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G = Q8×C21order 168 = 23·3·7

Direct product of C21 and Q8

direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary

Aliases: Q8×C21, C4.C42, C84.7C2, C28.7C6, C12.3C14, C42.24C22, C2.2(C2×C42), C6.7(C2×C14), C14.15(C2×C6), SmallGroup(168,41)

Series: Derived Chief Lower central Upper central

C1C2 — Q8×C21
C1C2C14C42C84 — Q8×C21
C1C2 — Q8×C21
C1C42 — Q8×C21

Generators and relations for Q8×C21
 G = < a,b,c | a21=b4=1, c2=b2, ab=ba, ac=ca, cbc-1=b-1 >


Smallest permutation representation of Q8×C21
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 63 36 109)(2 43 37 110)(3 44 38 111)(4 45 39 112)(5 46 40 113)(6 47 41 114)(7 48 42 115)(8 49 22 116)(9 50 23 117)(10 51 24 118)(11 52 25 119)(12 53 26 120)(13 54 27 121)(14 55 28 122)(15 56 29 123)(16 57 30 124)(17 58 31 125)(18 59 32 126)(19 60 33 106)(20 61 34 107)(21 62 35 108)(64 99 158 139)(65 100 159 140)(66 101 160 141)(67 102 161 142)(68 103 162 143)(69 104 163 144)(70 105 164 145)(71 85 165 146)(72 86 166 147)(73 87 167 127)(74 88 168 128)(75 89 148 129)(76 90 149 130)(77 91 150 131)(78 92 151 132)(79 93 152 133)(80 94 153 134)(81 95 154 135)(82 96 155 136)(83 97 156 137)(84 98 157 138)
(1 146 36 85)(2 147 37 86)(3 127 38 87)(4 128 39 88)(5 129 40 89)(6 130 41 90)(7 131 42 91)(8 132 22 92)(9 133 23 93)(10 134 24 94)(11 135 25 95)(12 136 26 96)(13 137 27 97)(14 138 28 98)(15 139 29 99)(16 140 30 100)(17 141 31 101)(18 142 32 102)(19 143 33 103)(20 144 34 104)(21 145 35 105)(43 166 110 72)(44 167 111 73)(45 168 112 74)(46 148 113 75)(47 149 114 76)(48 150 115 77)(49 151 116 78)(50 152 117 79)(51 153 118 80)(52 154 119 81)(53 155 120 82)(54 156 121 83)(55 157 122 84)(56 158 123 64)(57 159 124 65)(58 160 125 66)(59 161 126 67)(60 162 106 68)(61 163 107 69)(62 164 108 70)(63 165 109 71)

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

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

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

Q8×C21 is a maximal subgroup of   Q82D21  C217Q16  Q83D21

105 conjugacy classes

class 1  2 3A3B4A4B4C6A6B7A···7F12A···12F14A···14F21A···21L28A···28R42A···42L84A···84AJ
order1233444667···712···1214···1421···2128···2842···4284···84
size1111222111···12···21···11···12···21···12···2

105 irreducible representations

dim111111112222
type++-
imageC1C2C3C6C7C14C21C42Q8C3×Q8C7×Q8Q8×C21
kernelQ8×C21C84C7×Q8C28C3×Q8C12Q8C4C21C7C3C1
# reps1326618123612612

Matrix representation of Q8×C21 in GL2(𝔽43) generated by

90
09
,
929
1234
,
036
370
G:=sub<GL(2,GF(43))| [9,0,0,9],[9,12,29,34],[0,37,36,0] >;

Q8×C21 in GAP, Magma, Sage, TeX

Q_8\times C_{21}
% in TeX

G:=Group("Q8xC21");
// GroupNames label

G:=SmallGroup(168,41);
// by ID

G=gap.SmallGroup(168,41);
# by ID

G:=PCGroup([5,-2,-2,-3,-7,-2,420,861,426]);
// Polycyclic

G:=Group<a,b,c|a^21=b^4=1,c^2=b^2,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

Export

Subgroup lattice of Q8×C21 in TeX

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