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G = S3×C42⋊2C2order 192 = 26·3

Direct product of S3 and C42⋊2C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — S3×C42⋊2C2
 Chief series C1 — C3 — C6 — C2×C6 — C22×S3 — S3×C23 — S3×C22⋊C4 — S3×C42⋊2C2
 Lower central C3 — C2×C6 — S3×C42⋊2C2
 Upper central C1 — C22 — C42⋊2C2

Generators and relations for S3×C422C2
G = < a,b,c,d,e | a3=b2=c4=d4=e2=1, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece=cd2, ede=c2d-1 >

Subgroups: 624 in 246 conjugacy classes, 101 normal (16 characteristic)
C1, C2, C2, C3, C4, C22, C22, S3, S3, C6, C6, C2×C4, C2×C4, C23, C23, Dic3, C12, D6, D6, C2×C6, C2×C6, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C24, C4×S3, C2×Dic3, C2×C12, C22×S3, C22×S3, C22×C6, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C422C2, C422C2, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C6.D4, C4×C12, C3×C22⋊C4, C3×C4⋊C4, S3×C2×C4, S3×C23, C2×C422C2, S3×C42, C423S3, C23.8D6, S3×C22⋊C4, S3×C4⋊C4, C4⋊C4⋊S3, C3×C422C2, S3×C422C2
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C24, C22×S3, C422C2, C2×C4○D4, S3×C23, C2×C422C2, S3×C4○D4, S3×C422C2

Smallest permutation representation of S3×C422C2
On 48 points
Generators in S48
(1 24 27)(2 21 28)(3 22 25)(4 23 26)(5 43 32)(6 44 29)(7 41 30)(8 42 31)(9 47 37)(10 48 38)(11 45 39)(12 46 40)(13 33 18)(14 34 19)(15 35 20)(16 36 17)
(1 11)(2 12)(3 9)(4 10)(5 15)(6 16)(7 13)(8 14)(17 44)(18 41)(19 42)(20 43)(21 40)(22 37)(23 38)(24 39)(25 47)(26 48)(27 45)(28 46)(29 36)(30 33)(31 34)(32 35)
(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 35 9 30)(2 36 10 31)(3 33 11 32)(4 34 12 29)(5 22 18 45)(6 23 19 46)(7 24 20 47)(8 21 17 48)(13 39 43 25)(14 40 44 26)(15 37 41 27)(16 38 42 28)
(2 10)(4 12)(5 20)(6 8)(7 18)(13 41)(14 16)(15 43)(17 19)(21 48)(23 46)(26 40)(28 38)(29 31)(30 33)(32 35)(34 36)(42 44)

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

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

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

42 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 3 4A ··· 4F 4G 4H 4I 4J ··· 4O 4P 4Q 4R 6A 6B 6C 6D 12A ··· 12F 12G 12H 12I order 1 2 2 2 2 2 2 2 2 2 3 4 ··· 4 4 4 4 4 ··· 4 4 4 4 6 6 6 6 12 ··· 12 12 12 12 size 1 1 1 1 3 3 3 3 4 12 2 2 ··· 2 4 4 4 6 ··· 6 12 12 12 2 2 2 8 4 ··· 4 8 8 8

42 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 4 type + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 S3 D6 D6 D6 C4○D4 S3×C4○D4 kernel S3×C42⋊2C2 S3×C42 C42⋊3S3 C23.8D6 S3×C22⋊C4 S3×C4⋊C4 C4⋊C4⋊S3 C3×C42⋊2C2 C42⋊2C2 C42 C22⋊C4 C4⋊C4 D6 C2 # reps 1 1 1 3 3 3 3 1 1 1 3 3 12 6

Matrix representation of S3×C422C2 in GL6(𝔽13)

 0 12 0 0 0 0 1 12 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
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 8 3 0 0 0 0 0 5 0 0 0 0 0 0 8 0 0 0 0 0 0 8
,
 12 0 0 0 0 0 0 12 0 0 0 0 0 0 5 0 0 0 0 0 0 5 0 0 0 0 0 0 5 8 0 0 0 0 10 8
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 12 12 0 0 0 0 0 0 1 0 0 0 0 0 2 12

G:=sub<GL(6,GF(13))| [0,1,0,0,0,0,12,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,8,0,0,0,0,0,3,5,0,0,0,0,0,0,8,0,0,0,0,0,0,8],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,5,0,0,0,0,0,0,5,0,0,0,0,0,0,5,10,0,0,0,0,8,8],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,12,0,0,0,0,0,12,0,0,0,0,0,0,1,2,0,0,0,0,0,12] >;

S3×C422C2 in GAP, Magma, Sage, TeX

S_3\times C_4^2\rtimes_2C_2
% in TeX

G:=Group("S3xC4^2:2C2");
// GroupNames label

G:=SmallGroup(192,1262);
// by ID

G=gap.SmallGroup(192,1262);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,387,100,346,297,136,6278]);
// Polycyclic

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

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