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G = S3×C22×C4  order 96 = 25·3

Direct product of C22×C4 and S3

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

Aliases: S3×C22×C4, C12⋊3C23, C6.2C24, D6.9C23, C23.39D6, Dic3⋊3C23, C3⋊1(C23×C4), C6⋊1(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

Derived series
C1 — C3 — S3×C22×C4
Chief series
C1 — C3 — C6 — D6 — C22×S3 — S3×C23 — S3×C22×C4
Lower central
C3 — S3×C22×C4
Upper central
C1 — C22×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, C2, C3, C4, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, C23, C23, Dic3, C12, D6, C2×C6, C22×C4, C22×C4, C24, C4×S3, C2×Dic3, C2×C12, C22×S3, C22×C6, C23×C4, S3×C2×C4, C22×Dic3, C22×C12, S3×C23, S3×C22×C4
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D6, C22×C4, C24, C4×S3, C22×S3, C23×C4, S3×C2×C4, 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⋊C4⋊6C4  D6⋊6M4(2)  C42⋊10D6  C42⋊14D6  C4⋊C4⋊21D6  C4⋊C4⋊26D6  C4⋊C4⋊28D6  (C2×D4)⋊43D6
S3×C22×C4 is a maximal quotient of
 C24.35D6  C6.82+ 1+4  C42.87D6  C42⋊9D6  C42.188D6  C42.91D6  C42⋊13D6  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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