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## G = C2×S3×SD16order 192 = 26·3

### Direct product of C2, S3 and SD16

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C2×S3×SD16
 Chief series C1 — C3 — C6 — C12 — C4×S3 — S3×C2×C4 — C2×S3×D4 — C2×S3×SD16
 Lower central C3 — C6 — C12 — C2×S3×SD16
 Upper central C1 — C22 — C2×C4 — C2×SD16

Generators and relations for C2×S3×SD16
G = < a,b,c,d,e | a2=b3=c2=d8=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d3 >

Subgroups: 888 in 298 conjugacy classes, 111 normal (33 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, S3, S3, C6, C6, C6, C8, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C2×C8, C2×C8, SD16, SD16, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C24, C3⋊C8, C24, 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, C22×S3, C22×S3, C22×C6, C22×C8, C2×SD16, C2×SD16, C22×D4, C22×Q8, S3×C8, C24⋊C2, C2×C3⋊C8, D4.S3, Q82S3, C2×C24, C3×SD16, C2×Dic6, C2×Dic6, S3×C2×C4, S3×C2×C4, C2×D12, S3×D4, S3×D4, S3×Q8, S3×Q8, C2×C3⋊D4, C6×D4, C6×Q8, S3×C23, C22×SD16, S3×C2×C8, C2×C24⋊C2, S3×SD16, C2×D4.S3, C2×Q82S3, C6×SD16, C2×S3×D4, C2×S3×Q8, C2×S3×SD16
Quotients: C1, C2, C22, S3, D4, C23, D6, SD16, C2×D4, C24, C22×S3, C2×SD16, C22×D4, S3×D4, S3×C23, C22×SD16, S3×SD16, C2×S3×D4, C2×S3×SD16

Smallest permutation representation of C2×S3×SD16
On 48 points
Generators in S48
(1 15)(2 16)(3 9)(4 10)(5 11)(6 12)(7 13)(8 14)(17 27)(18 28)(19 29)(20 30)(21 31)(22 32)(23 25)(24 26)(33 47)(34 48)(35 41)(36 42)(37 43)(38 44)(39 45)(40 46)
(1 36 29)(2 37 30)(3 38 31)(4 39 32)(5 40 25)(6 33 26)(7 34 27)(8 35 28)(9 44 21)(10 45 22)(11 46 23)(12 47 24)(13 48 17)(14 41 18)(15 42 19)(16 43 20)
(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 9)(8 10)(17 38)(18 39)(19 40)(20 33)(21 34)(22 35)(23 36)(24 37)(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)
(1 15)(2 10)(3 13)(4 16)(5 11)(6 14)(7 9)(8 12)(17 31)(18 26)(19 29)(20 32)(21 27)(22 30)(23 25)(24 28)(33 41)(34 44)(35 47)(36 42)(37 45)(38 48)(39 43)(40 46)

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

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

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

42 conjugacy classes

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

42 irreducible representations

 dim 1 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 C2 S3 D4 D4 D4 D6 D6 D6 D6 SD16 S3×D4 S3×D4 S3×SD16 kernel C2×S3×SD16 S3×C2×C8 C2×C24⋊C2 S3×SD16 C2×D4.S3 C2×Q8⋊2S3 C6×SD16 C2×S3×D4 C2×S3×Q8 C2×SD16 C4×S3 C2×Dic3 C22×S3 C2×C8 SD16 C2×D4 C2×Q8 D6 C4 C22 C2 # reps 1 1 1 8 1 1 1 1 1 1 2 1 1 1 4 1 1 8 1 1 4

Matrix representation of C2×S3×SD16 in GL6(𝔽73)

 72 0 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 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 72 0 0 0 0 1 72
,
 72 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 0 0 1 0 0 0 0 1 0
,
 67 67 0 0 0 0 6 67 0 0 0 0 0 0 67 6 0 0 0 0 67 67 0 0 0 0 0 0 72 0 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 72 0 0 0 0 0 0 1 0 0 0 0 0 0 1

G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,72,72],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[67,6,0,0,0,0,67,67,0,0,0,0,0,0,67,67,0,0,0,0,6,67,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

C2×S3×SD16 in GAP, Magma, Sage, TeX

C_2\times S_3\times {\rm SD}_{16}
% in TeX

G:=Group("C2xS3xSD16");
// GroupNames label

G:=SmallGroup(192,1317);
// by ID

G=gap.SmallGroup(192,1317);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,185,136,438,235,102,6278]);
// Polycyclic

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

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