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## G = S3×C8.C22order 192 = 26·3

### Direct product of S3 and C8.C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — S3×C8.C22
 Chief series C1 — C3 — C6 — C12 — C4×S3 — S3×C2×C4 — C2×S3×Q8 — S3×C8.C22
 Lower central C3 — C6 — C12 — S3×C8.C22
 Upper central C1 — C2 — C2×C4 — C8.C22

Generators and relations for S3×C8.C22
G = < a,b,c,d,e | a3=b2=c8=d2=e2=1, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd=c3, ece=c5, ede=c4d >

Subgroups: 656 in 258 conjugacy classes, 101 normal (51 characteristic)
C1, C2, C2 [×6], C3, C4 [×2], C4 [×8], C22, C22 [×8], S3 [×2], S3 [×2], C6, C6 [×2], C8 [×2], C8 [×2], C2×C4, C2×C4 [×16], D4, D4 [×6], Q8, Q8 [×2], Q8 [×10], C23 [×2], Dic3 [×2], Dic3 [×3], C12 [×2], C12 [×3], D6 [×2], D6 [×5], C2×C6, C2×C6, C2×C8 [×2], M4(2), M4(2) [×3], SD16 [×2], SD16 [×6], Q16 [×2], Q16 [×6], C22×C4 [×3], C2×D4 [×2], C2×Q8, C2×Q8 [×9], C4○D4, C4○D4 [×5], C3⋊C8 [×2], C24 [×2], Dic6, Dic6 [×2], Dic6 [×6], C4×S3 [×4], C4×S3 [×7], D12, D12, C2×Dic3, C2×Dic3 [×2], C3⋊D4 [×3], C2×C12, C2×C12 [×2], C3×D4, C3×D4, C3×Q8, C3×Q8 [×2], C3×Q8, C22×S3, C22×S3, C2×M4(2), C2×SD16 [×2], C2×Q16 [×2], C8.C22, C8.C22 [×7], C22×Q8, C2×C4○D4, S3×C8 [×2], C8⋊S3 [×2], C24⋊C2 [×2], Dic12 [×2], C4.Dic3, D4.S3 [×2], Q82S3 [×2], C3⋊Q16 [×4], C3×M4(2), C3×SD16 [×2], C3×Q16 [×2], C2×Dic6, C2×Dic6, S3×C2×C4, S3×C2×C4 [×2], C4○D12, C4○D12, S3×D4, S3×D4, D42S3, D42S3, S3×Q8, S3×Q8 [×4], S3×Q8 [×2], Q83S3, C6×Q8, C3×C4○D4, C2×C8.C22, S3×M4(2), C8.D6, S3×SD16 [×2], D4.D6 [×2], S3×Q16 [×2], Q16⋊S3 [×2], Q8.11D6, Q8.14D6, C3×C8.C22, C2×S3×Q8, S3×C4○D4, S3×C8.C22
Quotients: C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D6 [×7], C2×D4 [×6], C24, C22×S3 [×7], C8.C22 [×2], C22×D4, S3×D4 [×2], S3×C23, C2×C8.C22, C2×S3×D4, S3×C8.C22

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

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

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

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

33 conjugacy classes

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

33 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 8 type + + + + + + + + + + + + + + + + + + + + + - + + - image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 S3 D4 D4 D4 D6 D6 D6 D6 D6 C8.C22 S3×D4 S3×D4 S3×C8.C22 kernel S3×C8.C22 S3×M4(2) C8.D6 S3×SD16 D4.D6 S3×Q16 Q16⋊S3 Q8.11D6 Q8.14D6 C3×C8.C22 C2×S3×Q8 S3×C4○D4 C8.C22 C4×S3 C2×Dic3 C22×S3 M4(2) SD16 Q16 C2×Q8 C4○D4 S3 C4 C22 C1 # reps 1 1 1 2 2 2 2 1 1 1 1 1 1 2 1 1 1 2 2 1 1 2 1 1 1

Matrix representation of S3×C8.C22 in GL6(𝔽73)

 72 72 0 0 0 0 1 0 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 72 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 22 51 66 14 0 0 11 0 0 7 0 0 7 0 0 51 0 0 7 66 11 51
,
 72 0 0 0 0 0 0 72 0 0 0 0 0 0 1 0 0 0 0 0 1 72 0 0 0 0 0 0 72 2 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 66 14 51 22 0 0 66 7 62 22 0 0 22 51 66 14 0 0 11 51 66 7

G:=sub<GL(6,GF(73))| [72,1,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,72,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,22,11,7,7,0,0,51,0,0,66,0,0,66,0,0,11,0,0,14,7,51,51],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,1,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,2,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,66,66,22,11,0,0,14,7,51,51,0,0,51,62,66,66,0,0,22,22,14,7] >;

S3×C8.C22 in GAP, Magma, Sage, TeX

S_3\times C_8.C_2^2
% in TeX

G:=Group("S3xC8.C2^2");
// GroupNames label

G:=SmallGroup(192,1335);
// by ID

G=gap.SmallGroup(192,1335);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,184,570,185,438,235,102,6278]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^2=c^8=d^2=e^2=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,d*c*d=c^3,e*c*e=c^5,e*d*e=c^4*d>;
// generators/relations

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