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## G = S3×D4⋊C4order 192 = 26·3

### Direct product of S3 and D4⋊C4

Series: Derived Chief Lower central Upper central

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

Generators and relations for S3×D4⋊C4
G = < a,b,c,d,e | a3=b2=c4=d2=e4=1, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd=ece-1=c-1, ede-1=cd >

Subgroups: 712 in 202 conjugacy classes, 63 normal (37 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, S3, C6, C6, C8, C2×C4, C2×C4, D4, D4, C23, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C2×D4, C2×D4, C24, C3⋊C8, C24, C4×S3, C4×S3, D12, D12, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×D4, C22×S3, C22×S3, C22×C6, D4⋊C4, D4⋊C4, C2×C4⋊C4, C22×C8, C22×D4, S3×C8, C2×C3⋊C8, Dic3⋊C4, C4⋊Dic3, C3×C4⋊C4, C2×C24, S3×C2×C4, S3×C2×C4, C2×D12, S3×D4, S3×D4, C2×C3⋊D4, C6×D4, S3×C23, C2×D4⋊C4, C6.D8, C2.D24, D4⋊Dic3, C3×D4⋊C4, S3×C4⋊C4, S3×C2×C8, C2×S3×D4, S3×D4⋊C4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D6, C22⋊C4, D8, SD16, C22×C4, C2×D4, C4×S3, C22×S3, D4⋊C4, C2×C22⋊C4, C2×D8, C2×SD16, S3×C2×C4, S3×D4, C2×D4⋊C4, S3×C22⋊C4, S3×D8, S3×SD16, S3×D4⋊C4

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

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

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

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

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 2 4 4 4 4 type + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C4 S3 D4 D4 D4 D6 D6 D6 D8 SD16 C4×S3 S3×D4 S3×D4 S3×D8 S3×SD16 kernel S3×D4⋊C4 C6.D8 C2.D24 D4⋊Dic3 C3×D4⋊C4 S3×C4⋊C4 S3×C2×C8 C2×S3×D4 S3×D4 D4⋊C4 C4×S3 C2×Dic3 C22×S3 C4⋊C4 C2×C8 C2×D4 D6 D6 D4 C4 C22 C2 C2 # reps 1 1 1 1 1 1 1 1 8 1 2 1 1 1 1 1 4 4 4 1 1 2 2

Matrix representation of S3×D4⋊C4 in GL4(𝔽73) generated by

 1 0 0 0 0 1 0 0 0 0 0 72 0 0 1 72
,
 1 0 0 0 0 1 0 0 0 0 0 72 0 0 72 0
,
 0 72 0 0 1 0 0 0 0 0 1 0 0 0 0 1
,
 0 72 0 0 72 0 0 0 0 0 72 0 0 0 0 72
,
 16 16 0 0 16 57 0 0 0 0 27 0 0 0 0 27
G:=sub<GL(4,GF(73))| [1,0,0,0,0,1,0,0,0,0,0,1,0,0,72,72],[1,0,0,0,0,1,0,0,0,0,0,72,0,0,72,0],[0,1,0,0,72,0,0,0,0,0,1,0,0,0,0,1],[0,72,0,0,72,0,0,0,0,0,72,0,0,0,0,72],[16,16,0,0,16,57,0,0,0,0,27,0,0,0,0,27] >;

S3×D4⋊C4 in GAP, Magma, Sage, TeX

S_3\times D_4\rtimes C_4
% in TeX

G:=Group("S3xD4:C4");
// GroupNames label

G:=SmallGroup(192,328);
// by ID

G=gap.SmallGroup(192,328);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,219,58,851,438,102,6278]);
// Polycyclic

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

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