Copied to
clipboard

## G = C2×C33⋊9D4order 432 = 24·33

### Direct product of C2 and C33⋊9D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32×C6 — C2×C33⋊9D4
 Chief series C1 — C3 — C32 — C33 — C32×C6 — C6×C3⋊S3 — C33⋊9D4 — C2×C33⋊9D4
 Lower central C33 — C32×C6 — C2×C33⋊9D4
 Upper central C1 — C22

Generators and relations for C2×C339D4
G = < a,b,c,d,e,f | a2=b3=c3=d3=e4=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, fbf=b-1, cd=dc, ece-1=c-1, cf=fc, ede-1=fdf=d-1, fef=e-1 >

Subgroups: 1560 in 306 conjugacy classes, 63 normal (15 characteristic)
C1, C2, C2, C2, C3, C3, C3, C4, C22, C22, S3, C6, C6, C6, C2×C4, D4, C23, C32, C32, C32, Dic3, C12, D6, C2×C6, C2×C6, C2×C6, C2×D4, C3×S3, C3⋊S3, C3×C6, C3×C6, C3×C6, D12, C2×Dic3, C3⋊D4, C2×C12, C22×S3, C22×C6, C33, C3×Dic3, C3⋊Dic3, S3×C6, C2×C3⋊S3, C2×C3⋊S3, C62, C62, C62, C2×D12, C2×C3⋊D4, C3×C3⋊S3, C32×C6, C32×C6, D6⋊S3, C3⋊D12, C6×Dic3, C2×C3⋊Dic3, S3×C2×C6, C22×C3⋊S3, C3×C3⋊Dic3, C6×C3⋊S3, C6×C3⋊S3, C3×C62, C2×D6⋊S3, C2×C3⋊D12, C339D4, C6×C3⋊Dic3, C2×C6×C3⋊S3, C2×C339D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, D12, C3⋊D4, C22×S3, S32, C2×D12, C2×C3⋊D4, D6⋊S3, C3⋊D12, C2×S32, C324D6, C2×D6⋊S3, C2×C3⋊D12, C339D4, C2×C324D6, C2×C339D4

Smallest permutation representation of C2×C339D4
On 48 points
Generators in S48
(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)(17 40)(18 37)(19 38)(20 39)(21 43)(22 44)(23 41)(24 42)(25 48)(26 45)(27 46)(28 47)(29 34)(30 35)(31 36)(32 33)
(1 44 45)(2 41 46)(3 42 47)(4 43 48)(5 22 26)(6 23 27)(7 24 28)(8 21 25)(9 32 37)(10 29 38)(11 30 39)(12 31 40)(13 33 18)(14 34 19)(15 35 20)(16 36 17)
(1 45 44)(2 41 46)(3 47 42)(4 43 48)(5 26 22)(6 23 27)(7 28 24)(8 21 25)(9 37 32)(10 29 38)(11 39 30)(12 31 40)(13 18 33)(14 34 19)(15 20 35)(16 36 17)
(1 44 45)(2 46 41)(3 42 47)(4 48 43)(5 22 26)(6 27 23)(7 24 28)(8 25 21)(9 37 32)(10 29 38)(11 39 30)(12 31 40)(13 18 33)(14 34 19)(15 20 35)(16 36 17)
(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 34)(2 33)(3 36)(4 35)(5 29)(6 32)(7 31)(8 30)(9 23)(10 22)(11 21)(12 24)(13 41)(14 44)(15 43)(16 42)(17 47)(18 46)(19 45)(20 48)(25 39)(26 38)(27 37)(28 40)

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

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

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

54 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 3D ··· 3H 4A 4B 6A ··· 6I 6J ··· 6X 6Y ··· 6AF 12A 12B 12C 12D order 1 2 2 2 2 2 2 2 3 3 3 3 ··· 3 4 4 6 ··· 6 6 ··· 6 6 ··· 6 12 12 12 12 size 1 1 1 1 18 18 18 18 2 2 2 4 ··· 4 18 18 2 ··· 2 4 ··· 4 18 ··· 18 18 18 18 18

54 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 type + + + + + + + + + + + + - + + image C1 C2 C2 C2 S3 S3 D4 D6 D6 D6 D12 C3⋊D4 S32 D6⋊S3 C3⋊D12 C2×S32 C32⋊4D6 C33⋊9D4 C2×C32⋊4D6 kernel C2×C33⋊9D4 C33⋊9D4 C6×C3⋊Dic3 C2×C6×C3⋊S3 C2×C3⋊Dic3 C22×C3⋊S3 C32×C6 C3⋊Dic3 C2×C3⋊S3 C62 C3×C6 C3×C6 C2×C6 C6 C6 C6 C22 C2 C2 # reps 1 4 1 2 1 2 2 2 4 3 4 8 3 2 4 3 2 4 2

Matrix representation of C2×C339D4 in GL6(𝔽13)

 12 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 0 0 0 0 0 0 12
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 12 0 0 0 0 1 12 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 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
,
 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 1 0 0 0 0 12 0
,
 1 0 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 4 11 0 0 0 0 2 9
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 12 0 0 0 0 12 0

G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,12,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,12,0,0,0,0,1,0],[1,12,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,2,0,0,0,0,11,9],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,12,0] >;

C2×C339D4 in GAP, Magma, Sage, TeX

C_2\times C_3^3\rtimes_9D_4
% in TeX

G:=Group("C2xC3^3:9D4");
// GroupNames label

G:=SmallGroup(432,694);
// by ID

G=gap.SmallGroup(432,694);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,141,1124,571,2028,14118]);
// Polycyclic

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

׿
×
𝔽