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## G = S3×C22⋊C8order 192 = 26·3

### Direct product of S3 and C22⋊C8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — S3×C22⋊C8
 Chief series C1 — C3 — C6 — C12 — C2×C12 — S3×C2×C4 — S3×C22×C4 — S3×C22⋊C8
 Lower central C3 — C6 — S3×C22⋊C8
 Upper central C1 — C2×C4 — C22⋊C8

Generators and relations for S3×C22⋊C8
G = < a,b,c,d,e | a3=b2=c2=d2=e8=1, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, ece-1=cd=dc, de=ed >

Subgroups: 512 in 202 conjugacy classes, 73 normal (33 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, S3, S3, C6, C6, C8, C2×C4, C2×C4, C23, C23, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C2×C6, C2×C8, C2×C8, C22×C4, C22×C4, C24, C3⋊C8, C24, C4×S3, C4×S3, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×S3, C22×S3, C22×S3, C22×C6, C22⋊C8, C22⋊C8, C22×C8, C23×C4, S3×C8, C2×C3⋊C8, C2×C24, S3×C2×C4, S3×C2×C4, C22×Dic3, C22×C12, S3×C23, C2×C22⋊C8, D6⋊C8, C12.55D4, C3×C22⋊C8, S3×C2×C8, S3×C22×C4, S3×C22⋊C8
Quotients: C1, C2, C4, C22, S3, C8, C2×C4, D4, C23, D6, C22⋊C4, C2×C8, M4(2), C22×C4, C2×D4, C4×S3, C22×S3, C22⋊C8, C2×C22⋊C4, C22×C8, C2×M4(2), S3×C8, S3×C2×C4, S3×D4, C2×C22⋊C8, S3×C22⋊C4, S3×C2×C8, S3×M4(2), S3×C22⋊C8

Smallest permutation representation of S3×C22⋊C8
On 48 points
Generators in S48
(1 19 43)(2 20 44)(3 21 45)(4 22 46)(5 23 47)(6 24 48)(7 17 41)(8 18 42)(9 30 37)(10 31 38)(11 32 39)(12 25 40)(13 26 33)(14 27 34)(15 28 35)(16 29 36)
(17 41)(18 42)(19 43)(20 44)(21 45)(22 46)(23 47)(24 48)(25 40)(26 33)(27 34)(28 35)(29 36)(30 37)(31 38)(32 39)
(1 5)(2 16)(3 7)(4 10)(6 12)(8 14)(9 13)(11 15)(17 21)(18 27)(19 23)(20 29)(22 31)(24 25)(26 30)(28 32)(33 37)(34 42)(35 39)(36 44)(38 46)(40 48)(41 45)(43 47)
(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 9)(8 10)(17 30)(18 31)(19 32)(20 25)(21 26)(22 27)(23 28)(24 29)(33 45)(34 46)(35 47)(36 48)(37 41)(38 42)(39 43)(40 44)
(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)

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

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

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

60 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 6A 6B 6C 6D 6E 8A ··· 8H 8I ··· 8P 12A 12B 12C 12D 12E 12F 24A ··· 24H 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 8 ··· 8 8 ··· 8 12 12 12 12 12 12 24 ··· 24 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 6 ··· 6 2 2 2 2 4 4 4 ··· 4

60 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C4 C4 C4 C8 S3 D4 D6 D6 M4(2) C4×S3 C4×S3 S3×C8 S3×D4 S3×M4(2) kernel S3×C22⋊C8 D6⋊C8 C12.55D4 C3×C22⋊C8 S3×C2×C8 S3×C22×C4 S3×C2×C4 C22×Dic3 S3×C23 C22×S3 C22⋊C8 C4×S3 C2×C8 C22×C4 D6 C2×C4 C23 C22 C4 C2 # reps 1 2 1 1 2 1 4 2 2 16 1 4 2 1 4 2 2 8 2 2

Matrix representation of S3×C22⋊C8 in GL4(𝔽73) generated by

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

S3×C22⋊C8 in GAP, Magma, Sage, TeX

S_3\times C_2^2\rtimes C_8
% in TeX

G:=Group("S3xC2^2:C8");
// GroupNames label

G:=SmallGroup(192,283);
// by ID

G=gap.SmallGroup(192,283);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,219,58,136,6278]);
// Polycyclic

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

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