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## G = C2×S3×C22⋊C4order 192 = 26·3

### Direct product of C2, S3 and C22⋊C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C2×S3×C22⋊C4
 Chief series C1 — C3 — C6 — C2×C6 — C22×S3 — S3×C23 — S3×C24 — C2×S3×C22⋊C4
 Lower central C3 — C6 — C2×S3×C22⋊C4
 Upper central C1 — C23 — C2×C22⋊C4

Generators and relations for C2×S3×C22⋊C4
G = < a,b,c,d,e,f | a2=b3=c2=d2=e2=f4=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, cbc=b-1, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, fdf-1=de=ed, ef=fe >

Subgroups: 1832 in 674 conjugacy classes, 207 normal (17 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, S3, S3, C6, C6, C6, C2×C4, C2×C4, C23, C23, C23, Dic3, C12, D6, D6, C2×C6, C2×C6, C2×C6, C22⋊C4, C22⋊C4, C22×C4, C22×C4, C24, C24, C4×S3, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×S3, C22×S3, C22×C6, C22×C6, C22×C6, C2×C22⋊C4, C2×C22⋊C4, C23×C4, C25, D6⋊C4, C6.D4, C3×C22⋊C4, S3×C2×C4, S3×C2×C4, C22×Dic3, C22×C12, S3×C23, S3×C23, S3×C23, C23×C6, C22×C22⋊C4, S3×C22⋊C4, C2×D6⋊C4, C2×C6.D4, C6×C22⋊C4, S3×C22×C4, S3×C24, C2×S3×C22⋊C4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D6, C22⋊C4, C22×C4, C2×D4, C24, C4×S3, C22×S3, C2×C22⋊C4, C23×C4, C22×D4, S3×C2×C4, S3×D4, S3×C23, C22×C22⋊C4, S3×C22⋊C4, S3×C22×C4, C2×S3×D4, C2×S3×C22⋊C4

Smallest permutation representation of C2×S3×C22⋊C4
On 48 points
Generators in S48
(1 27)(2 28)(3 25)(4 26)(5 40)(6 37)(7 38)(8 39)(9 36)(10 33)(11 34)(12 35)(13 29)(14 30)(15 31)(16 32)(17 44)(18 41)(19 42)(20 43)(21 48)(22 45)(23 46)(24 47)
(1 32 23)(2 29 24)(3 30 21)(4 31 22)(5 36 19)(6 33 20)(7 34 17)(8 35 18)(9 42 40)(10 43 37)(11 44 38)(12 41 39)(13 47 28)(14 48 25)(15 45 26)(16 46 27)
(1 6)(2 7)(3 8)(4 5)(9 45)(10 46)(11 47)(12 48)(13 44)(14 41)(15 42)(16 43)(17 29)(18 30)(19 31)(20 32)(21 35)(22 36)(23 33)(24 34)(25 39)(26 40)(27 37)(28 38)
(1 3)(2 38)(4 40)(5 26)(6 8)(7 28)(9 31)(10 12)(11 29)(13 34)(14 16)(15 36)(17 47)(18 20)(19 45)(21 23)(22 42)(24 44)(25 27)(30 32)(33 35)(37 39)(41 43)(46 48)
(1 39)(2 40)(3 37)(4 38)(5 28)(6 25)(7 26)(8 27)(9 29)(10 30)(11 31)(12 32)(13 36)(14 33)(15 34)(16 35)(17 45)(18 46)(19 47)(20 48)(21 43)(22 44)(23 41)(24 42)
(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)

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

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

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

60 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 2L ··· 2S 2T 2U 2V 2W 3 4A ··· 4H 4I ··· 4P 6A ··· 6G 6H 6I 6J 6K 12A ··· 12H order 1 2 ··· 2 2 2 2 2 2 ··· 2 2 2 2 2 3 4 ··· 4 4 ··· 4 6 ··· 6 6 6 6 6 12 ··· 12 size 1 1 ··· 1 2 2 2 2 3 ··· 3 6 6 6 6 2 2 ··· 2 6 ··· 6 2 ··· 2 4 4 4 4 4 ··· 4

60 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 4 type + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C4 S3 D4 D6 D6 D6 C4×S3 S3×D4 kernel C2×S3×C22⋊C4 S3×C22⋊C4 C2×D6⋊C4 C2×C6.D4 C6×C22⋊C4 S3×C22×C4 S3×C24 S3×C23 C2×C22⋊C4 C22×S3 C22⋊C4 C22×C4 C24 C23 C22 # reps 1 8 2 1 1 2 1 16 1 8 4 2 1 8 4

Matrix representation of C2×S3×C22⋊C4 in GL6(𝔽13)

 12 0 0 0 0 0 0 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
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 12 12 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 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 12 12 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 12 0 0 0 0 0 12 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 12 0 0 0 0 0 0 12
,
 8 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 11 0 0 0 0 1 12

G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,1,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,12,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,12,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[8,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,11,12] >;

C2×S3×C22⋊C4 in GAP, Magma, Sage, TeX

C_2\times S_3\times C_2^2\rtimes C_4
% in TeX

G:=Group("C2xS3xC2^2:C4");
// GroupNames label

G:=SmallGroup(192,1043);
// by ID

G=gap.SmallGroup(192,1043);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,297,80,6278]);
// Polycyclic

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

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