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## G = C2×S3×D8order 192 = 26·3

### Direct product of C2, S3 and D8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C2×S3×D8
 Chief series C1 — C3 — C6 — C12 — C4×S3 — S3×C2×C4 — C2×S3×D4 — C2×S3×D8
 Lower central C3 — C6 — C12 — C2×S3×D8
 Upper central C1 — C22 — C2×C4 — C2×D8

Generators and relations for C2×S3×D8
G = < a,b,c,d,e | a2=b3=c2=d8=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 1176 in 338 conjugacy classes, 111 normal (23 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, S3, S3, C6, C6, C6, C8, C8, C2×C4, C2×C4, D4, D4, C23, Dic3, C12, D6, D6, C2×C6, C2×C6, C2×C8, C2×C8, D8, D8, C22×C4, C2×D4, C2×D4, C24, C3⋊C8, C24, C4×S3, D12, D12, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C3×D4, C22×S3, C22×S3, C22×C6, C22×C8, C2×D8, C2×D8, C22×D4, S3×C8, D24, C2×C3⋊C8, D4⋊S3, C2×C24, C3×D8, S3×C2×C4, C2×D12, S3×D4, S3×D4, C2×C3⋊D4, C6×D4, S3×C23, C22×D8, S3×C2×C8, C2×D24, S3×D8, C2×D4⋊S3, C6×D8, C2×S3×D4, C2×S3×D8
Quotients: C1, C2, C22, S3, D4, C23, D6, D8, C2×D4, C24, C22×S3, C2×D8, C22×D4, S3×D4, S3×C23, C22×D8, S3×D8, C2×S3×D4, C2×S3×D8

Smallest permutation representation of C2×S3×D8
On 48 points
Generators in S48
(1 13)(2 14)(3 15)(4 16)(5 9)(6 10)(7 11)(8 12)(17 26)(18 27)(19 28)(20 29)(21 30)(22 31)(23 32)(24 25)(33 41)(34 42)(35 43)(36 44)(37 45)(38 46)(39 47)(40 48)
(1 39 21)(2 40 22)(3 33 23)(4 34 24)(5 35 17)(6 36 18)(7 37 19)(8 38 20)(9 43 26)(10 44 27)(11 45 28)(12 46 29)(13 47 30)(14 48 31)(15 41 32)(16 42 25)
(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)(17 39)(18 40)(19 33)(20 34)(21 35)(22 36)(23 37)(24 38)(25 46)(26 47)(27 48)(28 41)(29 42)(30 43)(31 44)(32 45)
(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)
(2 8)(3 7)(4 6)(10 16)(11 15)(12 14)(18 24)(19 23)(20 22)(25 27)(28 32)(29 31)(33 37)(34 36)(38 40)(41 45)(42 44)(46 48)

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

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

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

42 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 2L 2M 2N 2O 3 4A 4B 4C 4D 6A 6B 6C 6D 6E 6F 6G 8A 8B 8C 8D 8E 8F 8G 8H 12A 12B 24A 24B 24C 24D order 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 4 4 4 4 6 6 6 6 6 6 6 8 8 8 8 8 8 8 8 12 12 24 24 24 24 size 1 1 1 1 3 3 3 3 4 4 4 4 12 12 12 12 2 2 2 6 6 2 2 2 8 8 8 8 2 2 2 2 6 6 6 6 4 4 4 4 4 4

42 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 S3 D4 D4 D4 D6 D6 D6 D8 S3×D4 S3×D4 S3×D8 kernel C2×S3×D8 S3×C2×C8 C2×D24 S3×D8 C2×D4⋊S3 C6×D8 C2×S3×D4 C2×D8 C4×S3 C2×Dic3 C22×S3 C2×C8 D8 C2×D4 D6 C4 C22 C2 # reps 1 1 1 8 2 1 2 1 2 1 1 1 4 2 8 1 1 4

Matrix representation of C2×S3×D8 in GL6(𝔽73)

 72 0 0 0 0 0 0 72 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 72 0 0 0 0 1 72
,
 72 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 0 0 1 0 0 0 0 1 0
,
 16 57 0 0 0 0 16 16 0 0 0 0 0 0 57 57 0 0 0 0 16 57 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 72 0 0 0 0 0 0 1 0 0 0 0 0 0 72 0 0 0 0 0 0 1 0 0 0 0 0 0 1

G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,72,72],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[16,16,0,0,0,0,57,16,0,0,0,0,0,0,57,16,0,0,0,0,57,57,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

C2×S3×D8 in GAP, Magma, Sage, TeX

C_2\times S_3\times D_8
% in TeX

G:=Group("C2xS3xD8");
// GroupNames label

G:=SmallGroup(192,1313);
// by ID

G=gap.SmallGroup(192,1313);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,185,438,235,102,6278]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^3=c^2=d^8=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,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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