C21:C8 - GroupNames

(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)

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

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

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

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

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 Q₈⋊₂D₂₁ C₂₁⋊₇Q₁₆
C₂₁⋊C₈ is a maximal quotient of
C₂₁⋊C₁₆

48 conjugacy classes

class	1	2	3	4A	4B	6	7A	7B	7C	8A	8B	8C	8D	12A	12B	14A	14B	14C	21A	···	21F	28A	···	28F	42A	···	42F	84A	···	84L
order	1	2	3	4	4	6	7	7	7	8	8	8	8	12	12	14	14	14	21	···	21	28	···	28	42	···	42	84	···	84
size	1	1	2	1	1	2	2	2	2	21	21	21	21	2	2	2	2	2	2	···	2	2	···	2	2	···	2	2	···	2

48 irreducible representations

dim	1	1	1	1	2	2	2	2	2	2	2	2	2
type	+	+			+	-	+		-	+		-
image	C₁	C₂	C₄	C₈	S₃	Dic₃	D₇	C₃⋊C₈	Dic₇	D₂₁	C₇⋊C₈	Dic₂₁	C₂₁⋊C₈
kernel	C₂₁⋊C₈	C₈₄	C₄₂	C₂₁	C₂₈	C₁₄	C₁₂	C₇	C₆	C₄	C₃	C₂	C₁
# reps	1	1	2	4	1	1	3	2	3	6	6	6	12

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

40	20
21	30

27	34
0	14

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

G = C21⋊C8 order 168 = 23·3·7

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

G = C₂₁⋊C₈ order 168 = 2³·3·7

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