Copied to
clipboard

G = S3×C2×C12order 144 = 24·32

Direct product of C2×C12 and S3

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

Aliases: S3×C2×C12, C62.27C22, C61(C2×C12), C123(C2×C6), (C2×C12)⋊5C6, (C6×C12)⋊8C2, D6.4(C2×C6), (C2×C6).47D6, C31(C22×C12), (C3×C12)⋊8C22, Dic33(C2×C6), (C2×Dic3)⋊5C6, C22.9(S3×C6), C6.2(C22×C6), C325(C22×C4), (C6×Dic3)⋊11C2, (C22×S3).2C6, (C3×C6).20C23, C6.41(C22×S3), (S3×C6).13C22, (C3×Dic3)⋊10C22, C2.1(S3×C2×C6), (C3×C6)⋊4(C2×C4), (S3×C2×C6).4C2, (C2×C6).12(C2×C6), SmallGroup(144,159)

Series: Derived Chief Lower central Upper central

C1C3 — S3×C2×C12
C1C3C6C3×C6S3×C6S3×C2×C6 — S3×C2×C12
C3 — S3×C2×C12
C1C2×C12

Generators and relations for S3×C2×C12
 G = < a,b,c,d | a2=b12=c3=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 200 in 116 conjugacy classes, 70 normal (22 characteristic)
C1, C2, C2 [×2], C2 [×4], C3 [×2], C3, C4 [×2], C4 [×2], C22, C22 [×6], S3 [×4], C6 [×2], C6 [×4], C6 [×7], C2×C4, C2×C4 [×5], C23, C32, Dic3 [×2], C12 [×4], C12 [×4], D6 [×6], C2×C6 [×2], C2×C6 [×7], C22×C4, C3×S3 [×4], C3×C6, C3×C6 [×2], C4×S3 [×4], C2×Dic3, C2×C12 [×2], C2×C12 [×6], C22×S3, C22×C6, C3×Dic3 [×2], C3×C12 [×2], S3×C6 [×6], C62, S3×C2×C4, C22×C12, S3×C12 [×4], C6×Dic3, C6×C12, S3×C2×C6, S3×C2×C12
Quotients: C1, C2 [×7], C3, C4 [×4], C22 [×7], S3, C6 [×7], C2×C4 [×6], C23, C12 [×4], D6 [×3], C2×C6 [×7], C22×C4, C3×S3, C4×S3 [×2], C2×C12 [×6], C22×S3, C22×C6, S3×C6 [×3], S3×C2×C4, C22×C12, S3×C12 [×2], S3×C2×C6, S3×C2×C12

Smallest permutation representation of S3×C2×C12
On 48 points
Generators in S48
(1 22)(2 23)(3 24)(4 13)(5 14)(6 15)(7 16)(8 17)(9 18)(10 19)(11 20)(12 21)(25 40)(26 41)(27 42)(28 43)(29 44)(30 45)(31 46)(32 47)(33 48)(34 37)(35 38)(36 39)
(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 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)
(1 37)(2 38)(3 39)(4 40)(5 41)(6 42)(7 43)(8 44)(9 45)(10 46)(11 47)(12 48)(13 25)(14 26)(15 27)(16 28)(17 29)(18 30)(19 31)(20 32)(21 33)(22 34)(23 35)(24 36)

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

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

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

S3×C2×C12 is a maximal subgroup of
C12.77D12  C62.11C23  C62.20C23  D6⋊Dic6  C62.25C23  D66Dic6  D67Dic6  C62.49C23  C62.74C23  C62.75C23  D6⋊D12  D62D12  C127D12

72 conjugacy classes

class 1 2A2B2C2D2E2F2G3A3B3C3D3E4A4B4C4D4E4F4G4H6A···6F6G···6O6P···6W12A···12H12I···12T12U···12AB
order1222222233333444444446···66···66···612···1212···1212···12
size1111333311222111133331···12···23···31···12···23···3

72 irreducible representations

dim11111111111122222222
type++++++++
imageC1C2C2C2C2C3C4C6C6C6C6C12S3D6D6C3×S3C4×S3S3×C6S3×C6S3×C12
kernelS3×C2×C12S3×C12C6×Dic3C6×C12S3×C2×C6S3×C2×C4S3×C6C4×S3C2×Dic3C2×C12C22×S3D6C2×C12C12C2×C6C2×C4C6C4C22C2
# reps141112882221612124428

Matrix representation of S3×C2×C12 in GL3(𝔽13) generated by

1200
0120
0012
,
400
020
002
,
100
033
009
,
1200
0120
0111
G:=sub<GL(3,GF(13))| [12,0,0,0,12,0,0,0,12],[4,0,0,0,2,0,0,0,2],[1,0,0,0,3,0,0,3,9],[12,0,0,0,12,11,0,0,1] >;

S3×C2×C12 in GAP, Magma, Sage, TeX

S_3\times C_2\times C_{12}
% in TeX

G:=Group("S3xC2xC12");
// GroupNames label

G:=SmallGroup(144,159);
// by ID

G=gap.SmallGroup(144,159);
# by ID

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

G:=Group<a,b,c,d|a^2=b^12=c^3=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

׿
×
𝔽