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G = S3×C22×C4order 96 = 25·3

Direct product of C22×C4 and S3

direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary

Aliases: S3×C22×C4, C123C23, C6.2C24, D6.9C23, C23.39D6, Dic33C23, C31(C23×C4), C61(C22×C4), C2.1(S3×C23), C4(C22×Dic3), Dic3(C22×C4), (C22×C12)⋊10C2, (C2×C12)⋊14C22, (S3×C23).3C2, (C2×C6).63C23, (C2×Dic3)⋊12C22, (C22×Dic3)⋊10C2, C22.29(C22×S3), (C22×C6).44C22, (C22×S3).35C22, C4(S3×C2×C4), (C2×C4)(C4×S3), (C2×C6)⋊6(C2×C4), (C2×C4)2(C2×Dic3), (C2×C4)(C22×Dic3), (C22×C4)(C2×Dic3), (C22×C4)(C22×Dic3), (C2×C4)(S3×C2×C4), SmallGroup(96,206)

Series: Derived Chief Lower central Upper central

C1C3 — S3×C22×C4
C1C3C6D6C22×S3S3×C23 — S3×C22×C4
C3 — S3×C22×C4
C1C22×C4

Generators and relations for S3×C22×C4
 G = < a,b,c,d,e | a2=b2=c4=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 418 in 236 conjugacy classes, 145 normal (11 characteristic)
C1, C2, C2 [×6], C2 [×8], C3, C4 [×4], C4 [×4], C22 [×7], C22 [×28], S3 [×8], C6, C6 [×6], C2×C4 [×6], C2×C4 [×22], C23, C23 [×14], Dic3 [×4], C12 [×4], D6 [×28], C2×C6 [×7], C22×C4, C22×C4 [×13], C24, C4×S3 [×16], C2×Dic3 [×6], C2×C12 [×6], C22×S3 [×14], C22×C6, C23×C4, S3×C2×C4 [×12], C22×Dic3, C22×C12, S3×C23, S3×C22×C4
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], S3, C2×C4 [×28], C23 [×15], D6 [×7], C22×C4 [×14], C24, C4×S3 [×4], C22×S3 [×7], C23×C4, S3×C2×C4 [×6], S3×C23, S3×C22×C4

Smallest permutation representation of S3×C22×C4
On 48 points
Generators in S48
(1 36)(2 33)(3 34)(4 35)(5 27)(6 28)(7 25)(8 26)(9 31)(10 32)(11 29)(12 30)(13 39)(14 40)(15 37)(16 38)(17 43)(18 44)(19 41)(20 42)(21 47)(22 48)(23 45)(24 46)
(1 24)(2 21)(3 22)(4 23)(5 39)(6 40)(7 37)(8 38)(9 43)(10 44)(11 41)(12 42)(13 27)(14 28)(15 25)(16 26)(17 31)(18 32)(19 29)(20 30)(33 47)(34 48)(35 45)(36 46)
(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 16)(2 18 13)(3 19 14)(4 20 15)(5 47 10)(6 48 11)(7 45 12)(8 46 9)(21 32 27)(22 29 28)(23 30 25)(24 31 26)(33 44 39)(34 41 40)(35 42 37)(36 43 38)
(1 3)(2 4)(5 12)(6 9)(7 10)(8 11)(13 20)(14 17)(15 18)(16 19)(21 23)(22 24)(25 32)(26 29)(27 30)(28 31)(33 35)(34 36)(37 44)(38 41)(39 42)(40 43)(45 47)(46 48)

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

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

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

S3×C22×C4 is a maximal subgroup of
C22.58(S3×D4)  (C2×C4)⋊9D12  D6⋊C42  D6⋊(C4⋊C4)  D6⋊C4⋊C4  D6⋊M4(2)  C24.23D6  C4⋊(D6⋊C4)  D6⋊C46C4  D66M4(2)  C4210D6  C4214D6  C4⋊C421D6  C4⋊C426D6  C4⋊C428D6  (C2×D4)⋊43D6
S3×C22×C4 is a maximal quotient of
C24.35D6  C6.82+ 1+4  C42.87D6  C429D6  C42.188D6  C42.91D6  C4213D6  C42.108D6  C42.125D6  C42.126D6  M4(2)⋊26D6  M4(2)⋊28D6

48 conjugacy classes

class 1 2A···2G2H···2O 3 4A···4H4I···4P6A···6G12A···12H
order12···22···234···44···46···612···12
size11···13···321···13···32···22···2

48 irreducible representations

dim1111112222
type++++++++
imageC1C2C2C2C2C4S3D6D6C4×S3
kernelS3×C22×C4S3×C2×C4C22×Dic3C22×C12S3×C23C22×S3C22×C4C2×C4C23C22
# reps112111161618

Matrix representation of S3×C22×C4 in GL4(𝔽13) generated by

12000
0100
0010
0001
,
12000
01200
0010
0001
,
8000
0500
0080
0008
,
1000
0100
00012
00112
,
1000
0100
00121
0001
G:=sub<GL(4,GF(13))| [12,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[12,0,0,0,0,12,0,0,0,0,1,0,0,0,0,1],[8,0,0,0,0,5,0,0,0,0,8,0,0,0,0,8],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,12,12],[1,0,0,0,0,1,0,0,0,0,12,0,0,0,1,1] >;

S3×C22×C4 in GAP, Magma, Sage, TeX

S_3\times C_2^2\times C_4
% in TeX

G:=Group("S3xC2^2xC4");
// GroupNames label

G:=SmallGroup(96,206);
// by ID

G=gap.SmallGroup(96,206);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-3,69,2309]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^4=d^3=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations

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