direct product, metabelian, soluble, monomial, A-group
Aliases: C4×C33⋊C4, C33⋊3C42, C12⋊2(C32⋊C4), (C32×C12)⋊3C4, (C3×C12)⋊5Dic3, C3⋊Dic3⋊6Dic3, C32⋊5(C4×Dic3), C3⋊1(C4×C32⋊C4), C3⋊S3.7(C4×S3), (C4×C3⋊S3).11S3, (C3×C3⋊Dic3)⋊8C4, (C2×C3⋊S3).39D6, C6.10(C2×C32⋊C4), (C12×C3⋊S3).15C2, C2.2(C2×C33⋊C4), (C6×C3⋊S3).41C22, (C32×C6).17(C2×C4), (C2×C33⋊C4).8C2, (C3×C6).24(C2×Dic3), (C3×C3⋊S3).16(C2×C4), SmallGroup(432,637)
Series: Derived ►Chief ►Lower central ►Upper central
| C33 — C4×C33⋊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
►On 48 points
(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