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G = C3×C62order 108 = 22·33

Abelian group of type [3,6,6]

direct product, abelian, monomial

Aliases: C3×C62, SmallGroup(108,45)

Series: Derived Chief Lower central Upper central

C1 — C3×C62
C1C3C32C33C32×C6 — C3×C62
C1 — C3×C62
C1 — C3×C62

Generators and relations for C3×C62
 G = < a,b,c | a3=b6=c6=1, ab=ba, ac=ca, bc=cb >

Subgroups: 140, all normal (4 characteristic)
C1, C2 [×3], C3 [×13], C22, C6 [×39], C32 [×13], C2×C6 [×13], C3×C6 [×39], C33, C62 [×13], C32×C6 [×3], C3×C62
Quotients: C1, C2 [×3], C3 [×13], C22, C6 [×39], C32 [×13], C2×C6 [×13], C3×C6 [×39], C33, C62 [×13], C32×C6 [×3], C3×C62

Smallest permutation representation of C3×C62
Regular action on 108 points
Generators in S108
(1 13 74)(2 14 75)(3 15 76)(4 16 77)(5 17 78)(6 18 73)(7 88 26)(8 89 27)(9 90 28)(10 85 29)(11 86 30)(12 87 25)(19 68 32)(20 69 33)(21 70 34)(22 71 35)(23 72 36)(24 67 31)(37 47 81)(38 48 82)(39 43 83)(40 44 84)(41 45 79)(42 46 80)(49 99 96)(50 100 91)(51 101 92)(52 102 93)(53 97 94)(54 98 95)(55 62 105)(56 63 106)(57 64 107)(58 65 108)(59 66 103)(60 61 104)
(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)
(1 89 94 33 39 103)(2 90 95 34 40 104)(3 85 96 35 41 105)(4 86 91 36 42 106)(5 87 92 31 37 107)(6 88 93 32 38 108)(7 102 68 82 65 73)(8 97 69 83 66 74)(9 98 70 84 61 75)(10 99 71 79 62 76)(11 100 72 80 63 77)(12 101 67 81 64 78)(13 27 53 20 43 59)(14 28 54 21 44 60)(15 29 49 22 45 55)(16 30 50 23 46 56)(17 25 51 24 47 57)(18 26 52 19 48 58)

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

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

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

C3×C62 is a maximal subgroup of   C3315D4  C62⋊C9  C332A4

108 conjugacy classes

class 1 2A2B2C3A···3Z6A···6BZ
order12223···36···6
size11111···11···1

108 irreducible representations

dim1111
type++
imageC1C2C3C6
kernelC3×C62C32×C6C62C3×C6
# reps132678

Matrix representation of C3×C62 in GL3(𝔽7) generated by

400
040
004
,
100
020
006
,
100
030
005
G:=sub<GL(3,GF(7))| [4,0,0,0,4,0,0,0,4],[1,0,0,0,2,0,0,0,6],[1,0,0,0,3,0,0,0,5] >;

C3×C62 in GAP, Magma, Sage, TeX

C_3\times C_6^2
% in TeX

G:=Group("C3xC6^2");
// GroupNames label

G:=SmallGroup(108,45);
// by ID

G=gap.SmallGroup(108,45);
# by ID

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

G:=Group<a,b,c|a^3=b^6=c^6=1,a*b=b*a,a*c=c*a,b*c=c*b>;
// generators/relations

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