Copied to
clipboard

G = C6×Dic6order 144 = 24·32

Direct product of C6 and Dic6

direct product, metabelian, supersoluble, monomial

Aliases: C6×Dic6, C12.63D6, C62.26C22, C6⋊(C3×Q8), (C3×C6)⋊3Q8, C31(C6×Q8), C325(C2×Q8), C4.11(S3×C6), (C2×C12).5C6, (C6×C12).9C2, (C2×C6).46D6, (C2×C12).20S3, C12.11(C2×C6), C22.8(S3×C6), C6.1(C22×C6), C6.40(C22×S3), (C3×C6).19C23, (C6×Dic3).7C2, (C2×Dic3).3C6, Dic3.1(C2×C6), (C3×C12).40C22, (C3×Dic3).10C22, C2.3(S3×C2×C6), (C2×C4).4(C3×S3), (C2×C6).11(C2×C6), SmallGroup(144,158)

Series: Derived Chief Lower central Upper central

C1C6 — C6×Dic6
C1C3C6C3×C6C3×Dic3C6×Dic3 — C6×Dic6
C3C6 — C6×Dic6
C1C2×C6C2×C12

Generators and relations for C6×Dic6
 G = < a,b,c | a6=b12=1, c2=b6, ab=ba, ac=ca, cbc-1=b-1 >

Subgroups: 136 in 84 conjugacy classes, 54 normal (18 characteristic)
C1, C2, C2 [×2], C3 [×2], C3, C4 [×2], C4 [×4], C22, C6 [×2], C6 [×4], C6 [×3], C2×C4, C2×C4 [×2], Q8 [×4], C32, Dic3 [×4], C12 [×4], C12 [×6], C2×C6 [×2], C2×C6, C2×Q8, C3×C6, C3×C6 [×2], Dic6 [×4], C2×Dic3 [×2], C2×C12 [×2], C2×C12 [×3], C3×Q8 [×4], C3×Dic3 [×4], C3×C12 [×2], C62, C2×Dic6, C6×Q8, C3×Dic6 [×4], C6×Dic3 [×2], C6×C12, C6×Dic6
Quotients: C1, C2 [×7], C3, C22 [×7], S3, C6 [×7], Q8 [×2], C23, D6 [×3], C2×C6 [×7], C2×Q8, C3×S3, Dic6 [×2], C3×Q8 [×2], C22×S3, C22×C6, S3×C6 [×3], C2×Dic6, C6×Q8, C3×Dic6 [×2], S3×C2×C6, C6×Dic6

Smallest permutation representation of C6×Dic6
On 48 points
Generators in S48
(1 33 9 29 5 25)(2 34 10 30 6 26)(3 35 11 31 7 27)(4 36 12 32 8 28)(13 39 17 43 21 47)(14 40 18 44 22 48)(15 41 19 45 23 37)(16 42 20 46 24 38)
(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)
(1 19 7 13)(2 18 8 24)(3 17 9 23)(4 16 10 22)(5 15 11 21)(6 14 12 20)(25 41 31 47)(26 40 32 46)(27 39 33 45)(28 38 34 44)(29 37 35 43)(30 48 36 42)

G:=sub<Sym(48)| (1,33,9,29,5,25)(2,34,10,30,6,26)(3,35,11,31,7,27)(4,36,12,32,8,28)(13,39,17,43,21,47)(14,40,18,44,22,48)(15,41,19,45,23,37)(16,42,20,46,24,38), (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), (1,19,7,13)(2,18,8,24)(3,17,9,23)(4,16,10,22)(5,15,11,21)(6,14,12,20)(25,41,31,47)(26,40,32,46)(27,39,33,45)(28,38,34,44)(29,37,35,43)(30,48,36,42)>;

G:=Group( (1,33,9,29,5,25)(2,34,10,30,6,26)(3,35,11,31,7,27)(4,36,12,32,8,28)(13,39,17,43,21,47)(14,40,18,44,22,48)(15,41,19,45,23,37)(16,42,20,46,24,38), (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), (1,19,7,13)(2,18,8,24)(3,17,9,23)(4,16,10,22)(5,15,11,21)(6,14,12,20)(25,41,31,47)(26,40,32,46)(27,39,33,45)(28,38,34,44)(29,37,35,43)(30,48,36,42) );

G=PermutationGroup([(1,33,9,29,5,25),(2,34,10,30,6,26),(3,35,11,31,7,27),(4,36,12,32,8,28),(13,39,17,43,21,47),(14,40,18,44,22,48),(15,41,19,45,23,37),(16,42,20,46,24,38)], [(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)], [(1,19,7,13),(2,18,8,24),(3,17,9,23),(4,16,10,22),(5,15,11,21),(6,14,12,20),(25,41,31,47),(26,40,32,46),(27,39,33,45),(28,38,34,44),(29,37,35,43),(30,48,36,42)])

C6×Dic6 is a maximal subgroup of
C12.14D12  Dic6⋊Dic3  C6.Dic12  D12.32D6  Dic6.29D6  C62.9C23  C62.13C23  D66Dic6  C62.33C23  C12.28D12  Dic3⋊Dic6  C12.30D12  C62.43C23  D61Dic6  C62.58C23  C62.77C23  D12.33D6  S3×C6×Q8

54 conjugacy classes

class 1 2A2B2C3A3B3C3D3E4A4B4C4D4E4F6A···6F6G···6O12A···12P12Q···12X
order1222333334444446···66···612···1212···12
size1111112222266661···12···22···26···6

54 irreducible representations

dim111111112222222222
type+++++-++-
imageC1C2C2C2C3C6C6C6S3Q8D6D6C3×S3Dic6C3×Q8S3×C6S3×C6C3×Dic6
kernelC6×Dic6C3×Dic6C6×Dic3C6×C12C2×Dic6Dic6C2×Dic3C2×C12C2×C12C3×C6C12C2×C6C2×C4C6C6C4C22C2
# reps142128421221244428

Matrix representation of C6×Dic6 in GL4(𝔽13) generated by

10000
01000
0040
0004
,
10000
0400
0070
0002
,
0100
1000
0001
00120
G:=sub<GL(4,GF(13))| [10,0,0,0,0,10,0,0,0,0,4,0,0,0,0,4],[10,0,0,0,0,4,0,0,0,0,7,0,0,0,0,2],[0,1,0,0,1,0,0,0,0,0,0,12,0,0,1,0] >;

C6×Dic6 in GAP, Magma, Sage, TeX

C_6\times {\rm Dic}_6
% in TeX

G:=Group("C6xDic6");
// GroupNames label

G:=SmallGroup(144,158);
// by ID

G=gap.SmallGroup(144,158);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-2,-3,144,506,122,3461]);
// Polycyclic

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

׿
×
𝔽