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G = C7×C21order 147 = 3·72

Abelian group of type [7,21]

direct product, abelian, monomial, 7-elementary

Aliases: C7×C21, SmallGroup(147,6)

Series: Derived Chief Lower central Upper central

C1 — C7×C21
C1C7C72 — C7×C21
C1 — C7×C21
C1 — C7×C21

Generators and relations for C7×C21
 G = < a,b | a7=b21=1, ab=ba >


Smallest permutation representation of C7×C21
Regular action on 147 points
Generators in S147
(1 104 73 30 60 122 138)(2 105 74 31 61 123 139)(3 85 75 32 62 124 140)(4 86 76 33 63 125 141)(5 87 77 34 43 126 142)(6 88 78 35 44 106 143)(7 89 79 36 45 107 144)(8 90 80 37 46 108 145)(9 91 81 38 47 109 146)(10 92 82 39 48 110 147)(11 93 83 40 49 111 127)(12 94 84 41 50 112 128)(13 95 64 42 51 113 129)(14 96 65 22 52 114 130)(15 97 66 23 53 115 131)(16 98 67 24 54 116 132)(17 99 68 25 55 117 133)(18 100 69 26 56 118 134)(19 101 70 27 57 119 135)(20 102 71 28 58 120 136)(21 103 72 29 59 121 137)
(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)

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

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

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

C7×C21 is a maximal subgroup of   C7⋊D21  C72⋊C9  C723C9

147 conjugacy classes

class 1 3A3B7A···7AV21A···21CR
order1337···721···21
size1111···11···1

147 irreducible representations

dim1111
type+
imageC1C3C7C21
kernelC7×C21C72C21C7
# reps124896

Matrix representation of C7×C21 in GL2(𝔽43) generated by

350
016
,
130
021
G:=sub<GL(2,GF(43))| [35,0,0,16],[13,0,0,21] >;

C7×C21 in GAP, Magma, Sage, TeX

C_7\times C_{21}
% in TeX

G:=Group("C7xC21");
// GroupNames label

G:=SmallGroup(147,6);
// by ID

G=gap.SmallGroup(147,6);
# by ID

G:=PCGroup([3,-3,-7,-7]);
// Polycyclic

G:=Group<a,b|a^7=b^21=1,a*b=b*a>;
// generators/relations

Export

Subgroup lattice of C7×C21 in TeX

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