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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, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, Q8, C23, C10, C10, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C2×D4, C2×Q8, Dic5, C20, C20, C2×C10, C2×C10, C8⋊C4, D4⋊C4, D4⋊C4, Q8⋊C4, C4.4D4, C4⋊Q8, C52C8, C40, Dic10, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C5×D4, C22×C10, C42.28C22, C2×C52C8, C4×Dic5, C10.D4, C4⋊Dic5, C23.D5, 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, C22, D4, C23, D5, C2×D4, C4○D4, D10, 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 54 101 92)(2 45 102 83)(3 56 103 94)(4 47 104 85)(5 58 105 96)(6 49 106 87)(7 60 107 98)(8 51 108 89)(9 42 109 100)(10 53 110 91)(11 44 111 82)(12 55 112 93)(13 46 113 84)(14 57 114 95)(15 48 115 86)(16 59 116 97)(17 50 117 88)(18 41 118 99)(19 52 119 90)(20 43 120 81)(21 141 67 137)(22 152 68 128)(23 143 69 139)(24 154 70 130)(25 145 71 121)(26 156 72 132)(27 147 73 123)(28 158 74 134)(29 149 75 125)(30 160 76 136)(31 151 77 127)(32 142 78 138)(33 153 79 129)(34 144 80 140)(35 155 61 131)(36 146 62 122)(37 157 63 133)(38 148 64 124)(39 159 65 135)(40 150 66 126)
(1 155 101 131)(2 144 102 140)(3 153 103 129)(4 142 104 138)(5 151 105 127)(6 160 106 136)(7 149 107 125)(8 158 108 134)(9 147 109 123)(10 156 110 132)(11 145 111 121)(12 154 112 130)(13 143 113 139)(14 152 114 128)(15 141 115 137)(16 150 116 126)(17 159 117 135)(18 148 118 124)(19 157 119 133)(20 146 120 122)(21 86 67 48)(22 95 68 57)(23 84 69 46)(24 93 70 55)(25 82 71 44)(26 91 72 53)(27 100 73 42)(28 89 74 51)(29 98 75 60)(30 87 76 49)(31 96 77 58)(32 85 78 47)(33 94 79 56)(34 83 80 45)(35 92 61 54)(36 81 62 43)(37 90 63 52)(38 99 64 41)(39 88 65 50)(40 97 66 59)
(2 12)(4 14)(6 16)(8 18)(10 20)(21 36)(22 27)(23 38)(24 29)(25 40)(26 31)(28 33)(30 35)(32 37)(34 39)(41 94)(42 85)(43 96)(44 87)(45 98)(46 89)(47 100)(48 91)(49 82)(50 93)(51 84)(52 95)(53 86)(54 97)(55 88)(56 99)(57 90)(58 81)(59 92)(60 83)(61 76)(62 67)(63 78)(64 69)(65 80)(66 71)(68 73)(70 75)(72 77)(74 79)(102 112)(104 114)(106 116)(108 118)(110 120)(121 155)(122 146)(123 157)(124 148)(125 159)(126 150)(127 141)(128 152)(129 143)(130 154)(131 145)(132 156)(133 147)(134 158)(135 149)(136 160)(137 151)(138 142)(139 153)(140 144)

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

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

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

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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