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## G = C2×S3×C4○D4order 192 = 26·3

### Direct product of C2, S3 and C4○D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C2×S3×C4○D4
 Chief series C1 — C3 — C6 — D6 — C22×S3 — S3×C23 — S3×C22×C4 — C2×S3×C4○D4
 Lower central C3 — C6 — C2×S3×C4○D4
 Upper central C1 — C2×C4 — C2×C4○D4

Generators and relations for C2×S3×C4○D4
G = < a,b,c,d,e,f | a2=b3=c2=d4=f2=1, e2=d2, ab=ba, ac=ca, ad=da, ae=ea, af=fa, cbc=b-1, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef=d2e >

Subgroups: 1832 in 890 conjugacy classes, 455 normal (18 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C22, S3, S3, C6, C6, C6, C2×C4, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, Dic3, C12, D6, D6, C2×C6, C2×C6, C2×C6, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4, C4○D4, C24, Dic6, C4×S3, D12, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×Q8, C22×S3, C22×S3, C22×S3, C22×C6, C23×C4, C22×D4, C22×Q8, C2×C4○D4, C2×C4○D4, C2×Dic6, S3×C2×C4, S3×C2×C4, C2×D12, C4○D12, S3×D4, D42S3, S3×Q8, Q83S3, C22×Dic3, C2×C3⋊D4, C22×C12, C6×D4, C6×Q8, C3×C4○D4, S3×C23, C22×C4○D4, S3×C22×C4, C2×C4○D12, C2×S3×D4, C2×D42S3, C2×S3×Q8, C2×Q83S3, S3×C4○D4, C6×C4○D4, C2×S3×C4○D4
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C24, C22×S3, C2×C4○D4, C25, S3×C23, C22×C4○D4, S3×C4○D4, S3×C24, C2×S3×C4○D4

Smallest permutation representation of C2×S3×C4○D4
On 48 points
Generators in S48
(1 34)(2 35)(3 36)(4 33)(5 25)(6 26)(7 27)(8 28)(9 29)(10 30)(11 31)(12 32)(13 37)(14 38)(15 39)(16 40)(17 41)(18 42)(19 43)(20 44)(21 45)(22 46)(23 47)(24 48)
(1 16 17)(2 13 18)(3 14 19)(4 15 20)(5 10 47)(6 11 48)(7 12 45)(8 9 46)(21 27 32)(22 28 29)(23 25 30)(24 26 31)(33 39 44)(34 40 41)(35 37 42)(36 38 43)
(1 36)(2 33)(3 34)(4 35)(5 32)(6 29)(7 30)(8 31)(9 26)(10 27)(11 28)(12 25)(13 44)(14 41)(15 42)(16 43)(17 38)(18 39)(19 40)(20 37)(21 47)(22 48)(23 45)(24 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 46 3 48)(2 47 4 45)(5 15 7 13)(6 16 8 14)(9 19 11 17)(10 20 12 18)(21 35 23 33)(22 36 24 34)(25 39 27 37)(26 40 28 38)(29 43 31 41)(30 44 32 42)
(5 7)(6 8)(9 11)(10 12)(21 23)(22 24)(25 27)(26 28)(29 31)(30 32)(45 47)(46 48)

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

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

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

60 conjugacy classes

 class 1 2A 2B 2C 2D ··· 2I 2J 2K 2L 2M 2N ··· 2S 3 4A 4B 4C 4D 4E ··· 4J 4K 4L 4M 4N 4O ··· 4T 6A 6B 6C 6D ··· 6I 12A 12B 12C 12D 12E ··· 12J order 1 2 2 2 2 ··· 2 2 2 2 2 2 ··· 2 3 4 4 4 4 4 ··· 4 4 4 4 4 4 ··· 4 6 6 6 6 ··· 6 12 12 12 12 12 ··· 12 size 1 1 1 1 2 ··· 2 3 3 3 3 6 ··· 6 2 1 1 1 1 2 ··· 2 3 3 3 3 6 ··· 6 2 2 2 4 ··· 4 2 2 2 2 4 ··· 4

60 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 4 type + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 S3 D6 D6 D6 D6 C4○D4 S3×C4○D4 kernel C2×S3×C4○D4 S3×C22×C4 C2×C4○D12 C2×S3×D4 C2×D4⋊2S3 C2×S3×Q8 C2×Q8⋊3S3 S3×C4○D4 C6×C4○D4 C2×C4○D4 C22×C4 C2×D4 C2×Q8 C4○D4 D6 C2 # reps 1 3 3 3 3 1 1 16 1 1 3 3 1 8 8 4

Matrix representation of C2×S3×C4○D4 in GL4(𝔽13) generated by

 12 0 0 0 0 12 0 0 0 0 12 0 0 0 0 12
,
 12 12 0 0 1 0 0 0 0 0 1 0 0 0 0 1
,
 12 0 0 0 1 1 0 0 0 0 1 0 0 0 0 1
,
 12 0 0 0 0 12 0 0 0 0 5 0 0 0 0 5
,
 12 0 0 0 0 12 0 0 0 0 1 11 0 0 1 12
,
 1 0 0 0 0 1 0 0 0 0 1 0 0 0 1 12
G:=sub<GL(4,GF(13))| [12,0,0,0,0,12,0,0,0,0,12,0,0,0,0,12],[12,1,0,0,12,0,0,0,0,0,1,0,0,0,0,1],[12,1,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[12,0,0,0,0,12,0,0,0,0,5,0,0,0,0,5],[12,0,0,0,0,12,0,0,0,0,1,1,0,0,11,12],[1,0,0,0,0,1,0,0,0,0,1,1,0,0,0,12] >;

C2×S3×C4○D4 in GAP, Magma, Sage, TeX

C_2\times S_3\times C_4\circ D_4
% in TeX

G:=Group("C2xS3xC4oD4");
// GroupNames label

G:=SmallGroup(192,1520);
// by ID

G=gap.SmallGroup(192,1520);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,136,438,6278]);
// Polycyclic

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

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