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## G = C22×S3×D4order 192 = 26·3

### Direct product of C22, S3 and D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C22×S3×D4
 Chief series C1 — C3 — C6 — D6 — C22×S3 — S3×C23 — S3×C24 — C22×S3×D4
 Lower central C3 — C6 — C22×S3×D4
 Upper central C1 — C23 — C22×D4

Generators and relations for C22×S3×D4
G = < a,b,c,d,e,f | a2=b2=c3=d2=e4=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, dcd=c-1, ce=ec, cf=fc, de=ed, df=fd, fef=e-1 >

Subgroups: 3432 in 1362 conjugacy classes, 503 normal (15 characteristic)
C1, C2, C2 [×6], C2 [×24], C3, C4 [×4], C4 [×4], C22 [×15], C22 [×148], S3 [×8], S3 [×8], C6, C6 [×6], C6 [×8], C2×C4 [×6], C2×C4 [×22], D4 [×16], D4 [×48], C23, C23 [×12], C23 [×170], Dic3 [×4], C12 [×4], D6 [×36], D6 [×88], C2×C6 [×15], C2×C6 [×24], C22×C4, C22×C4 [×13], C2×D4 [×12], C2×D4 [×100], C24 [×2], C24 [×43], C4×S3 [×16], D12 [×16], C2×Dic3 [×6], C3⋊D4 [×32], C2×C12 [×6], C3×D4 [×16], C22×S3 [×58], C22×S3 [×104], C22×C6, C22×C6 [×12], C22×C6 [×8], C23×C4, C22×D4, C22×D4 [×27], C25 [×2], S3×C2×C4 [×12], C2×D12 [×12], S3×D4 [×64], C22×Dic3, C2×C3⋊D4 [×24], C22×C12, C6×D4 [×12], S3×C23, S3×C23 [×26], S3×C23 [×16], C23×C6 [×2], D4×C23, S3×C22×C4, C22×D12, C2×S3×D4 [×24], C22×C3⋊D4 [×2], D4×C2×C6, S3×C24 [×2], C22×S3×D4
Quotients: C1, C2 [×31], C22 [×155], S3, D4 [×8], C23 [×155], D6 [×15], C2×D4 [×28], C24 [×31], C22×S3 [×35], C22×D4 [×14], C25, S3×D4 [×4], S3×C23 [×15], D4×C23, C2×S3×D4 [×6], S3×C24, C22×S3×D4

Smallest permutation representation of C22×S3×D4
On 48 points
Generators in S48
(1 44)(2 41)(3 42)(4 43)(5 45)(6 46)(7 47)(8 48)(9 19)(10 20)(11 17)(12 18)(13 28)(14 25)(15 26)(16 27)(21 35)(22 36)(23 33)(24 34)(29 40)(30 37)(31 38)(32 39)
(1 8)(2 5)(3 6)(4 7)(9 21)(10 22)(11 23)(12 24)(13 29)(14 30)(15 31)(16 32)(17 33)(18 34)(19 35)(20 36)(25 37)(26 38)(27 39)(28 40)(41 45)(42 46)(43 47)(44 48)
(1 17 16)(2 18 13)(3 19 14)(4 20 15)(5 34 29)(6 35 30)(7 36 31)(8 33 32)(9 25 42)(10 26 43)(11 27 44)(12 28 41)(21 37 46)(22 38 47)(23 39 48)(24 40 45)
(1 42)(2 43)(3 44)(4 41)(5 47)(6 48)(7 45)(8 46)(9 16)(10 13)(11 14)(12 15)(17 25)(18 26)(19 27)(20 28)(21 32)(22 29)(23 30)(24 31)(33 37)(34 38)(35 39)(36 40)
(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 42)(2 41)(3 44)(4 43)(5 45)(6 48)(7 47)(8 46)(9 17)(10 20)(11 19)(12 18)(13 28)(14 27)(15 26)(16 25)(21 33)(22 36)(23 35)(24 34)(29 40)(30 39)(31 38)(32 37)

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

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

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

60 conjugacy classes

 class 1 2A ··· 2G 2H ··· 2O 2P ··· 2W 2X ··· 2AE 3 4A 4B 4C 4D 4E 4F 4G 4H 6A ··· 6G 6H ··· 6O 12A 12B 12C 12D order 1 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2 3 4 4 4 4 4 4 4 4 6 ··· 6 6 ··· 6 12 12 12 12 size 1 1 ··· 1 2 ··· 2 3 ··· 3 6 ··· 6 2 2 2 2 2 6 6 6 6 2 ··· 2 4 ··· 4 4 4 4 4

60 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 4 type + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 S3 D4 D6 D6 D6 S3×D4 kernel C22×S3×D4 S3×C22×C4 C22×D12 C2×S3×D4 C22×C3⋊D4 D4×C2×C6 S3×C24 C22×D4 C22×S3 C22×C4 C2×D4 C24 C22 # reps 1 1 1 24 2 1 2 1 8 1 12 2 4

Matrix representation of C22×S3×D4 in GL5(ℤ)

 1 0 0 0 0 0 -1 0 0 0 0 0 -1 0 0 0 0 0 1 0 0 0 0 0 1
,
 -1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 -1 0 0 0 0 0 -1
,
 1 0 0 0 0 0 0 -1 0 0 0 1 -1 0 0 0 0 0 1 0 0 0 0 0 1
,
 -1 0 0 0 0 0 0 -1 0 0 0 -1 0 0 0 0 0 0 -1 0 0 0 0 0 -1
,
 -1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 -1 0
,
 -1 0 0 0 0 0 -1 0 0 0 0 0 -1 0 0 0 0 0 -1 0 0 0 0 0 1

G:=sub<GL(5,Integers())| [1,0,0,0,0,0,-1,0,0,0,0,0,-1,0,0,0,0,0,1,0,0,0,0,0,1],[-1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,-1,0,0,0,0,0,-1],[1,0,0,0,0,0,0,1,0,0,0,-1,-1,0,0,0,0,0,1,0,0,0,0,0,1],[-1,0,0,0,0,0,0,-1,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,-1],[-1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,1,0],[-1,0,0,0,0,0,-1,0,0,0,0,0,-1,0,0,0,0,0,-1,0,0,0,0,0,1] >;

C22×S3×D4 in GAP, Magma, Sage, TeX

C_2^2\times S_3\times D_4
% in TeX

G:=Group("C2^2xS3xD4");
// GroupNames label

G:=SmallGroup(192,1514);
// by ID

G=gap.SmallGroup(192,1514);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,235,6278]);
// Polycyclic

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

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