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G = C20⋊Q8⋊C2order 320 = 26·5

4th semidirect product of C20⋊Q8 and C2 acting faithfully

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C20⋊Q84C2, C4⋊C4.6D10, C408C417C2, D4⋊C4.8D5, (C2×D4).20D10, C20.4(C4○D4), (C2×C8).167D10, C10.Q161C2, C4.22(C4○D20), (C2×Dic5).24D4, D4⋊Dic5.4C2, C22.167(D4×D5), C2.12(D8⋊D5), C4.48(D42D5), C20.44D416C2, C10.28(C8⋊C22), (C2×C20).205C23, (C2×C40).180C22, C20.17D4.4C2, C2.9(SD16⋊D5), (D4×C10).26C22, C4⋊Dic5.64C22, C10.24(C4.4D4), C10.26(C8.C22), (C4×Dic5).15C22, C52(C42.28C22), (C2×Dic10).56C22, C2.14(Dic5.5D4), (C5×D4⋊C4).9C2, (C2×C10).218(C2×D4), (C5×C4⋊C4).10C22, (C2×C52C8).11C22, (C2×C4).312(C22×D5), SmallGroup(320,392)

Series: Derived Chief Lower central Upper central

C1C2×C20 — C20⋊Q8⋊C2
C1C5C10C20C2×C20C4×Dic5C20⋊Q8 — C20⋊Q8⋊C2
C5C10C2×C20 — C20⋊Q8⋊C2
C1C22C2×C4D4⋊C4

Generators and relations for C20⋊Q8⋊C2
 G = < a,b,c,d | a20=b4=d2=1, c2=b2, bab-1=dad=a11, cac-1=a9, cbc-1=b-1, dbd=a15b-1, dcd=a10b2c >

Subgroups: 398 in 100 conjugacy classes, 37 normal (all characteristic)
C1, C2 [×3], C2, C4 [×2], C4 [×5], C22, C22 [×3], C5, C8 [×2], C2×C4, C2×C4 [×5], D4 [×2], Q8 [×4], C23, C10 [×3], C10, C42, C22⋊C4 [×2], C4⋊C4, C4⋊C4 [×2], C2×C8, C2×C8, C2×D4, C2×Q8 [×2], Dic5 [×4], C20 [×2], C20, C2×C10, C2×C10 [×3], C8⋊C4, D4⋊C4, D4⋊C4, Q8⋊C4 [×2], C4.4D4, C4⋊Q8, C52C8, C40, Dic10 [×4], C2×Dic5 [×2], C2×Dic5 [×2], C2×C20, C2×C20, C5×D4 [×2], C22×C10, C42.28C22, C2×C52C8, C4×Dic5, C10.D4, C4⋊Dic5, C23.D5 [×2], C5×C4⋊C4, C2×C40, C2×Dic10, C2×Dic10, D4×C10, C10.Q16, C408C4, C20.44D4, D4⋊Dic5, C5×D4⋊C4, C20⋊Q8, C20.17D4, C20⋊Q8⋊C2
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, D5, C2×D4, C4○D4 [×2], D10 [×3], C4.4D4, C8⋊C22, C8.C22, C22×D5, C42.28C22, C4○D20, D4×D5, D42D5, Dic5.5D4, D8⋊D5, SD16⋊D5, C20⋊Q8⋊C2

Smallest permutation representation of C20⋊Q8⋊C2
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 28 113 46)(2 39 114 57)(3 30 115 48)(4 21 116 59)(5 32 117 50)(6 23 118 41)(7 34 119 52)(8 25 120 43)(9 36 101 54)(10 27 102 45)(11 38 103 56)(12 29 104 47)(13 40 105 58)(14 31 106 49)(15 22 107 60)(16 33 108 51)(17 24 109 42)(18 35 110 53)(19 26 111 44)(20 37 112 55)(61 124 91 156)(62 135 92 147)(63 126 93 158)(64 137 94 149)(65 128 95 160)(66 139 96 151)(67 130 97 142)(68 121 98 153)(69 132 99 144)(70 123 100 155)(71 134 81 146)(72 125 82 157)(73 136 83 148)(74 127 84 159)(75 138 85 150)(76 129 86 141)(77 140 87 152)(78 131 88 143)(79 122 89 154)(80 133 90 145)
(1 91 113 61)(2 100 114 70)(3 89 115 79)(4 98 116 68)(5 87 117 77)(6 96 118 66)(7 85 119 75)(8 94 120 64)(9 83 101 73)(10 92 102 62)(11 81 103 71)(12 90 104 80)(13 99 105 69)(14 88 106 78)(15 97 107 67)(16 86 108 76)(17 95 109 65)(18 84 110 74)(19 93 111 63)(20 82 112 72)(21 121 59 153)(22 130 60 142)(23 139 41 151)(24 128 42 160)(25 137 43 149)(26 126 44 158)(27 135 45 147)(28 124 46 156)(29 133 47 145)(30 122 48 154)(31 131 49 143)(32 140 50 152)(33 129 51 141)(34 138 52 150)(35 127 53 159)(36 136 54 148)(37 125 55 157)(38 134 56 146)(39 123 57 155)(40 132 58 144)
(2 12)(4 14)(6 16)(8 18)(10 20)(21 54)(22 45)(23 56)(24 47)(25 58)(26 49)(27 60)(28 51)(29 42)(30 53)(31 44)(32 55)(33 46)(34 57)(35 48)(36 59)(37 50)(38 41)(39 52)(40 43)(61 81)(62 92)(63 83)(64 94)(65 85)(66 96)(67 87)(68 98)(69 89)(70 100)(71 91)(72 82)(73 93)(74 84)(75 95)(76 86)(77 97)(78 88)(79 99)(80 90)(102 112)(104 114)(106 116)(108 118)(110 120)(121 126)(122 137)(123 128)(124 139)(125 130)(127 132)(129 134)(131 136)(133 138)(135 140)(141 146)(142 157)(143 148)(144 159)(145 150)(147 152)(149 154)(151 156)(153 158)(155 160)

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

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

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

44 conjugacy classes

class 1 2A2B2C2D4A4B4C4D4E4F4G5A5B8A8B8C8D10A···10F10G10H10I10J20A20B20C20D20E20F20G20H40A···40H
order12222444444455888810···1010101010202020202020202040···40
size1111822820204040224420202···28888444488884···4

44 irreducible representations

dim111111112222222444444
type++++++++++++++--+-
imageC1C2C2C2C2C2C2C2D4D5C4○D4D10D10D10C4○D20C8⋊C22C8.C22D42D5D4×D5D8⋊D5SD16⋊D5
kernelC20⋊Q8⋊C2C10.Q16C408C4C20.44D4D4⋊Dic5C5×D4⋊C4C20⋊Q8C20.17D4C2×Dic5D4⋊C4C20C4⋊C4C2×C8C2×D4C4C10C10C4C22C2C2
# reps111111112242228112244

Matrix representation of C20⋊Q8⋊C2 in GL8(𝔽41)

01000000
4034000000
00010000
0040340000
0000712837
000033401332
000030040
0000213135
,
00100000
00010000
400000000
040000000
0000221720
000037391328
00003332220
00003112119
,
0038170000
003830000
3817000000
383000000
00001211427
00003740132
00009222118
00003222120
,
10000000
01000000
004000000
000400000
00001000
00000100
00003838400
000030040

G:=sub<GL(8,GF(41))| [0,40,0,0,0,0,0,0,1,34,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,1,34,0,0,0,0,0,0,0,0,7,33,3,21,0,0,0,0,1,40,0,3,0,0,0,0,28,13,0,1,0,0,0,0,37,32,40,35],[0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,2,37,3,3,0,0,0,0,21,39,33,11,0,0,0,0,7,13,22,21,0,0,0,0,20,28,20,19],[0,0,38,38,0,0,0,0,0,0,17,3,0,0,0,0,38,38,0,0,0,0,0,0,17,3,0,0,0,0,0,0,0,0,0,0,1,37,9,3,0,0,0,0,21,40,22,22,0,0,0,0,14,13,21,21,0,0,0,0,27,2,18,20],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,38,3,0,0,0,0,0,1,38,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40] >;

C20⋊Q8⋊C2 in GAP, Magma, Sage, TeX

C_{20}\rtimes Q_8\rtimes C_2
% in TeX

G:=Group("C20:Q8:C2");
// GroupNames label

G:=SmallGroup(320,392);
// by ID

G=gap.SmallGroup(320,392);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,253,120,1094,135,570,297,136,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^20=b^4=d^2=1,c^2=b^2,b*a*b^-1=d*a*d=a^11,c*a*c^-1=a^9,c*b*c^-1=b^-1,d*b*d=a^15*b^-1,d*c*d=a^10*b^2*c>;
// generators/relations

׿
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