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## G = C32×C12order 108 = 22·33

### Abelian group of type [3,3,12]

Aliases: C32×C12, SmallGroup(108,35)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32×C12
 Chief series C1 — C2 — C6 — C3×C6 — C32×C6 — C32×C12
 Lower central C1 — C32×C12
 Upper central C1 — C32×C12

Generators and relations for C32×C12
G = < a,b,c | a3=b3=c12=1, ab=ba, ac=ca, bc=cb >

Subgroups: 84, all normal (6 characteristic)
C1, C2, C3 [×13], C4, C6 [×13], C32 [×13], C12 [×13], C3×C6 [×13], C33, C3×C12 [×13], C32×C6, C32×C12
Quotients: C1, C2, C3 [×13], C4, C6 [×13], C32 [×13], C12 [×13], C3×C6 [×13], C33, C3×C12 [×13], C32×C6, C32×C12

Smallest permutation representation of C32×C12
Regular action on 108 points
Generators in S108
(1 107 83)(2 108 84)(3 97 73)(4 98 74)(5 99 75)(6 100 76)(7 101 77)(8 102 78)(9 103 79)(10 104 80)(11 105 81)(12 106 82)(13 36 89)(14 25 90)(15 26 91)(16 27 92)(17 28 93)(18 29 94)(19 30 95)(20 31 96)(21 32 85)(22 33 86)(23 34 87)(24 35 88)(37 57 72)(38 58 61)(39 59 62)(40 60 63)(41 49 64)(42 50 65)(43 51 66)(44 52 67)(45 53 68)(46 54 69)(47 55 70)(48 56 71)
(1 39 19)(2 40 20)(3 41 21)(4 42 22)(5 43 23)(6 44 24)(7 45 13)(8 46 14)(9 47 15)(10 48 16)(11 37 17)(12 38 18)(25 102 54)(26 103 55)(27 104 56)(28 105 57)(29 106 58)(30 107 59)(31 108 60)(32 97 49)(33 98 50)(34 99 51)(35 100 52)(36 101 53)(61 94 82)(62 95 83)(63 96 84)(64 85 73)(65 86 74)(66 87 75)(67 88 76)(68 89 77)(69 90 78)(70 91 79)(71 92 80)(72 93 81)
(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)

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

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

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

C32×C12 is a maximal subgroup of   C337C8  C338Q8  C3312D4

108 conjugacy classes

 class 1 2 3A ··· 3Z 4A 4B 6A ··· 6Z 12A ··· 12AZ order 1 2 3 ··· 3 4 4 6 ··· 6 12 ··· 12 size 1 1 1 ··· 1 1 1 1 ··· 1 1 ··· 1

108 irreducible representations

 dim 1 1 1 1 1 1 type + + image C1 C2 C3 C4 C6 C12 kernel C32×C12 C32×C6 C3×C12 C33 C3×C6 C32 # reps 1 1 26 2 26 52

Matrix representation of C32×C12 in GL3(𝔽13) generated by

 1 0 0 0 9 0 0 0 1
,
 9 0 0 0 9 0 0 0 1
,
 11 0 0 0 2 0 0 0 2
G:=sub<GL(3,GF(13))| [1,0,0,0,9,0,0,0,1],[9,0,0,0,9,0,0,0,1],[11,0,0,0,2,0,0,0,2] >;

C32×C12 in GAP, Magma, Sage, TeX

C_3^2\times C_{12}
% in TeX

G:=Group("C3^2xC12");
// GroupNames label

G:=SmallGroup(108,35);
// by ID

G=gap.SmallGroup(108,35);
# by ID

G:=PCGroup([5,-2,-3,-3,-3,-2,270]);
// Polycyclic

G:=Group<a,b,c|a^3=b^3=c^12=1,a*b=b*a,a*c=c*a,b*c=c*b>;
// generators/relations

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