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G = C32xDic6order 216 = 23·33

Direct product of C32 and Dic6

direct product, metabelian, supersoluble, monomial

Aliases: C32xDic6, C33:6Q8, C6.1C62, C3:(Q8xC32), C4.(S3xC32), C6.35(S3xC6), C12.1(C3xC6), (C3xC6).67D6, Dic3.(C3xC6), C32:5(C3xQ8), C12.17(C3xS3), (C3xC12).12C6, (C3xC12).21S3, (C32xC12).2C2, (C3xDic3).4C6, (C32xC6).16C22, (C32xDic3).3C2, C2.3(S3xC3xC6), (C3xC6).24(C2xC6), SmallGroup(216,135)

Series: Derived Chief Lower central Upper central

C1C6 — C32xDic6
C1C3C6C3xC6C32xC6C32xDic3 — C32xDic6
C3C6 — C32xDic6
C1C3xC6C3xC12

Generators and relations for C32xDic6
 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, C3xC6, C3xC6, C3xC6, Dic6, C3xQ8, C33, C3xDic3, C3xC12, C3xC12, C3xC12, C32xC6, C3xDic6, Q8xC32, C32xDic3, C32xC12, C32xDic6
Quotients: C1, C2, C3, C22, S3, C6, Q8, C32, D6, C2xC6, C3xS3, C3xC6, Dic6, C3xQ8, S3xC6, C62, S3xC32, C3xDic6, Q8xC32, S3xC3xC6, C32xDic6

Smallest permutation representation of C32xDic6
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)]])

C32xDic6 is a maximal subgroup of
C33:13SD16  C33:15SD16  C33:6Q16  C33:7Q16  C12.39S32  C12.40S32  C32:9(S3xQ8)  S3xQ8xC32

81 conjugacy classes

class 1  2 3A···3H3I···3Q4A4B4C6A···6H6I···6Q12A···12Z12AA···12AP
order123···33···34446···66···612···1212···12
size111···12···22661···12···22···26···6

81 irreducible representations

dim11111122222222
type++++-+-
imageC1C2C2C3C6C6S3Q8D6C3xS3Dic6C3xQ8S3xC6C3xDic6
kernelC32xDic6C32xDic3C32xC12C3xDic6C3xDic3C3xC12C3xC12C33C3xC6C12C32C32C6C3
# reps1218168111828816

Matrix representation of C32xDic6 in GL6(F13)

900000
090000
003000
000300
000010
000001
,
100000
010000
009000
000900
000010
000001
,
1020000
040000
004000
0091000
000001
0000120
,
670000
870000
0012800
000100
000034
0000410

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] >;

C32xDic6 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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