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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, C3, C4, C4, C22, C22, S3, S3, C6, C6, C8, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, Q8, C23, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C2×C8, M4(2), M4(2), SD16, SD16, Q16, Q16, C22×C4, C2×D4, C2×Q8, C2×Q8, C4○D4, C4○D4, C3⋊C8, C24, Dic6, Dic6, Dic6, C4×S3, C4×S3, D12, D12, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C3×Q8, C3×Q8, C22×S3, C22×S3, C2×M4(2), C2×SD16, C2×Q16, C8.C22, C8.C22, C22×Q8, C2×C4○D4, S3×C8, C8⋊S3, C24⋊C2, Dic12, C4.Dic3, D4.S3, Q82S3, C3⋊Q16, C3×M4(2), C3×SD16, C3×Q16, C2×Dic6, C2×Dic6, S3×C2×C4, S3×C2×C4, C4○D12, C4○D12, S3×D4, S3×D4, D42S3, D42S3, S3×Q8, S3×Q8, S3×Q8, Q83S3, C6×Q8, C3×C4○D4, C2×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, S3×C8.C22
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C24, C22×S3, C8.C22, C22×D4, S3×D4, 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 28 45)(2 29 46)(3 30 47)(4 31 48)(5 32 41)(6 25 42)(7 26 43)(8 27 44)(9 35 19)(10 36 20)(11 37 21)(12 38 22)(13 39 23)(14 40 24)(15 33 17)(16 34 18)
(9 35)(10 36)(11 37)(12 38)(13 39)(14 40)(15 33)(16 34)(25 42)(26 43)(27 44)(28 45)(29 46)(30 47)(31 48)(32 41)
(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)(9 15)(11 13)(12 16)(17 19)(18 22)(21 23)(25 27)(26 30)(29 31)(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 25)(10 30)(11 27)(12 32)(13 29)(14 26)(15 31)(16 28)(33 48)(34 45)(35 42)(36 47)(37 44)(38 41)(39 46)(40 43)

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

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

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

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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