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## G = S3×C4.4D4order 192 = 26·3

### Direct product of S3 and C4.4D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — S3×C4.4D4
 Chief series C1 — C3 — C6 — C2×C6 — C22×S3 — S3×C23 — S3×C22⋊C4 — S3×C4.4D4
 Lower central C3 — C2×C6 — S3×C4.4D4
 Upper central C1 — C22 — C4.4D4

Generators and relations for S3×C4.4D4
G = < a,b,c,d,e | a3=b2=c4=d4=1, e2=c2, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece-1=c-1, ede-1=c2d-1 >

Subgroups: 944 in 330 conjugacy classes, 111 normal (29 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, S3, S3, C6, C6, C6, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C42, C42, C22⋊C4, C22⋊C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C24, Dic6, C4×S3, C4×S3, D12, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×Q8, C22×S3, C22×S3, C22×S3, C22×C6, C2×C42, C2×C22⋊C4, C4.4D4, C4.4D4, C22×D4, C22×Q8, C4×Dic3, C4×Dic3, D6⋊C4, C6.D4, C4×C12, C3×C22⋊C4, C2×Dic6, C2×Dic6, S3×C2×C4, S3×C2×C4, C2×D12, S3×D4, S3×Q8, C2×C3⋊D4, C6×D4, C6×Q8, S3×C23, C2×C4.4D4, S3×C42, C427S3, S3×C22⋊C4, C23.11D6, C23.12D6, C12.23D4, C3×C4.4D4, C2×S3×D4, C2×S3×Q8, S3×C4.4D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, C24, C22×S3, C4.4D4, C22×D4, C2×C4○D4, S3×D4, S3×C23, C2×C4.4D4, C2×S3×D4, S3×C4○D4, S3×C4.4D4

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

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

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

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

42 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 3 4A ··· 4F 4G 4H 4I ··· 4N 4O 4P 6A 6B 6C 6D 6E 12A ··· 12F 12G 12H 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 12 ··· 12 12 12 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 4 ··· 4 8 8

42 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 S3 D4 D6 D6 D6 D6 C4○D4 S3×D4 S3×C4○D4 kernel S3×C4.4D4 S3×C42 C42⋊7S3 S3×C22⋊C4 C23.11D6 C23.12D6 C12.23D4 C3×C4.4D4 C2×S3×D4 C2×S3×Q8 C4.4D4 C4×S3 C42 C22⋊C4 C2×D4 C2×Q8 D6 C4 C2 # reps 1 1 1 4 4 1 1 1 1 1 1 4 1 4 1 1 8 2 4

Matrix representation of S3×C4.4D4 in GL6(𝔽13)

 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 12 0 0 0 0 1 12
,
 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 1 0
,
 0 8 0 0 0 0 8 0 0 0 0 0 0 0 1 3 0 0 0 0 8 12 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 5 0 0 0 0 0 0 5 0 0 0 0 0 0 5 2 0 0 0 0 1 8 0 0 0 0 0 0 12 0 0 0 0 0 0 12
,
 0 1 0 0 0 0 12 0 0 0 0 0 0 0 8 0 0 0 0 0 12 5 0 0 0 0 0 0 12 0 0 0 0 0 0 12

G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,12,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[0,8,0,0,0,0,8,0,0,0,0,0,0,0,1,8,0,0,0,0,3,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[5,0,0,0,0,0,0,5,0,0,0,0,0,0,5,1,0,0,0,0,2,8,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[0,12,0,0,0,0,1,0,0,0,0,0,0,0,8,12,0,0,0,0,0,5,0,0,0,0,0,0,12,0,0,0,0,0,0,12] >;

S3×C4.4D4 in GAP, Magma, Sage, TeX

S_3\times C_4._4D_4
% in TeX

G:=Group("S3xC4.4D4");
// GroupNames label

G:=SmallGroup(192,1232);
// by ID

G=gap.SmallGroup(192,1232);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,387,100,346,136,6278]);
// Polycyclic

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

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