Copied to
clipboard

## G = C3×S3×C4○D4order 288 = 25·32

### Direct product of C3, S3 and C4○D4

Series: Derived Chief Lower central Upper central

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

Generators and relations for C3×S3×C4○D4
G = < a,b,c,d,e,f | a3=b3=c2=d4=f2=1, e2=d2, 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, de=ed, df=fd, fef=d2e >

Subgroups: 706 in 348 conjugacy classes, 174 normal (32 characteristic)
C1, C2, C2 [×8], C3 [×2], C3, C4, C4 [×3], C4 [×4], C22 [×3], C22 [×10], S3 [×2], S3 [×3], C6 [×2], C6 [×15], C2×C4 [×3], C2×C4 [×13], D4 [×3], D4 [×9], Q8, Q8 [×3], C23 [×3], C32, Dic3, Dic3 [×3], C12 [×2], C12 [×6], C12 [×8], D6, D6 [×3], D6 [×6], C2×C6 [×6], C2×C6 [×13], C22×C4 [×3], C2×D4 [×3], C2×Q8, C4○D4, C4○D4 [×7], C3×S3 [×2], C3×S3 [×3], C3×C6, C3×C6 [×3], Dic6 [×3], C4×S3, C4×S3 [×9], D12 [×3], C2×Dic3 [×3], C3⋊D4 [×6], C2×C12 [×6], C2×C12 [×16], C3×D4 [×6], C3×D4 [×12], C3×Q8 [×2], C3×Q8 [×4], C22×S3 [×3], C22×C6 [×3], C2×C4○D4, C3×Dic3, C3×Dic3 [×3], C3×C12, C3×C12 [×3], S3×C6, S3×C6 [×3], S3×C6 [×6], C62 [×3], S3×C2×C4 [×3], C4○D12 [×3], S3×D4 [×3], D42S3 [×3], S3×Q8, Q83S3, C22×C12 [×3], C6×D4 [×3], C6×Q8, C3×C4○D4 [×2], C3×C4○D4 [×8], C3×Dic6 [×3], S3×C12, S3×C12 [×9], C3×D12 [×3], C6×Dic3 [×3], C3×C3⋊D4 [×6], C6×C12 [×3], D4×C32 [×3], Q8×C32, S3×C2×C6 [×3], S3×C4○D4, C6×C4○D4, S3×C2×C12 [×3], C3×C4○D12 [×3], C3×S3×D4 [×3], C3×D42S3 [×3], C3×S3×Q8, C3×Q83S3, C32×C4○D4, C3×S3×C4○D4
Quotients: C1, C2 [×15], C3, C22 [×35], S3, C6 [×15], C23 [×15], D6 [×7], C2×C6 [×35], C4○D4 [×2], C24, C3×S3, C22×S3 [×7], C22×C6 [×15], C2×C4○D4, S3×C6 [×7], C3×C4○D4 [×2], S3×C23, C23×C6, S3×C2×C6 [×7], S3×C4○D4, C6×C4○D4, S3×C22×C6, C3×S3×C4○D4

Smallest permutation representation of C3×S3×C4○D4
On 48 points
Generators in S48
(1 6 19)(2 7 20)(3 8 17)(4 5 18)(9 23 25)(10 24 26)(11 21 27)(12 22 28)(13 43 47)(14 44 48)(15 41 45)(16 42 46)(29 35 37)(30 36 38)(31 33 39)(32 34 40)
(1 6 19)(2 7 20)(3 8 17)(4 5 18)(9 23 25)(10 24 26)(11 21 27)(12 22 28)(13 47 43)(14 48 44)(15 45 41)(16 46 42)(29 37 35)(30 38 36)(31 39 33)(32 40 34)
(1 35)(2 36)(3 33)(4 34)(5 40)(6 37)(7 38)(8 39)(9 43)(10 44)(11 41)(12 42)(13 25)(14 26)(15 27)(16 28)(17 31)(18 32)(19 29)(20 30)(21 45)(22 46)(23 47)(24 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 2 3 4)(5 6 7 8)(9 12 11 10)(13 16 15 14)(17 18 19 20)(21 24 23 22)(25 28 27 26)(29 30 31 32)(33 34 35 36)(37 38 39 40)(41 44 43 42)(45 48 47 46)
(1 11)(2 12)(3 9)(4 10)(5 24)(6 21)(7 22)(8 23)(13 31)(14 32)(15 29)(16 30)(17 25)(18 26)(19 27)(20 28)(33 43)(34 44)(35 41)(36 42)(37 45)(38 46)(39 47)(40 48)

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

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

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

90 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 3A 3B 3C 3D 3E 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 6A 6B 6C ··· 6K 6L 6M 6N 6O 6P ··· 6X 6Y ··· 6AD 12A 12B 12C 12D 12E ··· 12P 12Q 12R 12S 12T 12U ··· 12AC 12AD ··· 12AI order 1 2 2 2 2 2 2 2 2 2 3 3 3 3 3 4 4 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 12 12 12 12 12 ··· 12 12 ··· 12 size 1 1 2 2 2 3 3 6 6 6 1 1 2 2 2 1 1 2 2 2 3 3 6 6 6 1 1 2 ··· 2 3 3 3 3 4 ··· 4 6 ··· 6 1 1 1 1 2 ··· 2 3 3 3 3 4 ··· 4 6 ··· 6

90 irreducible representations

 dim 1 1 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 C2 C2 C3 C6 C6 C6 C6 C6 C6 C6 S3 D6 D6 D6 C4○D4 C3×S3 S3×C6 S3×C6 S3×C6 C3×C4○D4 S3×C4○D4 C3×S3×C4○D4 kernel C3×S3×C4○D4 S3×C2×C12 C3×C4○D12 C3×S3×D4 C3×D4⋊2S3 C3×S3×Q8 C3×Q8⋊3S3 C32×C4○D4 S3×C4○D4 S3×C2×C4 C4○D12 S3×D4 D4⋊2S3 S3×Q8 Q8⋊3S3 C3×C4○D4 C3×C4○D4 C2×C12 C3×D4 C3×Q8 C3×S3 C4○D4 C2×C4 D4 Q8 S3 C3 C1 # reps 1 3 3 3 3 1 1 1 2 6 6 6 6 2 2 2 1 3 3 1 4 2 6 6 2 8 2 4

Matrix representation of C3×S3×C4○D4 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 8 0 0 0 0 8
,
 1 0 0 0 0 1 0 0 0 0 8 0 0 0 3 5
,
 1 0 0 0 0 1 0 0 0 0 12 1 0 0 0 1
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],[1,0,0,0,0,1,0,0,0,0,8,0,0,0,0,8],[1,0,0,0,0,1,0,0,0,0,8,3,0,0,0,5],[1,0,0,0,0,1,0,0,0,0,12,0,0,0,1,1] >;

C3×S3×C4○D4 in GAP, Magma, Sage, TeX

C_3\times S_3\times C_4\circ D_4
% in TeX

G:=Group("C3xS3xC4oD4");
// GroupNames label

G:=SmallGroup(288,998);
// by ID

G=gap.SmallGroup(288,998);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-3,268,794,9414]);
// Polycyclic

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

׿
×
𝔽