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## G = C32×D12order 216 = 23·33

### Direct product of C32 and D12

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C32×D12
 Chief series C1 — C3 — C6 — C3×C6 — C32×C6 — S3×C3×C6 — C32×D12
 Lower central C3 — C6 — C32×D12
 Upper central C1 — C3×C6 — C3×C12

Generators and relations for C32×D12
G = < a,b,c,d | a3=b3=c12=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 252 in 120 conjugacy classes, 54 normal (14 characteristic)
C1, C2, C2, C3, C3, C3, C4, C22, S3, C6, C6, C6, D4, C32, C32, C32, C12, C12, C12, D6, C2×C6, C3×S3, C3×C6, C3×C6, C3×C6, D12, C3×D4, C33, C3×C12, C3×C12, C3×C12, S3×C6, C62, S3×C32, C32×C6, C3×D12, D4×C32, C32×C12, S3×C3×C6, C32×D12
Quotients: C1, C2, C3, C22, S3, C6, D4, C32, D6, C2×C6, C3×S3, C3×C6, D12, C3×D4, S3×C6, C62, S3×C32, C3×D12, D4×C32, S3×C3×C6, C32×D12

Smallest permutation representation of C32×D12
On 72 points
Generators in S72
(1 5 9)(2 6 10)(3 7 11)(4 8 12)(13 17 21)(14 18 22)(15 19 23)(16 20 24)(25 33 29)(26 34 30)(27 35 31)(28 36 32)(37 45 41)(38 46 42)(39 47 43)(40 48 44)(49 57 53)(50 58 54)(51 59 55)(52 60 56)(61 65 69)(62 66 70)(63 67 71)(64 68 72)
(1 71 19)(2 72 20)(3 61 21)(4 62 22)(5 63 23)(6 64 24)(7 65 13)(8 66 14)(9 67 15)(10 68 16)(11 69 17)(12 70 18)(25 53 48)(26 54 37)(27 55 38)(28 56 39)(29 57 40)(30 58 41)(31 59 42)(32 60 43)(33 49 44)(34 50 45)(35 51 46)(36 52 47)
(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)
(1 52)(2 51)(3 50)(4 49)(5 60)(6 59)(7 58)(8 57)(9 56)(10 55)(11 54)(12 53)(13 30)(14 29)(15 28)(16 27)(17 26)(18 25)(19 36)(20 35)(21 34)(22 33)(23 32)(24 31)(37 69)(38 68)(39 67)(40 66)(41 65)(42 64)(43 63)(44 62)(45 61)(46 72)(47 71)(48 70)

G:=sub<Sym(72)| (1,5,9)(2,6,10)(3,7,11)(4,8,12)(13,17,21)(14,18,22)(15,19,23)(16,20,24)(25,33,29)(26,34,30)(27,35,31)(28,36,32)(37,45,41)(38,46,42)(39,47,43)(40,48,44)(49,57,53)(50,58,54)(51,59,55)(52,60,56)(61,65,69)(62,66,70)(63,67,71)(64,68,72), (1,71,19)(2,72,20)(3,61,21)(4,62,22)(5,63,23)(6,64,24)(7,65,13)(8,66,14)(9,67,15)(10,68,16)(11,69,17)(12,70,18)(25,53,48)(26,54,37)(27,55,38)(28,56,39)(29,57,40)(30,58,41)(31,59,42)(32,60,43)(33,49,44)(34,50,45)(35,51,46)(36,52,47), (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), (1,52)(2,51)(3,50)(4,49)(5,60)(6,59)(7,58)(8,57)(9,56)(10,55)(11,54)(12,53)(13,30)(14,29)(15,28)(16,27)(17,26)(18,25)(19,36)(20,35)(21,34)(22,33)(23,32)(24,31)(37,69)(38,68)(39,67)(40,66)(41,65)(42,64)(43,63)(44,62)(45,61)(46,72)(47,71)(48,70)>;

G:=Group( (1,5,9)(2,6,10)(3,7,11)(4,8,12)(13,17,21)(14,18,22)(15,19,23)(16,20,24)(25,33,29)(26,34,30)(27,35,31)(28,36,32)(37,45,41)(38,46,42)(39,47,43)(40,48,44)(49,57,53)(50,58,54)(51,59,55)(52,60,56)(61,65,69)(62,66,70)(63,67,71)(64,68,72), (1,71,19)(2,72,20)(3,61,21)(4,62,22)(5,63,23)(6,64,24)(7,65,13)(8,66,14)(9,67,15)(10,68,16)(11,69,17)(12,70,18)(25,53,48)(26,54,37)(27,55,38)(28,56,39)(29,57,40)(30,58,41)(31,59,42)(32,60,43)(33,49,44)(34,50,45)(35,51,46)(36,52,47), (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), (1,52)(2,51)(3,50)(4,49)(5,60)(6,59)(7,58)(8,57)(9,56)(10,55)(11,54)(12,53)(13,30)(14,29)(15,28)(16,27)(17,26)(18,25)(19,36)(20,35)(21,34)(22,33)(23,32)(24,31)(37,69)(38,68)(39,67)(40,66)(41,65)(42,64)(43,63)(44,62)(45,61)(46,72)(47,71)(48,70) );

G=PermutationGroup([[(1,5,9),(2,6,10),(3,7,11),(4,8,12),(13,17,21),(14,18,22),(15,19,23),(16,20,24),(25,33,29),(26,34,30),(27,35,31),(28,36,32),(37,45,41),(38,46,42),(39,47,43),(40,48,44),(49,57,53),(50,58,54),(51,59,55),(52,60,56),(61,65,69),(62,66,70),(63,67,71),(64,68,72)], [(1,71,19),(2,72,20),(3,61,21),(4,62,22),(5,63,23),(6,64,24),(7,65,13),(8,66,14),(9,67,15),(10,68,16),(11,69,17),(12,70,18),(25,53,48),(26,54,37),(27,55,38),(28,56,39),(29,57,40),(30,58,41),(31,59,42),(32,60,43),(33,49,44),(34,50,45),(35,51,46),(36,52,47)], [(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)], [(1,52),(2,51),(3,50),(4,49),(5,60),(6,59),(7,58),(8,57),(9,56),(10,55),(11,54),(12,53),(13,30),(14,29),(15,28),(16,27),(17,26),(18,25),(19,36),(20,35),(21,34),(22,33),(23,32),(24,31),(37,69),(38,68),(39,67),(40,66),(41,65),(42,64),(43,63),(44,62),(45,61),(46,72),(47,71),(48,70)]])

C32×D12 is a maximal subgroup of
C336D8  C337D8  C3312SD16  C3314SD16  (C3×D12)⋊S3  D12⋊(C3⋊S3)  C12⋊S32  S3×D4×C32

81 conjugacy classes

 class 1 2A 2B 2C 3A ··· 3H 3I ··· 3Q 4 6A ··· 6H 6I ··· 6Q 6R ··· 6AG 12A ··· 12Z order 1 2 2 2 3 ··· 3 3 ··· 3 4 6 ··· 6 6 ··· 6 6 ··· 6 12 ··· 12 size 1 1 6 6 1 ··· 1 2 ··· 2 2 1 ··· 1 2 ··· 2 6 ··· 6 2 ··· 2

81 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + + + + + image C1 C2 C2 C3 C6 C6 S3 D4 D6 C3×S3 D12 C3×D4 S3×C6 C3×D12 kernel C32×D12 C32×C12 S3×C3×C6 C3×D12 C3×C12 S3×C6 C3×C12 C33 C3×C6 C12 C32 C32 C6 C3 # reps 1 1 2 8 8 16 1 1 1 8 2 8 8 16

Matrix representation of C32×D12 in GL3(𝔽13) generated by

 9 0 0 0 3 0 0 0 3
,
 1 0 0 0 9 0 0 0 9
,
 12 0 0 0 11 0 0 0 6
,
 12 0 0 0 0 6 0 11 0
G:=sub<GL(3,GF(13))| [9,0,0,0,3,0,0,0,3],[1,0,0,0,9,0,0,0,9],[12,0,0,0,11,0,0,0,6],[12,0,0,0,0,11,0,6,0] >;

C32×D12 in GAP, Magma, Sage, TeX

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

G:=Group("C3^2xD12");
// GroupNames label

G:=SmallGroup(216,137);
// by ID

G=gap.SmallGroup(216,137);
# by ID

G:=PCGroup([6,-2,-2,-3,-3,-2,-3,457,223,5189]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^3=c^12=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

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