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G = C21⋊C8order 168 = 23·3·7

1st semidirect product of C21 and C8 acting via C8/C4=C2

metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary

Aliases: C211C8, C6.Dic7, C84.2C2, C42.1C4, C28.2S3, C4.2D21, C12.2D7, C2.Dic21, C14.Dic3, C3⋊(C7⋊C8), C7⋊(C3⋊C8), SmallGroup(168,5)

Series: Derived Chief Lower central Upper central

C1C21 — C21⋊C8
C1C7C21C42C84 — C21⋊C8
C21 — C21⋊C8
C1C4

Generators and relations for C21⋊C8
 G = < a,b | a21=b8=1, bab-1=a-1 >

21C8
7C3⋊C8
3C7⋊C8

Smallest permutation representation of C21⋊C8
Regular action on 168 points
Generators in S168
(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)(148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168)
(1 163 67 115 26 146 49 104)(2 162 68 114 27 145 50 103)(3 161 69 113 28 144 51 102)(4 160 70 112 29 143 52 101)(5 159 71 111 30 142 53 100)(6 158 72 110 31 141 54 99)(7 157 73 109 32 140 55 98)(8 156 74 108 33 139 56 97)(9 155 75 107 34 138 57 96)(10 154 76 106 35 137 58 95)(11 153 77 126 36 136 59 94)(12 152 78 125 37 135 60 93)(13 151 79 124 38 134 61 92)(14 150 80 123 39 133 62 91)(15 149 81 122 40 132 63 90)(16 148 82 121 41 131 43 89)(17 168 83 120 42 130 44 88)(18 167 84 119 22 129 45 87)(19 166 64 118 23 128 46 86)(20 165 65 117 24 127 47 85)(21 164 66 116 25 147 48 105)

G:=sub<Sym(168)| (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)(148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168), (1,163,67,115,26,146,49,104)(2,162,68,114,27,145,50,103)(3,161,69,113,28,144,51,102)(4,160,70,112,29,143,52,101)(5,159,71,111,30,142,53,100)(6,158,72,110,31,141,54,99)(7,157,73,109,32,140,55,98)(8,156,74,108,33,139,56,97)(9,155,75,107,34,138,57,96)(10,154,76,106,35,137,58,95)(11,153,77,126,36,136,59,94)(12,152,78,125,37,135,60,93)(13,151,79,124,38,134,61,92)(14,150,80,123,39,133,62,91)(15,149,81,122,40,132,63,90)(16,148,82,121,41,131,43,89)(17,168,83,120,42,130,44,88)(18,167,84,119,22,129,45,87)(19,166,64,118,23,128,46,86)(20,165,65,117,24,127,47,85)(21,164,66,116,25,147,48,105)>;

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)(148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168), (1,163,67,115,26,146,49,104)(2,162,68,114,27,145,50,103)(3,161,69,113,28,144,51,102)(4,160,70,112,29,143,52,101)(5,159,71,111,30,142,53,100)(6,158,72,110,31,141,54,99)(7,157,73,109,32,140,55,98)(8,156,74,108,33,139,56,97)(9,155,75,107,34,138,57,96)(10,154,76,106,35,137,58,95)(11,153,77,126,36,136,59,94)(12,152,78,125,37,135,60,93)(13,151,79,124,38,134,61,92)(14,150,80,123,39,133,62,91)(15,149,81,122,40,132,63,90)(16,148,82,121,41,131,43,89)(17,168,83,120,42,130,44,88)(18,167,84,119,22,129,45,87)(19,166,64,118,23,128,46,86)(20,165,65,117,24,127,47,85)(21,164,66,116,25,147,48,105) );

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),(148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168)], [(1,163,67,115,26,146,49,104),(2,162,68,114,27,145,50,103),(3,161,69,113,28,144,51,102),(4,160,70,112,29,143,52,101),(5,159,71,111,30,142,53,100),(6,158,72,110,31,141,54,99),(7,157,73,109,32,140,55,98),(8,156,74,108,33,139,56,97),(9,155,75,107,34,138,57,96),(10,154,76,106,35,137,58,95),(11,153,77,126,36,136,59,94),(12,152,78,125,37,135,60,93),(13,151,79,124,38,134,61,92),(14,150,80,123,39,133,62,91),(15,149,81,122,40,132,63,90),(16,148,82,121,41,131,43,89),(17,168,83,120,42,130,44,88),(18,167,84,119,22,129,45,87),(19,166,64,118,23,128,46,86),(20,165,65,117,24,127,47,85),(21,164,66,116,25,147,48,105)])

C21⋊C8 is a maximal subgroup of
D7×C3⋊C8  S3×C7⋊C8  C28.32D6  D6.Dic7  C21⋊D8  C28.D6  C42.D4  C21⋊Q16  C8×D21  C56⋊S3  C84.C4  D4⋊D21  D4.D21  Q82D21  C217Q16
C21⋊C8 is a maximal quotient of
C21⋊C16

48 conjugacy classes

class 1  2  3 4A4B 6 7A7B7C8A8B8C8D12A12B14A14B14C21A···21F28A···28F42A···42F84A···84L
order1234467778888121214141421···2128···2842···4284···84
size11211222221212121222222···22···22···22···2

48 irreducible representations

dim1111222222222
type+++-+-+-
imageC1C2C4C8S3Dic3D7C3⋊C8Dic7D21C7⋊C8Dic21C21⋊C8
kernelC21⋊C8C84C42C21C28C14C12C7C6C4C3C2C1
# reps11241132366612

Matrix representation of C21⋊C8 in GL2(𝔽41) generated by

4020
2130
,
2734
014
G:=sub<GL(2,GF(41))| [40,21,20,30],[27,0,34,14] >;

C21⋊C8 in GAP, Magma, Sage, TeX

C_{21}\rtimes C_8
% in TeX

G:=Group("C21:C8");
// GroupNames label

G:=SmallGroup(168,5);
// by ID

G=gap.SmallGroup(168,5);
# by ID

G:=PCGroup([5,-2,-2,-2,-3,-7,10,26,323,3604]);
// Polycyclic

G:=Group<a,b|a^21=b^8=1,b*a*b^-1=a^-1>;
// generators/relations

Export

Subgroup lattice of C21⋊C8 in TeX

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