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G = C422S3order 96 = 25·3

1st semidirect product of C42 and S3 acting via S3/C3=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C422S3, (C4×S3)⋊3C4, (C4×C12)⋊9C2, C4(D6⋊C4), C4.22(C4×S3), D6⋊C4.7C2, D6.3(C2×C4), (C2×C4).96D6, C12.25(C2×C4), (C4×Dic3)⋊8C2, C6.3(C4○D4), C6.3(C22×C4), C42(Dic3⋊C4), Dic3⋊C417C2, C31(C42⋊C2), C2.2(C4○D12), (C2×C6).13C23, Dic3.5(C2×C4), (C2×C12).73C22, C22.10(C22×S3), (C22×S3).14C22, (C2×Dic3).24C22, C2.5(S3×C2×C4), (S3×C2×C4).8C2, (C2×C4)(D6⋊C4), (C2×C4)(Dic3⋊C4), SmallGroup(96,79)

Series: Derived Chief Lower central Upper central

C1C6 — C422S3
C1C3C6C2×C6C22×S3S3×C2×C4 — C422S3
C3C6 — C422S3
C1C2×C4C42

Generators and relations for C422S3
 G = < a,b,c,d | a4=b4=c3=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=a2b, dcd=c-1 >

Subgroups: 154 in 76 conjugacy classes, 41 normal (15 characteristic)
C1, C2, C2 [×2], C2 [×2], C3, C4 [×2], C4 [×6], C22, C22 [×4], S3 [×2], C6, C6 [×2], C2×C4, C2×C4 [×2], C2×C4 [×7], C23, Dic3 [×2], Dic3 [×2], C12 [×2], C12 [×2], D6 [×2], D6 [×2], C2×C6, C42, C42, C22⋊C4 [×2], C4⋊C4 [×2], C22×C4, C4×S3 [×4], C2×Dic3, C2×Dic3 [×2], C2×C12, C2×C12 [×2], C22×S3, C42⋊C2, C4×Dic3, Dic3⋊C4 [×2], D6⋊C4 [×2], C4×C12, S3×C2×C4, C422S3
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], C23, D6 [×3], C22×C4, C4○D4 [×2], C4×S3 [×2], C22×S3, C42⋊C2, S3×C2×C4, C4○D12 [×2], C422S3

Smallest permutation representation of C422S3
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 13 29 23)(2 14 30 24)(3 15 31 21)(4 16 32 22)(5 27 43 33)(6 28 44 34)(7 25 41 35)(8 26 42 36)(9 38 48 20)(10 39 45 17)(11 40 46 18)(12 37 47 19)
(1 39 5)(2 40 6)(3 37 7)(4 38 8)(9 36 22)(10 33 23)(11 34 24)(12 35 21)(13 45 27)(14 46 28)(15 47 25)(16 48 26)(17 43 29)(18 44 30)(19 41 31)(20 42 32)
(5 39)(6 40)(7 37)(8 38)(9 34)(10 35)(11 36)(12 33)(13 15)(14 16)(17 43)(18 44)(19 41)(20 42)(21 23)(22 24)(25 45)(26 46)(27 47)(28 48)

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

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

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

C422S3 is a maximal subgroup of
D6.C42  C42.243D6  D6.4C42  C42.185D6  C423D6  C42.30D6  C42.31D6  C4×C4○D12  C42.277D6  S3×C42⋊C2  C42.188D6  C4212D6  C42.93D6  C42.94D6  C42.96D6  C42.97D6  C42.102D6  C42.104D6  C4213D6  C42.108D6  C4214D6  C4218D6  C42.115D6  C42.116D6  C42.122D6  C42.125D6  C42.126D6  C42.232D6  C42.131D6  C42.132D6  C42.133D6  C42.134D6  C42.138D6  C4220D6  C42.141D6  C4223D6  C42.144D6  C42.148D6  C42.151D6  C42.155D6  C42.156D6  C42.160D6  C4226D6  C42.162D6  C42.163D6  C4228D6  C42.168D6  C42.171D6  C42.174D6  C42.176D6  C42.178D6  C422D9  C62.6C23  C62.25C23  C62.44C23  C62.47C23  C12216C2  (S3×C20)⋊7C4  (C4×D15)⋊10C4  D6.(C4×D5)  D30.C2⋊C4  C422D15  (C4×S3)⋊F5
C422S3 is a maximal quotient of
C3⋊(C428C4)  C3⋊(C425C4)  C6.(C4×D4)  C22.58(S3×D4)  D6⋊(C4⋊C4)  D6⋊C45C4  C42.282D6  C42.243D6  C42.182D6  C42.185D6  C4×Dic3⋊C4  C426Dic3  (C2×C42).6S3  C4×D6⋊C4  (C2×C42)⋊3S3  C422D9  C62.6C23  C62.25C23  C62.44C23  C62.47C23  C12216C2  (S3×C20)⋊7C4  (C4×D15)⋊10C4  D6.(C4×D5)  D30.C2⋊C4  C422D15  (C4×S3)⋊F5

36 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C4D4E4F4G4H4I···4N6A6B6C12A···12L
order1222223444444444···466612···12
size1111662111122226···62222···2

36 irreducible representations

dim111111122222
type++++++++
imageC1C2C2C2C2C2C4S3D6C4○D4C4×S3C4○D12
kernelC422S3C4×Dic3Dic3⋊C4D6⋊C4C4×C12S3×C2×C4C4×S3C42C2×C4C6C4C2
# reps112211813448

Matrix representation of C422S3 in GL3(𝔽13) generated by

1200
050
005
,
800
029
0411
,
100
0012
0112
,
1200
001
010
G:=sub<GL(3,GF(13))| [12,0,0,0,5,0,0,0,5],[8,0,0,0,2,4,0,9,11],[1,0,0,0,0,1,0,12,12],[12,0,0,0,0,1,0,1,0] >;

C422S3 in GAP, Magma, Sage, TeX

C_4^2\rtimes_2S_3
% in TeX

G:=Group("C4^2:2S3");
// GroupNames label

G:=SmallGroup(96,79);
// by ID

G=gap.SmallGroup(96,79);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-3,103,362,50,2309]);
// Polycyclic

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

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