Copied to
clipboard

## G = C3×S3×C22⋊C4order 288 = 25·32

### Direct product of C3, S3 and C22⋊C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C3×S3×C22⋊C4
 Chief series C1 — C3 — C6 — C2×C6 — C62 — S3×C2×C6 — S3×C22×C6 — C3×S3×C22⋊C4
 Lower central C3 — C6 — C3×S3×C22⋊C4
 Upper central C1 — C2×C6 — C3×C22⋊C4

Generators and relations for C3×S3×C22⋊C4
G = < a,b,c,d,e,f | a3=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: 706 in 281 conjugacy classes, 98 normal (30 characteristic)
C1, C2, C2 [×2], C2 [×8], C3 [×2], C3, C4 [×4], C22, C22 [×2], C22 [×20], S3 [×4], S3 [×2], C6 [×2], C6 [×4], C6 [×15], C2×C4 [×2], C2×C4 [×6], C23, C23 [×10], C32, Dic3 [×2], C12 [×8], D6 [×8], D6 [×10], C2×C6 [×2], C2×C6 [×4], C2×C6 [×27], C22⋊C4, C22⋊C4 [×3], C22×C4 [×2], C24, C3×S3 [×4], C3×S3 [×2], C3×C6, C3×C6 [×2], C3×C6 [×2], C4×S3 [×4], C2×Dic3 [×2], C2×C12 [×4], C2×C12 [×8], C22×S3 [×2], C22×S3 [×4], C22×S3 [×4], C22×C6 [×2], C22×C6 [×11], C2×C22⋊C4, C3×Dic3 [×2], C3×C12 [×2], S3×C6 [×8], S3×C6 [×10], C62, C62 [×2], C62 [×2], D6⋊C4 [×2], C6.D4, C3×C22⋊C4 [×2], C3×C22⋊C4 [×4], S3×C2×C4 [×2], C22×C12 [×2], S3×C23, C23×C6, S3×C12 [×4], C6×Dic3 [×2], C6×C12 [×2], S3×C2×C6 [×2], S3×C2×C6 [×4], S3×C2×C6 [×4], C2×C62, S3×C22⋊C4, C6×C22⋊C4, C3×D6⋊C4 [×2], C3×C6.D4, C32×C22⋊C4, S3×C2×C12 [×2], S3×C22×C6, C3×S3×C22⋊C4
Quotients: C1, C2 [×7], C3, C4 [×4], C22 [×7], S3, C6 [×7], C2×C4 [×6], D4 [×4], C23, C12 [×4], D6 [×3], C2×C6 [×7], C22⋊C4 [×4], C22×C4, C2×D4 [×2], C3×S3, C4×S3 [×2], C2×C12 [×6], C3×D4 [×4], C22×S3, C22×C6, C2×C22⋊C4, S3×C6 [×3], C3×C22⋊C4 [×4], S3×C2×C4, S3×D4 [×2], C22×C12, C6×D4 [×2], S3×C12 [×2], S3×C2×C6, S3×C22⋊C4, C6×C22⋊C4, S3×C2×C12, C3×S3×D4 [×2], C3×S3×C22⋊C4

Smallest permutation representation of C3×S3×C22⋊C4
On 48 points
Generators in S48
(1 37 23)(2 38 24)(3 39 21)(4 40 22)(5 36 46)(6 33 47)(7 34 48)(8 35 45)(9 29 27)(10 30 28)(11 31 25)(12 32 26)(13 43 17)(14 44 18)(15 41 19)(16 42 20)
(1 37 23)(2 38 24)(3 39 21)(4 40 22)(5 46 36)(6 47 33)(7 48 34)(8 45 35)(9 29 27)(10 30 28)(11 31 25)(12 32 26)(13 17 43)(14 18 44)(15 19 41)(16 20 42)
(1 47)(2 48)(3 45)(4 46)(5 40)(6 37)(7 38)(8 39)(9 44)(10 41)(11 42)(12 43)(13 26)(14 27)(15 28)(16 25)(17 32)(18 29)(19 30)(20 31)(21 35)(22 36)(23 33)(24 34)
(2 28)(4 26)(5 43)(7 41)(10 38)(12 40)(13 46)(15 48)(17 36)(19 34)(22 32)(24 30)
(1 27)(2 28)(3 25)(4 26)(5 43)(6 44)(7 41)(8 42)(9 37)(10 38)(11 39)(12 40)(13 46)(14 47)(15 48)(16 45)(17 36)(18 33)(19 34)(20 35)(21 31)(22 32)(23 29)(24 30)
(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)| (1,37,23)(2,38,24)(3,39,21)(4,40,22)(5,36,46)(6,33,47)(7,34,48)(8,35,45)(9,29,27)(10,30,28)(11,31,25)(12,32,26)(13,43,17)(14,44,18)(15,41,19)(16,42,20), (1,37,23)(2,38,24)(3,39,21)(4,40,22)(5,46,36)(6,47,33)(7,48,34)(8,45,35)(9,29,27)(10,30,28)(11,31,25)(12,32,26)(13,17,43)(14,18,44)(15,19,41)(16,20,42), (1,47)(2,48)(3,45)(4,46)(5,40)(6,37)(7,38)(8,39)(9,44)(10,41)(11,42)(12,43)(13,26)(14,27)(15,28)(16,25)(17,32)(18,29)(19,30)(20,31)(21,35)(22,36)(23,33)(24,34), (2,28)(4,26)(5,43)(7,41)(10,38)(12,40)(13,46)(15,48)(17,36)(19,34)(22,32)(24,30), (1,27)(2,28)(3,25)(4,26)(5,43)(6,44)(7,41)(8,42)(9,37)(10,38)(11,39)(12,40)(13,46)(14,47)(15,48)(16,45)(17,36)(18,33)(19,34)(20,35)(21,31)(22,32)(23,29)(24,30), (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( (1,37,23)(2,38,24)(3,39,21)(4,40,22)(5,36,46)(6,33,47)(7,34,48)(8,35,45)(9,29,27)(10,30,28)(11,31,25)(12,32,26)(13,43,17)(14,44,18)(15,41,19)(16,42,20), (1,37,23)(2,38,24)(3,39,21)(4,40,22)(5,46,36)(6,47,33)(7,48,34)(8,45,35)(9,29,27)(10,30,28)(11,31,25)(12,32,26)(13,17,43)(14,18,44)(15,19,41)(16,20,42), (1,47)(2,48)(3,45)(4,46)(5,40)(6,37)(7,38)(8,39)(9,44)(10,41)(11,42)(12,43)(13,26)(14,27)(15,28)(16,25)(17,32)(18,29)(19,30)(20,31)(21,35)(22,36)(23,33)(24,34), (2,28)(4,26)(5,43)(7,41)(10,38)(12,40)(13,46)(15,48)(17,36)(19,34)(22,32)(24,30), (1,27)(2,28)(3,25)(4,26)(5,43)(6,44)(7,41)(8,42)(9,37)(10,38)(11,39)(12,40)(13,46)(14,47)(15,48)(16,45)(17,36)(18,33)(19,34)(20,35)(21,31)(22,32)(23,29)(24,30), (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([(1,37,23),(2,38,24),(3,39,21),(4,40,22),(5,36,46),(6,33,47),(7,34,48),(8,35,45),(9,29,27),(10,30,28),(11,31,25),(12,32,26),(13,43,17),(14,44,18),(15,41,19),(16,42,20)], [(1,37,23),(2,38,24),(3,39,21),(4,40,22),(5,46,36),(6,47,33),(7,48,34),(8,45,35),(9,29,27),(10,30,28),(11,31,25),(12,32,26),(13,17,43),(14,18,44),(15,19,41),(16,20,42)], [(1,47),(2,48),(3,45),(4,46),(5,40),(6,37),(7,38),(8,39),(9,44),(10,41),(11,42),(12,43),(13,26),(14,27),(15,28),(16,25),(17,32),(18,29),(19,30),(20,31),(21,35),(22,36),(23,33),(24,34)], [(2,28),(4,26),(5,43),(7,41),(10,38),(12,40),(13,46),(15,48),(17,36),(19,34),(22,32),(24,30)], [(1,27),(2,28),(3,25),(4,26),(5,43),(6,44),(7,41),(8,42),(9,37),(10,38),(11,39),(12,40),(13,46),(14,47),(15,48),(16,45),(17,36),(18,33),(19,34),(20,35),(21,31),(22,32),(23,29),(24,30)], [(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)])

90 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 3A 3B 3C 3D 3E 4A 4B 4C 4D 4E 4F 4G 4H 6A ··· 6F 6G ··· 6S 6T ··· 6AA 6AB ··· 6AG 6AH 6AI 6AJ 6AK 12A ··· 12H 12I ··· 12T 12U ··· 12AB order 1 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 4 4 4 4 4 4 4 4 6 ··· 6 6 ··· 6 6 ··· 6 6 ··· 6 6 6 6 6 12 ··· 12 12 ··· 12 12 ··· 12 size 1 1 1 1 2 2 3 3 3 3 6 6 1 1 2 2 2 2 2 2 2 6 6 6 6 1 ··· 1 2 ··· 2 3 ··· 3 4 ··· 4 6 6 6 6 2 ··· 2 4 ··· 4 6 ··· 6

90 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C3 C4 C6 C6 C6 C6 C6 C12 S3 D4 D6 D6 C3×S3 C3×D4 C4×S3 S3×C6 S3×C6 S3×C12 S3×D4 C3×S3×D4 kernel C3×S3×C22⋊C4 C3×D6⋊C4 C3×C6.D4 C32×C22⋊C4 S3×C2×C12 S3×C22×C6 S3×C22⋊C4 S3×C2×C6 D6⋊C4 C6.D4 C3×C22⋊C4 S3×C2×C4 S3×C23 C22×S3 C3×C22⋊C4 S3×C6 C2×C12 C22×C6 C22⋊C4 D6 C2×C6 C2×C4 C23 C22 C6 C2 # reps 1 2 1 1 2 1 2 8 4 2 2 4 2 16 1 4 2 1 2 8 4 4 2 8 2 4

Matrix representation of C3×S3×C22⋊C4 in GL4(𝔽13) generated by

 9 0 0 0 0 9 0 0 0 0 1 0 0 0 0 1
,
 9 0 0 0 0 3 0 0 0 0 1 0 0 0 0 1
,
 0 1 0 0 1 0 0 0 0 0 12 0 0 0 0 12
,
 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 12
,
 1 0 0 0 0 1 0 0 0 0 12 0 0 0 0 12
,
 8 0 0 0 0 8 0 0 0 0 0 11 0 0 6 0
G:=sub<GL(4,GF(13))| [9,0,0,0,0,9,0,0,0,0,1,0,0,0,0,1],[9,0,0,0,0,3,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,12,0,0,0,0,12],[1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,12],[1,0,0,0,0,1,0,0,0,0,12,0,0,0,0,12],[8,0,0,0,0,8,0,0,0,0,0,6,0,0,11,0] >;

C3×S3×C22⋊C4 in GAP, Magma, Sage, TeX

C_3\times S_3\times C_2^2\rtimes C_4
% in TeX

G:=Group("C3xS3xC2^2:C4");
// GroupNames label

G:=SmallGroup(288,651);
// by ID

G=gap.SmallGroup(288,651);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,555,142,9414]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^3=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

׿
×
𝔽