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## G = S3×C4.10D4order 192 = 26·3

### Direct product of S3 and C4.10D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — S3×C4.10D4
 Chief series C1 — C3 — C6 — C12 — C2×C12 — S3×C2×C4 — C2×S3×Q8 — S3×C4.10D4
 Lower central C3 — C6 — C2×C6 — S3×C4.10D4
 Upper central C1 — C2 — C2×C4 — C4.10D4

Generators and relations for S3×C4.10D4
G = < a,b,c,d,e | a3=b2=c4=1, d4=c2, e2=dcd-1=c-1, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, ce=ec, ede-1=c-1d3 >

Subgroups: 368 in 146 conjugacy classes, 53 normal (21 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, S3, C6, C6, C8, C2×C4, C2×C4, C2×C4, Q8, C23, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C2×C8, M4(2), M4(2), C22×C4, C2×Q8, C2×Q8, C3⋊C8, C24, Dic6, C4×S3, C4×S3, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C3×Q8, C22×S3, C4.10D4, C4.10D4, C2×M4(2), C22×Q8, S3×C8, C8⋊S3, C4.Dic3, C3×M4(2), C2×Dic6, C2×Dic6, S3×C2×C4, S3×C2×C4, S3×Q8, C6×Q8, C2×C4.10D4, C12.47D4, C12.10D4, C3×C4.10D4, S3×M4(2), C2×S3×Q8, S3×C4.10D4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D6, C22⋊C4, C22×C4, C2×D4, C4×S3, C22×S3, C4.10D4, C2×C22⋊C4, S3×C2×C4, S3×D4, C2×C4.10D4, S3×C22⋊C4, S3×C4.10D4

Smallest permutation representation of S3×C4.10D4
On 48 points
Generators in S48
(1 27 35)(2 28 36)(3 29 37)(4 30 38)(5 31 39)(6 32 40)(7 25 33)(8 26 34)(9 48 19)(10 41 20)(11 42 21)(12 43 22)(13 44 23)(14 45 24)(15 46 17)(16 47 18)
(17 46)(18 47)(19 48)(20 41)(21 42)(22 43)(23 44)(24 45)(25 33)(26 34)(27 35)(28 36)(29 37)(30 38)(31 39)(32 40)
(1 3 5 7)(2 8 6 4)(9 11 13 15)(10 16 14 12)(17 19 21 23)(18 24 22 20)(25 27 29 31)(26 32 30 28)(33 35 37 39)(34 40 38 36)(41 47 45 43)(42 44 46 48)
(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 16 7 10 5 12 3 14)(2 9 4 15 6 13 8 11)(17 40 23 34 21 36 19 38)(18 33 20 39 22 37 24 35)(25 41 31 43 29 45 27 47)(26 42 28 48 30 46 32 44)

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

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

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

33 conjugacy classes

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

33 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 4 4 8 type + + + + + + + + + + - + - image C1 C2 C2 C2 C2 C2 C4 C4 S3 D4 D6 D6 C4×S3 C4.10D4 S3×D4 S3×C4.10D4 kernel S3×C4.10D4 C12.47D4 C12.10D4 C3×C4.10D4 S3×M4(2) C2×S3×Q8 C2×Dic6 S3×C2×C4 C4.10D4 C4×S3 M4(2) C2×Q8 C2×C4 S3 C4 C1 # reps 1 2 1 1 2 1 4 4 1 4 2 1 4 2 2 1

Matrix representation of S3×C4.10D4 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 0 1 0 0 0 0 72 0 0 0 0 0 0 0 0 72 0 0 0 0 1 0
,
 72 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 72 0 0 0
,
 72 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 53 16 0 0 0 0 16 20 0 0 57 53 0 0 0 0 53 16 0 0

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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,72,0],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,72,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,57,53,0,0,0,0,53,16,0,0,53,16,0,0,0,0,16,20,0,0] >;

S3×C4.10D4 in GAP, Magma, Sage, TeX

S_3\times C_4._{10}D_4
% in TeX

G:=Group("S3xC4.10D4");
// GroupNames label

G:=SmallGroup(192,309);
// by ID

G=gap.SmallGroup(192,309);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,219,58,570,136,438,6278]);
// Polycyclic

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

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