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G = (C2×C20).290D4order 320 = 26·5

264th non-split extension by C2×C20 of D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: (C2×C20).290D4, C2.9(C202D4), (C22×D5).42D4, C22.249(D4×D5), C10.92(C4⋊D4), (C22×C4).105D10, C55(C23.11D4), C2.7(C20.23D4), C10.54(C4.4D4), (C23×D5).22C22, C23.382(C22×D5), C10.10C4220C2, C10.29(C422C2), C22.110(C4○D20), (C22×C20).393C22, (C22×C10).357C23, C22.53(Q82D5), C2.22(D10.13D4), C22.104(D42D5), C10.64(C22.D4), (C22×Dic5).62C22, C2.15(C23.23D10), (C2×C4⋊C4)⋊11D5, (C10×C4⋊C4)⋊27C2, (C2×C10).337(C2×D4), (C2×C4).43(C5⋊D4), C2.14(C4⋊C4⋊D5), C22.142(C2×C5⋊D4), (C2×D10⋊C4).17C2, (C2×C10).192(C4○D4), SmallGroup(320,620)

Series: Derived Chief Lower central Upper central

C1C22×C10 — (C2×C20).290D4
C1C5C10C2×C10C22×C10C23×D5C2×D10⋊C4 — (C2×C20).290D4
C5C22×C10 — (C2×C20).290D4
C1C23C2×C4⋊C4

Generators and relations for (C2×C20).290D4
 G = < a,b,c,d | a2=b20=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=ab-1, dbd=ab9, dcd=ac-1 >

Subgroups: 726 in 170 conjugacy classes, 57 normal (25 characteristic)
C1, C2 [×3], C2 [×4], C2 [×2], C4 [×7], C22 [×3], C22 [×4], C22 [×10], C5, C2×C4 [×2], C2×C4 [×17], C23, C23 [×8], D5 [×2], C10 [×3], C10 [×4], C22⋊C4 [×6], C4⋊C4 [×2], C22×C4, C22×C4 [×2], C22×C4 [×3], C24, Dic5 [×3], C20 [×4], D10 [×10], C2×C10 [×3], C2×C10 [×4], C2.C42 [×3], C2×C22⋊C4 [×3], C2×C4⋊C4, C2×Dic5 [×9], C2×C20 [×2], C2×C20 [×8], C22×D5 [×2], C22×D5 [×6], C22×C10, C23.11D4, D10⋊C4 [×6], C5×C4⋊C4 [×2], C22×Dic5, C22×Dic5 [×2], C22×C20, C22×C20 [×2], C23×D5, C10.10C42, C10.10C42 [×2], C2×D10⋊C4, C2×D10⋊C4 [×2], C10×C4⋊C4, (C2×C20).290D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, D5, C2×D4 [×2], C4○D4 [×5], D10 [×3], C4⋊D4, C22.D4 [×3], C4.4D4, C422C2 [×2], C5⋊D4 [×2], C22×D5, C23.11D4, C4○D20 [×2], D4×D5, D42D5, Q82D5 [×2], C2×C5⋊D4, D10.13D4 [×2], C4⋊C4⋊D5 [×2], C23.23D10, C202D4, C20.23D4, (C2×C20).290D4

Smallest permutation representation of (C2×C20).290D4
On 160 points
Generators in S160
(1 118)(2 119)(3 120)(4 101)(5 102)(6 103)(7 104)(8 105)(9 106)(10 107)(11 108)(12 109)(13 110)(14 111)(15 112)(16 113)(17 114)(18 115)(19 116)(20 117)(21 128)(22 129)(23 130)(24 131)(25 132)(26 133)(27 134)(28 135)(29 136)(30 137)(31 138)(32 139)(33 140)(34 121)(35 122)(36 123)(37 124)(38 125)(39 126)(40 127)(41 91)(42 92)(43 93)(44 94)(45 95)(46 96)(47 97)(48 98)(49 99)(50 100)(51 81)(52 82)(53 83)(54 84)(55 85)(56 86)(57 87)(58 88)(59 89)(60 90)(61 145)(62 146)(63 147)(64 148)(65 149)(66 150)(67 151)(68 152)(69 153)(70 154)(71 155)(72 156)(73 157)(74 158)(75 159)(76 160)(77 141)(78 142)(79 143)(80 144)
(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)
(1 158 128 47)(2 73 129 96)(3 156 130 45)(4 71 131 94)(5 154 132 43)(6 69 133 92)(7 152 134 41)(8 67 135 90)(9 150 136 59)(10 65 137 88)(11 148 138 57)(12 63 139 86)(13 146 140 55)(14 61 121 84)(15 144 122 53)(16 79 123 82)(17 142 124 51)(18 77 125 100)(19 160 126 49)(20 75 127 98)(21 97 118 74)(22 46 119 157)(23 95 120 72)(24 44 101 155)(25 93 102 70)(26 42 103 153)(27 91 104 68)(28 60 105 151)(29 89 106 66)(30 58 107 149)(31 87 108 64)(32 56 109 147)(33 85 110 62)(34 54 111 145)(35 83 112 80)(36 52 113 143)(37 81 114 78)(38 50 115 141)(39 99 116 76)(40 48 117 159)
(2 107)(3 19)(4 105)(5 17)(6 103)(7 15)(8 101)(9 13)(10 119)(12 117)(14 115)(16 113)(18 111)(20 109)(22 137)(23 39)(24 135)(25 37)(26 133)(27 35)(28 131)(29 33)(30 129)(32 127)(34 125)(36 123)(38 121)(40 139)(41 80)(42 153)(43 78)(44 151)(45 76)(46 149)(47 74)(48 147)(49 72)(50 145)(51 70)(52 143)(53 68)(54 141)(55 66)(56 159)(57 64)(58 157)(59 62)(60 155)(61 100)(63 98)(65 96)(67 94)(69 92)(71 90)(73 88)(75 86)(77 84)(79 82)(81 154)(83 152)(85 150)(87 148)(89 146)(91 144)(93 142)(95 160)(97 158)(99 156)(102 114)(104 112)(106 110)(116 120)(122 134)(124 132)(126 130)(136 140)

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

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

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

62 conjugacy classes

class 1 2A···2G2H2I4A···4F4G···4L5A5B10A···10N20A···20X
order12···2224···44···45510···1020···20
size11···120204···420···20222···24···4

62 irreducible representations

dim11112222222444
type+++++++++-+
imageC1C2C2C2D4D4D5C4○D4D10C5⋊D4C4○D20D4×D5D42D5Q82D5
kernel(C2×C20).290D4C10.10C42C2×D10⋊C4C10×C4⋊C4C2×C20C22×D5C2×C4⋊C4C2×C10C22×C4C2×C4C22C22C22C22
# reps1331222106816224

Matrix representation of (C2×C20).290D4 in GL6(𝔽41)

100000
010000
0040000
0004000
000010
000001
,
14220000
19270000
00352300
00182000
00004040
00003635
,
010000
100000
00132800
00322800
0000203
0000321
,
100000
010000
0063500
00403500
000067
00003635

G:=sub<GL(6,GF(41))| [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],[14,19,0,0,0,0,22,27,0,0,0,0,0,0,35,18,0,0,0,0,23,20,0,0,0,0,0,0,40,36,0,0,0,0,40,35],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,13,32,0,0,0,0,28,28,0,0,0,0,0,0,20,3,0,0,0,0,3,21],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,6,40,0,0,0,0,35,35,0,0,0,0,0,0,6,36,0,0,0,0,7,35] >;

(C2×C20).290D4 in GAP, Magma, Sage, TeX

(C_2\times C_{20})._{290}D_4
% in TeX

G:=Group("(C2xC20).290D4");
// GroupNames label

G:=SmallGroup(320,620);
// by ID

G=gap.SmallGroup(320,620);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,232,254,387,100,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^20=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=a*b^-1,d*b*d=a*b^9,d*c*d=a*c^-1>;
// generators/relations

׿
×
𝔽