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G = C2×C62order 124 = 22·31

Abelian group of type [2,62]

direct product, abelian, monomial, 2-elementary

Aliases: C2×C62, SmallGroup(124,4)

Series: Derived Chief Lower central Upper central

C1 — C2×C62
C1C31C62 — C2×C62
C1 — C2×C62
C1 — C2×C62

Generators and relations for C2×C62
 G = < a,b | a2=b62=1, ab=ba >


Smallest permutation representation of C2×C62
Regular action on 124 points
Generators in S124
(1 112)(2 113)(3 114)(4 115)(5 116)(6 117)(7 118)(8 119)(9 120)(10 121)(11 122)(12 123)(13 124)(14 63)(15 64)(16 65)(17 66)(18 67)(19 68)(20 69)(21 70)(22 71)(23 72)(24 73)(25 74)(26 75)(27 76)(28 77)(29 78)(30 79)(31 80)(32 81)(33 82)(34 83)(35 84)(36 85)(37 86)(38 87)(39 88)(40 89)(41 90)(42 91)(43 92)(44 93)(45 94)(46 95)(47 96)(48 97)(49 98)(50 99)(51 100)(52 101)(53 102)(54 103)(55 104)(56 105)(57 106)(58 107)(59 108)(60 109)(61 110)(62 111)
(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)

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

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

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

C2×C62 is a maximal subgroup of   C31⋊D4  C31⋊A4

124 conjugacy classes

class 1 2A2B2C31A···31AD62A···62CL
order122231···3162···62
size11111···11···1

124 irreducible representations

dim1111
type++
imageC1C2C31C62
kernelC2×C62C62C22C2
# reps133090

Matrix representation of C2×C62 in GL2(𝔽311) generated by

3100
0310
,
2930
0146
G:=sub<GL(2,GF(311))| [310,0,0,310],[293,0,0,146] >;

C2×C62 in GAP, Magma, Sage, TeX

C_2\times C_{62}
% in TeX

G:=Group("C2xC62");
// GroupNames label

G:=SmallGroup(124,4);
// by ID

G=gap.SmallGroup(124,4);
# by ID

G:=PCGroup([3,-2,-2,-31]);
// Polycyclic

G:=Group<a,b|a^2=b^62=1,a*b=b*a>;
// generators/relations

Export

Subgroup lattice of C2×C62 in TeX

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