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## G = S3×SD32order 192 = 26·3

### Direct product of S3 and SD32

Series: Derived Chief Lower central Upper central

 Derived series C1 — C24 — S3×SD32
 Chief series C1 — C3 — C6 — C12 — C24 — S3×C8 — S3×D8 — S3×SD32
 Lower central C3 — C6 — C12 — C24 — S3×SD32
 Upper central C1 — C2 — C4 — C8 — SD32

Generators and relations for S3×SD32
G = < a,b,c,d | a3=b2=c16=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c7 >

Subgroups: 364 in 90 conjugacy classes, 33 normal (31 characteristic)
C1, C2, C2, C3, C4, C4, C22, S3, S3, C6, C6, C8, C8, C2×C4, D4, Q8, C23, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C16, C16, C2×C8, D8, D8, Q16, Q16, C2×D4, C2×Q8, C3⋊C8, C24, Dic6, C4×S3, C4×S3, D12, C3⋊D4, C3×D4, C3×Q8, C22×S3, C2×C16, SD32, SD32, C2×D8, C2×Q16, C3⋊C16, C48, S3×C8, D24, Dic12, D4⋊S3, C3⋊Q16, C3×D8, C3×Q16, S3×D4, S3×Q8, C2×SD32, S3×C16, C48⋊C2, D8.S3, C8.6D6, C3×SD32, S3×D8, S3×Q16, S3×SD32
Quotients: C1, C2, C22, S3, D4, C23, D6, D8, C2×D4, C22×S3, SD32, C2×D8, S3×D4, C2×SD32, S3×D8, S3×SD32

Smallest permutation representation of S3×SD32
On 48 points
Generators in S48
(1 44 25)(2 45 26)(3 46 27)(4 47 28)(5 48 29)(6 33 30)(7 34 31)(8 35 32)(9 36 17)(10 37 18)(11 38 19)(12 39 20)(13 40 21)(14 41 22)(15 42 23)(16 43 24)
(1 9)(2 10)(3 11)(4 12)(5 13)(6 14)(7 15)(8 16)(17 44)(18 45)(19 46)(20 47)(21 48)(22 33)(23 34)(24 35)(25 36)(26 37)(27 38)(28 39)(29 40)(30 41)(31 42)(32 43)
(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 8)(3 15)(4 6)(5 13)(7 11)(10 16)(12 14)(18 24)(19 31)(20 22)(21 29)(23 27)(26 32)(28 30)(33 47)(34 38)(35 45)(37 43)(39 41)(40 48)(42 46)

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

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

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

33 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 4D 6A 6B 8A 8B 8C 8D 12A 12B 16A 16B 16C 16D 16E 16F 16G 16H 24A 24B 48A 48B 48C 48D order 1 2 2 2 2 2 3 4 4 4 4 6 6 8 8 8 8 12 12 16 16 16 16 16 16 16 16 24 24 48 48 48 48 size 1 1 3 3 8 24 2 2 6 8 24 2 16 2 2 6 6 4 16 2 2 2 2 6 6 6 6 4 4 4 4 4 4

33 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 S3 D4 D4 D6 D6 D6 D8 D8 SD32 S3×D4 S3×D8 S3×SD32 kernel S3×SD32 S3×C16 C48⋊C2 D8.S3 C8.6D6 C3×SD32 S3×D8 S3×Q16 SD32 C3⋊C8 C4×S3 C16 D8 Q16 Dic3 D6 S3 C4 C2 C1 # reps 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 8 1 2 4

Matrix representation of S3×SD32 in GL4(𝔽7) generated by

 3 0 4 6 1 2 6 5 4 0 2 4 1 0 3 5
,
 0 2 3 2 5 4 4 0 2 6 1 1 3 1 5 2
,
 3 4 0 5 6 3 6 6 5 5 0 4 4 3 5 3
,
 6 0 1 1 2 6 2 5 1 0 4 3 6 0 4 5
G:=sub<GL(4,GF(7))| [3,1,4,1,0,2,0,0,4,6,2,3,6,5,4,5],[0,5,2,3,2,4,6,1,3,4,1,5,2,0,1,2],[3,6,5,4,4,3,5,3,0,6,0,5,5,6,4,3],[6,2,1,6,0,6,0,0,1,2,4,4,1,5,3,5] >;

S3×SD32 in GAP, Magma, Sage, TeX

S_3\times {\rm SD}_{32}
% in TeX

G:=Group("S3xSD32");
// GroupNames label

G:=SmallGroup(192,472);
// by ID

G=gap.SmallGroup(192,472);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,135,184,346,185,192,851,438,102,6278]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^2=c^16=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^7>;
// generators/relations

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