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## G = S3×C22×C4order 96 = 25·3

### Direct product of C22×C4 and S3

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

 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 [×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 ··· 2G 2H ··· 2O 3 4A ··· 4H 4I ··· 4P 6A ··· 6G 12A ··· 12H order 1 2 ··· 2 2 ··· 2 3 4 ··· 4 4 ··· 4 6 ··· 6 12 ··· 12 size 1 1 ··· 1 3 ··· 3 2 1 ··· 1 3 ··· 3 2 ··· 2 2 ··· 2

48 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 type + + + + + + + + image C1 C2 C2 C2 C2 C4 S3 D6 D6 C4×S3 kernel S3×C22×C4 S3×C2×C4 C22×Dic3 C22×C12 S3×C23 C22×S3 C22×C4 C2×C4 C23 C22 # reps 1 12 1 1 1 16 1 6 1 8

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

 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 12 0 0 1 12
,
 1 0 0 0 0 1 0 0 0 0 12 1 0 0 0 1
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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