58 73)(49 64 59 74)(50 65 60 75)(81 151 91 141)(82 152 92 142)(83 153 93 143)(84 154 94 144)(85 155 95 145)(86 156 96 146)(87 157 97 147)(88 158 98 148)(89 159 99 149)(90 160 100 150)
(1 70 100 123)(2 69 81 122)(3 68 82 121)(4 67 83 140)(5 66 84 139)(6 65 85 138)(7 64 86 137)(8 63 87 136)(9 62 88 135)(10 61 89 134)(11 80 90 133)(12 79 91 132)(13 78 92 131)(14 77 93 130)(15 76 94 129)(16 75 95 128)(17 74 96 127)(18 73 97 126)(19 72 98 125)(20 71 99 124)(21 113 44 151)(22 112 45 150)(23 111 46 149)(24 110 47 148)(25 109 48 147)(26 108 49 146)(27 107 50 145)(28 106 51 144)(29 105 52 143)(30 104 53 142)(31 103 54 141)(32 102 55 160)(33 101 56 159)(34 120 57 158)(35 119 58 157)(36 118 59 156)(37 117 60 155)(38 116 41 154)(39 115 42 153)(40 114 43 152)
(1 6)(2 17)(3 8)(4 19)(5 10)(7 12)(9 14)(11 16)(13 18)(15 20)(21 44)(22 55)(23 46)(24 57)(25 48)(26 59)(27 50)(28 41)(29 52)(30 43)(31 54)(32 45)(33 56)(34 47)(35 58)(36 49)(37 60)(38 51)(39 42)(40 53)(61 134)(62 125)(63 136)(64 127)(65 138)(66 129)(67 140)(68 131)(69 122)(70 133)(71 124)(72 135)(73 126)(74 137)(75 128)(76 139)(77 130)(78 121)(79 132)(80 123)(81 96)(82 87)(83 98)(84 89)(85 100)(86 91)(88 93)(90 95)(92 97)(94 99)(101 116)(102 107)(103 118)(104 109)(105 120)(106 111)(108 113)(110 115)(112 117)(114 119)(141 156)(142 147)(143 158)(144 149)(145 160)(146 151)(148 153)(150 155)(152 157)(154 159)

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

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

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

68 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E4F4G4H4I4J4K4L5A5B8A···8H10A···10F10G···10R20A···20H20I···20T
order12222222444444444444558···810···1010···1020···2020···20
size1111224411112244202020202210···102···24···42···24···4

68 irreducible representations

dim111111122222222224
type++++++++++++-
imageC1C2C2C2C2C2C4D4D4D5D10D10D10Dic5C4○D8C5⋊D4C5⋊D4D4.8D10
kernelC20.(C2×D4)D4⋊Dic5Q8⋊Dic5C22×C52C8C23.21D10C10×C4○D4C5×C4○D4C2×C20C22×C10C2×C4○D4C22×C4C2×D4C2×Q8C4○D4C10C2×C4C23C2
# reps1221118312222881248

Matrix representation of C20.(C2×D4) in GL4(𝔽41) generated by

23000
02500
00040
0010
,
40000
04000
00320
00032
,
0100
40000
001229
002929
,
40000
04000
0001
0010
G:=sub<GL(4,GF(41))| [23,0,0,0,0,25,0,0,0,0,0,1,0,0,40,0],[40,0,0,0,0,40,0,0,0,0,32,0,0,0,0,32],[0,40,0,0,1,0,0,0,0,0,12,29,0,0,29,29],[40,0,0,0,0,40,0,0,0,0,0,1,0,0,1,0] >;

C20.(C2×D4) in GAP, Magma, Sage, TeX

C_{20}.(C_2\times D_4)
% in TeX

G:=Group("C20.(C2xD4)");
// GroupNames label

G:=SmallGroup(320,860);
// by ID

G=gap.SmallGroup(320,860);
# by ID

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

G:=Group<a,b,c,d|a^20=c^4=d^2=1,b^2=a^10,a*b=b*a,c*a*c^-1=a^-1,d*a*d=a^11,b*c=c*b,b*d=d*b,d*c*d=a^5*c^-1>;
// generators/relations

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×
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