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G = C32×C12order 108 = 22·33

Abelian group of type [3,3,12]

direct product, abelian, monomial, 3-elementary

Aliases: C32×C12, SmallGroup(108,35)

Series: Derived Chief Lower central Upper central

C1 — C32×C12
C1C2C6C3×C6C32×C6 — C32×C12
C1 — C32×C12
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···3Z4A4B6A···6Z12A···12AZ
order123···3446···612···12
size111···1111···11···1

108 irreducible representations

dim111111
type++
imageC1C2C3C4C6C12
kernelC32×C12C32×C6C3×C12C33C3×C6C32
# reps112622652

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

100
090
001
,
900
090
001
,
1100
020
002
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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