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G = C4xC33:C4order 432 = 24·33

Direct product of C4 and C33:C4

direct product, metabelian, soluble, monomial, A-group

Aliases: C4xC33:C4, C33:3C42, C12:2(C32:C4), (C32xC12):3C4, (C3xC12):5Dic3, C3:Dic3:6Dic3, C32:5(C4xDic3), C3:1(C4xC32:C4), C3:S3.7(C4xS3), (C4xC3:S3).11S3, (C3xC3:Dic3):8C4, (C2xC3:S3).39D6, C6.10(C2xC32:C4), (C12xC3:S3).15C2, C2.2(C2xC33:C4), (C6xC3:S3).41C22, (C32xC6).17(C2xC4), (C2xC33:C4).8C2, (C3xC6).24(C2xDic3), (C3xC3:S3).16(C2xC4), SmallGroup(432,637)

Series: Derived Chief Lower central Upper central

C1C33 — C4xC33:C4
C1C3C33C3xC3:S3C6xC3:S3C2xC33:C4 — C4xC33:C4
C33 — C4xC33:C4
C1C4

Generators and relations for C4xC33: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, C2xC4, C32, C32, Dic3, C12, C12, D6, C2xC6, C42, C3xS3, C3:S3, C3xC6, C3xC6, C4xS3, C2xDic3, C2xC12, C33, C3xDic3, C3:Dic3, C3xC12, C3xC12, C32:C4, S3xC6, C2xC3:S3, C4xDic3, C3xC3:S3, C32xC6, S3xC12, C4xC3:S3, C2xC32:C4, C3xC3:Dic3, C32xC12, C33:C4, C6xC3:S3, C4xC32:C4, C12xC3:S3, C2xC33:C4, C4xC33:C4
Quotients: C1, C2, C4, C22, S3, C2xC4, Dic3, D6, C42, C4xS3, C2xDic3, C32:C4, C4xDic3, C2xC32:C4, C33:C4, C4xC32:C4, C2xC33:C4, C4xC33:C4

Smallest permutation representation of C4xC33: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 2A2B2C3A3B···3G4A4B4C4D4E···4L6A6B···6G6H6I12A12B12C···12N12O12P
order122233···344444···466···666121212···121212
size119924···4119927···2724···41818224···41818

48 irreducible representations

dim11111122222444444
type++++--+++
imageC1C2C2C4C4C4S3Dic3Dic3D6C4xS3C32:C4C2xC32:C4C33:C4C4xC32:C4C2xC33:C4C4xC33:C4
kernelC4xC33:C4C12xC3:S3C2xC33:C4C3xC3:Dic3C32xC12C33:C4C4xC3:S3C3:Dic3C3xC12C2xC3:S3C3:S3C12C6C4C3C2C1
# reps11222811114224448

Matrix representation of C4xC33:C4 in GL4(F13) generated by

8000
0800
0080
0008
,
3000
0900
0030
0009
,
9000
0300
0010
0001
,
9000
0900
0030
0003
,
0050
0005
0500
5000
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] >;

C4xC33: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

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