Copied to
clipboard

## G = C4×C33⋊C4order 432 = 24·33

### Direct product of C4 and C33⋊C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C33 — C4×C33⋊C4
 Chief series C1 — C3 — C33 — C3×C3⋊S3 — C6×C3⋊S3 — C2×C33⋊C4 — C4×C33⋊C4
 Lower central C33 — C4×C33⋊C4
 Upper central C1 — C4

Generators and relations for C4×C33⋊C4
G = < a,b,c,d,e | a4=b3=c3=d3=e4=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, ebe-1=bc-1, cd=dc, ece-1=b-1c-1, ede-1=d-1 >

Subgroups: 584 in 104 conjugacy classes, 29 normal (19 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C2×C4, C32, C32, Dic3, C12, C12, D6, C2×C6, C42, C3×S3, C3⋊S3, C3×C6, C3×C6, C4×S3, C2×Dic3, C2×C12, C33, C3×Dic3, C3⋊Dic3, C3×C12, C3×C12, C32⋊C4, S3×C6, C2×C3⋊S3, C4×Dic3, C3×C3⋊S3, C32×C6, S3×C12, C4×C3⋊S3, C2×C32⋊C4, C3×C3⋊Dic3, C32×C12, C33⋊C4, C6×C3⋊S3, C4×C32⋊C4, C12×C3⋊S3, C2×C33⋊C4, C4×C33⋊C4
Quotients: C1, C2, C4, C22, S3, C2×C4, Dic3, D6, C42, C4×S3, C2×Dic3, C32⋊C4, C4×Dic3, C2×C32⋊C4, C33⋊C4, C4×C32⋊C4, C2×C33⋊C4, C4×C33⋊C4

Smallest permutation representation of C4×C33⋊C4
On 48 points
Generators in S48
(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 5 9)(2 6 10)(3 7 11)(4 8 12)(13 43 47)(14 44 48)(15 41 45)(16 42 46)(17 21 27)(18 22 28)(19 23 25)(20 24 26)(29 38 35)(30 39 36)(31 40 33)(32 37 34)
(1 9 5)(2 10 6)(3 11 7)(4 12 8)(17 27 21)(18 28 22)(19 25 23)(20 26 24)
(1 9 5)(2 10 6)(3 11 7)(4 12 8)(13 47 43)(14 48 44)(15 45 41)(16 46 42)(17 21 27)(18 22 28)(19 23 25)(20 24 26)(29 38 35)(30 39 36)(31 40 33)(32 37 34)
(1 44 19 36)(2 41 20 33)(3 42 17 34)(4 43 18 35)(5 14 25 30)(6 15 26 31)(7 16 27 32)(8 13 28 29)(9 48 23 39)(10 45 24 40)(11 46 21 37)(12 47 22 38)

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

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

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

48 conjugacy classes

 class 1 2A 2B 2C 3A 3B ··· 3G 4A 4B 4C 4D 4E ··· 4L 6A 6B ··· 6G 6H 6I 12A 12B 12C ··· 12N 12O 12P order 1 2 2 2 3 3 ··· 3 4 4 4 4 4 ··· 4 6 6 ··· 6 6 6 12 12 12 ··· 12 12 12 size 1 1 9 9 2 4 ··· 4 1 1 9 9 27 ··· 27 2 4 ··· 4 18 18 2 2 4 ··· 4 18 18

48 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 4 4 4 4 4 4 type + + + + - - + + + image C1 C2 C2 C4 C4 C4 S3 Dic3 Dic3 D6 C4×S3 C32⋊C4 C2×C32⋊C4 C33⋊C4 C4×C32⋊C4 C2×C33⋊C4 C4×C33⋊C4 kernel C4×C33⋊C4 C12×C3⋊S3 C2×C33⋊C4 C3×C3⋊Dic3 C32×C12 C33⋊C4 C4×C3⋊S3 C3⋊Dic3 C3×C12 C2×C3⋊S3 C3⋊S3 C12 C6 C4 C3 C2 C1 # reps 1 1 2 2 2 8 1 1 1 1 4 2 2 4 4 4 8

Matrix representation of C4×C33⋊C4 in GL4(𝔽13) generated by

 8 0 0 0 0 8 0 0 0 0 8 0 0 0 0 8
,
 3 0 0 0 0 9 0 0 0 0 3 0 0 0 0 9
,
 9 0 0 0 0 3 0 0 0 0 1 0 0 0 0 1
,
 9 0 0 0 0 9 0 0 0 0 3 0 0 0 0 3
,
 0 0 5 0 0 0 0 5 0 5 0 0 5 0 0 0
G:=sub<GL(4,GF(13))| [8,0,0,0,0,8,0,0,0,0,8,0,0,0,0,8],[3,0,0,0,0,9,0,0,0,0,3,0,0,0,0,9],[9,0,0,0,0,3,0,0,0,0,1,0,0,0,0,1],[9,0,0,0,0,9,0,0,0,0,3,0,0,0,0,3],[0,0,0,5,0,0,5,0,5,0,0,0,0,5,0,0] >;

C4×C33⋊C4 in GAP, Magma, Sage, TeX

C_4\times C_3^3\rtimes C_4
% in TeX

G:=Group("C4xC3^3:C4");
// GroupNames label

G:=SmallGroup(432,637);
// by ID

G=gap.SmallGroup(432,637);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,3,-3,28,64,2804,298,2693,1027,14118]);
// Polycyclic

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

׿
×
𝔽