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G = C52C8.D4order 320 = 26·5

1st non-split extension by C52C8 of D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C52C8.1D4, C4⋊C4.30D10, C4.169(D4×D5), C51(C8.D4), Q8⋊C419D5, C20.127(C2×D4), (C2×C8).178D10, (C2×Q8).22D10, C20.22(C4○D4), C4.35(C4○D20), D102Q8.4C2, D103Q8.7C2, C20.Q813C2, (C2×Dic5).42D4, (C22×D5).30D4, C22.206(D4×D5), C20.44D425C2, C10.26(C4⋊D4), (C2×C20).256C23, (C2×C40).203C22, (Q8×C10).39C22, C2.29(D10⋊D4), C2.18(Q16⋊D5), C2.19(SD16⋊D5), C10.64(C8.C22), C4⋊Dic5.100C22, (C2×Dic10).78C22, (C2×C5⋊Q16)⋊6C2, (C2×C4×D5).32C22, (C5×Q8⋊C4)⋊25C2, (C2×C10).269(C2×D4), (C2×C8⋊D5).11C2, (C5×C4⋊C4).57C22, (C2×C52C8).46C22, (C2×C4).363(C22×D5), SmallGroup(320,443)

Series: Derived Chief Lower central Upper central

C1C2×C20 — C52C8.D4
C1C5C10C20C2×C20C2×C4×D5D102Q8 — C52C8.D4
C5C10C2×C20 — C52C8.D4
C1C22C2×C4Q8⋊C4

Generators and relations for C52C8.D4
 G = < a,b,c,d | a5=b8=c4=d2=1, bab-1=dad=a-1, ac=ca, cbc-1=b3, dbd=b5, dcd=b4c-1 >

Subgroups: 438 in 110 conjugacy classes, 39 normal (37 characteristic)
C1, C2 [×3], C2, C4 [×2], C4 [×5], C22, C22 [×3], C5, C8 [×3], C2×C4, C2×C4 [×7], Q8 [×4], C23, D5, C10 [×3], C22⋊C4 [×2], C4⋊C4, C4⋊C4 [×3], C2×C8, C2×C8, M4(2) [×2], Q16 [×2], C22×C4, C2×Q8, C2×Q8, Dic5 [×3], C20 [×2], C20 [×2], D10 [×3], C2×C10, Q8⋊C4, Q8⋊C4, C4.Q8, C22⋊Q8 [×2], C2×M4(2), C2×Q16, C52C8 [×2], C40, Dic10 [×2], C4×D5 [×2], C2×Dic5, C2×Dic5 [×2], C2×C20, C2×C20 [×2], C5×Q8 [×2], C22×D5, C8.D4, C8⋊D5 [×2], C2×C52C8, C10.D4, C4⋊Dic5, C4⋊Dic5, D10⋊C4 [×2], C5⋊Q16 [×2], C5×C4⋊C4, C2×C40, C2×Dic10, C2×C4×D5, Q8×C10, C20.Q8, C20.44D4, C5×Q8⋊C4, D102Q8, C2×C8⋊D5, C2×C5⋊Q16, D103Q8, C52C8.D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, D5, C2×D4 [×2], C4○D4, D10 [×3], C4⋊D4, C8.C22 [×2], C22×D5, C8.D4, C4○D20, D4×D5 [×2], D10⋊D4, SD16⋊D5, Q16⋊D5, C52C8.D4

Smallest permutation representation of C52C8.D4
On 160 points
Generators in S160
(1 52 35 99 110)(2 111 100 36 53)(3 54 37 101 112)(4 105 102 38 55)(5 56 39 103 106)(6 107 104 40 49)(7 50 33 97 108)(8 109 98 34 51)(9 44 124 71 87)(10 88 72 125 45)(11 46 126 65 81)(12 82 66 127 47)(13 48 128 67 83)(14 84 68 121 41)(15 42 122 69 85)(16 86 70 123 43)(17 93 159 73 113)(18 114 74 160 94)(19 95 153 75 115)(20 116 76 154 96)(21 89 155 77 117)(22 118 78 156 90)(23 91 157 79 119)(24 120 80 158 92)(25 150 58 139 134)(26 135 140 59 151)(27 152 60 141 136)(28 129 142 61 145)(29 146 62 143 130)(30 131 144 63 147)(31 148 64 137 132)(32 133 138 57 149)
(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 12 135 96)(2 15 136 91)(3 10 129 94)(4 13 130 89)(5 16 131 92)(6 11 132 95)(7 14 133 90)(8 9 134 93)(17 51 87 139)(18 54 88 142)(19 49 81 137)(20 52 82 140)(21 55 83 143)(22 50 84 138)(23 53 85 141)(24 56 86 144)(25 159 109 44)(26 154 110 47)(27 157 111 42)(28 160 112 45)(29 155 105 48)(30 158 106 43)(31 153 107 46)(32 156 108 41)(33 68 57 118)(34 71 58 113)(35 66 59 116)(36 69 60 119)(37 72 61 114)(38 67 62 117)(39 70 63 120)(40 65 64 115)(73 98 124 150)(74 101 125 145)(75 104 126 148)(76 99 127 151)(77 102 128 146)(78 97 121 149)(79 100 122 152)(80 103 123 147)
(1 131)(2 136)(3 133)(4 130)(5 135)(6 132)(7 129)(8 134)(9 13)(11 15)(17 155)(18 160)(19 157)(20 154)(21 159)(22 156)(23 153)(24 158)(25 51)(26 56)(27 53)(28 50)(29 55)(30 52)(31 49)(32 54)(33 145)(34 150)(35 147)(36 152)(37 149)(38 146)(39 151)(40 148)(41 84)(42 81)(43 86)(44 83)(45 88)(46 85)(47 82)(48 87)(57 101)(58 98)(59 103)(60 100)(61 97)(62 102)(63 99)(64 104)(65 122)(66 127)(67 124)(68 121)(69 126)(70 123)(71 128)(72 125)(73 117)(74 114)(75 119)(76 116)(77 113)(78 118)(79 115)(80 120)(89 93)(91 95)(105 143)(106 140)(107 137)(108 142)(109 139)(110 144)(111 141)(112 138)

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

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

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

44 conjugacy classes

class 1 2A2B2C2D4A4B4C4D4E4F4G5A5B8A8B8C8D10A···10F20A20B20C20D20E···20L40A···40H
order12222444444455888810···102020202020···2040···40
size1111202288204040224420202···244448···84···4

44 irreducible representations

dim1111111122222222244444
type+++++++++++++++-++-
imageC1C2C2C2C2C2C2C2D4D4D4D5C4○D4D10D10D10C4○D20C8.C22D4×D5D4×D5SD16⋊D5Q16⋊D5
kernelC52C8.D4C20.Q8C20.44D4C5×Q8⋊C4D102Q8C2×C8⋊D5C2×C5⋊Q16D103Q8C52C8C2×Dic5C22×D5Q8⋊C4C20C4⋊C4C2×C8C2×Q8C4C10C4C22C2C2
# reps1111111121122222822244

Matrix representation of C52C8.D4 in GL6(𝔽41)

100000
010000
0004000
001600
00180640
0002310
,
3200000
690000
003131016
00991627
0026182912
0032142213
,
1540000
5260000
002962729
000231427
004131835
0044012
,
4000000
2810000
0063500
00403500
00233810
00015640

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,18,0,0,0,40,6,0,23,0,0,0,0,6,1,0,0,0,0,40,0],[32,6,0,0,0,0,0,9,0,0,0,0,0,0,31,9,26,32,0,0,31,9,18,14,0,0,0,16,29,22,0,0,16,27,12,13],[15,5,0,0,0,0,4,26,0,0,0,0,0,0,29,0,4,4,0,0,6,23,13,4,0,0,27,14,18,0,0,0,29,27,35,12],[40,28,0,0,0,0,0,1,0,0,0,0,0,0,6,40,23,0,0,0,35,35,38,15,0,0,0,0,1,6,0,0,0,0,0,40] >;

C52C8.D4 in GAP, Magma, Sage, TeX

C_5\rtimes_2C_8.D_4
% in TeX

G:=Group("C5:2C8.D4");
// GroupNames label

G:=SmallGroup(320,443);
// by ID

G=gap.SmallGroup(320,443);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,344,1094,135,184,570,297,136,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^5=b^8=c^4=d^2=1,b*a*b^-1=d*a*d=a^-1,a*c=c*a,c*b*c^-1=b^3,d*b*d=b^5,d*c*d=b^4*c^-1>;
// generators/relations

׿
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