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G = Dic3×C12order 144 = 24·32

Direct product of C12 and Dic3

direct product, metacyclic, supersoluble, monomial, A-group

Aliases: Dic3×C12, C122C12, C323C42, C62.16C22, C3⋊(C4×C12), (C3×C12)⋊5C4, C2.2(S3×C12), C6.23(C4×S3), (C2×C12).8C6, C6.7(C2×C12), (C2×C6).41D6, (C2×C12).22S3, (C6×C12).11C2, C22.3(S3×C6), C2.2(C6×Dic3), (C2×Dic3).4C6, (C6×Dic3).8C2, C6.19(C2×Dic3), (C2×C4).6(C3×S3), (C2×C6).6(C2×C6), (C3×C6).19(C2×C4), SmallGroup(144,76)

Series: Derived Chief Lower central Upper central

C1C3 — Dic3×C12
C1C3C6C2×C6C62C6×Dic3 — Dic3×C12
C3 — Dic3×C12
C1C2×C12

Generators and relations for Dic3×C12
 G = < a,b,c | a12=b6=1, c2=b3, ab=ba, ac=ca, cbc-1=b-1 >

Subgroups: 104 in 68 conjugacy classes, 46 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], C32, Dic3 [×4], C12 [×4], C12 [×6], C2×C6 [×2], C2×C6, C42, C3×C6, C3×C6 [×2], C2×Dic3 [×2], C2×C12 [×2], C2×C12 [×3], C3×Dic3 [×4], C3×C12 [×2], C62, C4×Dic3, C4×C12, C6×Dic3 [×2], C6×C12, Dic3×C12
Quotients: C1, C2 [×3], C3, C4 [×6], C22, S3, C6 [×3], C2×C4 [×3], Dic3 [×2], C12 [×6], D6, C2×C6, C42, C3×S3, C4×S3 [×2], C2×Dic3, C2×C12 [×3], C3×Dic3 [×2], S3×C6, C4×Dic3, C4×C12, S3×C12 [×2], C6×Dic3, Dic3×C12

Smallest permutation representation of Dic3×C12
On 48 points
Generators in S48
(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 17 5 21 9 13)(2 18 6 22 10 14)(3 19 7 23 11 15)(4 20 8 24 12 16)(25 39 33 47 29 43)(26 40 34 48 30 44)(27 41 35 37 31 45)(28 42 36 38 32 46)
(1 40 21 30)(2 41 22 31)(3 42 23 32)(4 43 24 33)(5 44 13 34)(6 45 14 35)(7 46 15 36)(8 47 16 25)(9 48 17 26)(10 37 18 27)(11 38 19 28)(12 39 20 29)

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

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

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

Dic3×C12 is a maximal subgroup of
C3⋊C8⋊Dic3  D122Dic3  C12.80D12  C12.81D12  C62.6C23  Dic35Dic6  Dic36Dic6  Dic3.D12  C62.25C23  C62.29C23  C12.27D12  C12.28D12  Dic3⋊Dic6  C62.37C23  C62.38C23  C62.39C23  C62.44C23  C62.47C23  Dic34D12  Dic35D12  C12⋊D12  C123Dic6  S3×C4×C12

72 conjugacy classes

class 1 2A2B2C3A3B3C3D3E4A4B4C4D4E···4L6A···6F6G···6O12A···12H12I···12T12U···12AJ
order12223333344444···46···66···612···1212···1212···12
size11111122211113···31···12···21···12···23···3

72 irreducible representations

dim111111111122222222
type++++-+
imageC1C2C2C3C4C4C6C6C12C12S3Dic3D6C3×S3C4×S3C3×Dic3S3×C6S3×C12
kernelDic3×C12C6×Dic3C6×C12C4×Dic3C3×Dic3C3×C12C2×Dic3C2×C12Dic3C12C2×C12C12C2×C6C2×C4C6C4C22C2
# reps1212844216812124428

Matrix representation of Dic3×C12 in GL3(𝔽13) generated by

900
020
002
,
1200
092
003
,
800
0120
031
G:=sub<GL(3,GF(13))| [9,0,0,0,2,0,0,0,2],[12,0,0,0,9,0,0,2,3],[8,0,0,0,12,3,0,0,1] >;

Dic3×C12 in GAP, Magma, Sage, TeX

{\rm Dic}_3\times C_{12}
% in TeX

G:=Group("Dic3xC12");
// GroupNames label

G:=SmallGroup(144,76);
// by ID

G=gap.SmallGroup(144,76);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-2,-3,72,151,3461]);
// Polycyclic

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

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