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## G = C32×Dic6order 216 = 23·33

### Direct product of C32 and Dic6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C32×Dic6
 Chief series C1 — C3 — C6 — C3×C6 — C32×C6 — C32×Dic3 — C32×Dic6
 Lower central C3 — C6 — C32×Dic6
 Upper central C1 — C3×C6 — C3×C12

Generators and relations for C32×Dic6
G = < a,b,c,d | a3=b3=c12=1, d2=c6, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 156 in 96 conjugacy classes, 54 normal (14 characteristic)
C1, C2, C3, C3, C3, C4, C4, C6, C6, C6, Q8, C32, C32, C32, Dic3, C12, C12, C12, C3×C6, C3×C6, C3×C6, Dic6, C3×Q8, C33, C3×Dic3, C3×C12, C3×C12, C3×C12, C32×C6, C3×Dic6, Q8×C32, C32×Dic3, C32×C12, C32×Dic6
Quotients: C1, C2, C3, C22, S3, C6, Q8, C32, D6, C2×C6, C3×S3, C3×C6, Dic6, C3×Q8, S3×C6, C62, S3×C32, C3×Dic6, Q8×C32, S3×C3×C6, C32×Dic6

Smallest permutation representation of C32×Dic6
On 72 points
Generators in S72
(1 9 5)(2 10 6)(3 11 7)(4 12 8)(13 21 17)(14 22 18)(15 23 19)(16 24 20)(25 29 33)(26 30 34)(27 31 35)(28 32 36)(37 41 45)(38 42 46)(39 43 47)(40 44 48)(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 53 20)(2 54 21)(3 55 22)(4 56 23)(5 57 24)(6 58 13)(7 59 14)(8 60 15)(9 49 16)(10 50 17)(11 51 18)(12 52 19)(25 38 71)(26 39 72)(27 40 61)(28 41 62)(29 42 63)(30 43 64)(31 44 65)(32 45 66)(33 46 67)(34 47 68)(35 48 69)(36 37 70)
(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 30 7 36)(2 29 8 35)(3 28 9 34)(4 27 10 33)(5 26 11 32)(6 25 12 31)(13 71 19 65)(14 70 20 64)(15 69 21 63)(16 68 22 62)(17 67 23 61)(18 66 24 72)(37 53 43 59)(38 52 44 58)(39 51 45 57)(40 50 46 56)(41 49 47 55)(42 60 48 54)

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

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

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

C32×Dic6 is a maximal subgroup of
C3313SD16  C3315SD16  C336Q16  C337Q16  C12.39S32  C12.40S32  C329(S3×Q8)  S3×Q8×C32

81 conjugacy classes

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

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 Q8 D6 C3×S3 Dic6 C3×Q8 S3×C6 C3×Dic6 kernel C32×Dic6 C32×Dic3 C32×C12 C3×Dic6 C3×Dic3 C3×C12 C3×C12 C33 C3×C6 C12 C32 C32 C6 C3 # reps 1 2 1 8 16 8 1 1 1 8 2 8 8 16

Matrix representation of C32×Dic6 in GL6(𝔽13)

 9 0 0 0 0 0 0 9 0 0 0 0 0 0 3 0 0 0 0 0 0 3 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 9 0 0 0 0 0 0 9 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 10 2 0 0 0 0 0 4 0 0 0 0 0 0 4 0 0 0 0 0 9 10 0 0 0 0 0 0 0 1 0 0 0 0 12 0
,
 6 7 0 0 0 0 8 7 0 0 0 0 0 0 12 8 0 0 0 0 0 1 0 0 0 0 0 0 3 4 0 0 0 0 4 10

G:=sub<GL(6,GF(13))| [9,0,0,0,0,0,0,9,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[10,0,0,0,0,0,2,4,0,0,0,0,0,0,4,9,0,0,0,0,0,10,0,0,0,0,0,0,0,12,0,0,0,0,1,0],[6,8,0,0,0,0,7,7,0,0,0,0,0,0,12,0,0,0,0,0,8,1,0,0,0,0,0,0,3,4,0,0,0,0,4,10] >;

C32×Dic6 in GAP, Magma, Sage, TeX

C_3^2\times {\rm Dic}_6
% in TeX

G:=Group("C3^2xDic6");
// GroupNames label

G:=SmallGroup(216,135);
// by ID

G=gap.SmallGroup(216,135);
# by ID

G:=PCGroup([6,-2,-2,-3,-3,-2,-3,216,457,223,5189]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^3=c^12=1,d^2=c^6,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

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