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## G = S3×C8○D4order 192 = 26·3

### Direct product of S3 and C8○D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — S3×C8○D4
 Chief series C1 — C3 — C6 — C12 — C4×S3 — S3×C2×C4 — S3×C4○D4 — S3×C8○D4
 Lower central C3 — C6 — S3×C8○D4
 Upper central C1 — C8 — C8○D4

Generators and relations for S3×C8○D4
G = < a,b,c,d,e | a3=b2=c8=e2=1, d2=c4, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=c4d >

Subgroups: 512 in 266 conjugacy classes, 149 normal (24 characteristic)
C1, C2, C2, C3, C4, C4, C4, C22, C22, S3, S3, C6, C6, C8, C8, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, Dic3, Dic3, C12, C12, D6, D6, D6, C2×C6, C2×C8, C2×C8, M4(2), M4(2), C22×C4, C2×D4, C2×Q8, C4○D4, C4○D4, C3⋊C8, C3⋊C8, C24, C24, Dic6, C4×S3, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C3×Q8, C22×S3, C22×C8, C2×M4(2), C8○D4, C8○D4, C2×C4○D4, S3×C8, S3×C8, C8⋊S3, C2×C3⋊C8, C4.Dic3, C2×C24, C3×M4(2), S3×C2×C4, C4○D12, S3×D4, D42S3, S3×Q8, Q83S3, C3×C4○D4, C2×C8○D4, S3×C2×C8, C8○D12, S3×M4(2), D12.C4, D4.Dic3, C3×C8○D4, S3×C4○D4, S3×C8○D4
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D6, C22×C4, C24, C4×S3, C22×S3, C8○D4, C23×C4, S3×C2×C4, S3×C23, C2×C8○D4, S3×C22×C4, S3×C8○D4

Smallest permutation representation of S3×C8○D4
On 48 points
Generators in S48
(1 26 35)(2 27 36)(3 28 37)(4 29 38)(5 30 39)(6 31 40)(7 32 33)(8 25 34)(9 44 23)(10 45 24)(11 46 17)(12 47 18)(13 48 19)(14 41 20)(15 42 21)(16 43 22)
(17 46)(18 47)(19 48)(20 41)(21 42)(22 43)(23 44)(24 45)(25 34)(26 35)(27 36)(28 37)(29 38)(30 39)(31 40)(32 33)
(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 15 5 11)(2 16 6 12)(3 9 7 13)(4 10 8 14)(17 35 21 39)(18 36 22 40)(19 37 23 33)(20 38 24 34)(25 41 29 45)(26 42 30 46)(27 43 31 47)(28 44 32 48)
(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)(41 45)(42 46)(43 47)(44 48)

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

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

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

60 conjugacy classes

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

60 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 type + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C4 C4 C4 C4 S3 D6 D6 D6 C4×S3 C4×S3 C8○D4 S3×C8○D4 kernel S3×C8○D4 S3×C2×C8 C8○D12 S3×M4(2) D12.C4 D4.Dic3 C3×C8○D4 S3×C4○D4 S3×D4 D4⋊2S3 S3×Q8 Q8⋊3S3 C8○D4 C2×C8 M4(2) C4○D4 D4 Q8 S3 C1 # reps 1 3 3 3 3 1 1 1 6 6 2 2 1 3 3 1 6 2 8 4

Matrix representation of S3×C8○D4 in GL4(𝔽73) generated by

 0 72 0 0 1 72 0 0 0 0 1 0 0 0 0 1
,
 1 72 0 0 0 72 0 0 0 0 1 0 0 0 0 1
,
 27 0 0 0 0 27 0 0 0 0 22 0 0 0 0 22
,
 72 0 0 0 0 72 0 0 0 0 0 72 0 0 1 0
,
 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 72
G:=sub<GL(4,GF(73))| [0,1,0,0,72,72,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,72,72,0,0,0,0,1,0,0,0,0,1],[27,0,0,0,0,27,0,0,0,0,22,0,0,0,0,22],[72,0,0,0,0,72,0,0,0,0,0,1,0,0,72,0],[1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,72] >;

S3×C8○D4 in GAP, Magma, Sage, TeX

S_3\times C_8\circ D_4
% in TeX

G:=Group("S3xC8oD4");
// GroupNames label

G:=SmallGroup(192,1308);
// by ID

G=gap.SmallGroup(192,1308);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,387,80,102,6278]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^2=c^8=e^2=1,d^2=c^4,b*a*b=a^-1,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=c^4*d>;
// generators/relations

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