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## G = C3×C62order 108 = 22·33

### Abelian group of type [3,6,6]

Aliases: C3×C62, SmallGroup(108,45)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C62
 Chief series C1 — C3 — C32 — C33 — C32×C6 — C3×C62
 Lower central C1 — C3×C62
 Upper central 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 2A 2B 2C 3A ··· 3Z 6A ··· 6BZ order 1 2 2 2 3 ··· 3 6 ··· 6 size 1 1 1 1 1 ··· 1 1 ··· 1

108 irreducible representations

 dim 1 1 1 1 type + + image C1 C2 C3 C6 kernel C3×C62 C32×C6 C62 C3×C6 # reps 1 3 26 78

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

 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
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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