non-abelian, soluble, monomial
Aliases: C6.14S3≀C2, C33⋊3(C4⋊C4), C33⋊C4⋊2C4, C3⋊S3.1Dic6, (C32×C6).8D4, C6.D6.2S3, C2.3(C33⋊D4), C32⋊3(Dic3⋊C4), C3⋊S3.3(C4×S3), (C3×C3⋊S3).3Q8, (C2×C3⋊S3).12D6, C3⋊1(C3⋊S3.Q8), C33⋊9(C2×C4).3C2, (C6×C3⋊S3).8C22, (C3×C6).14(C3⋊D4), (C2×C33⋊C4).3C2, (C3×C6.D6).4C2, (C3×C3⋊S3).10(C2×C4), SmallGroup(432,581)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C33⋊C4⋊C4
G = < a,b,c,d,e | a3=b3=c3=d4=e4=1, ab=ba, ac=ca, dad-1=ab-1, ae=ea, bc=cb, dbd-1=ebe-1=a-1b-1, dcd-1=c-1, ce=ec, ede-1=d-1 >
Subgroups: 580 in 96 conjugacy classes, 21 normal (19 characteristic)
C1, C2, C2, C3, C3, C4, C22, S3, C6, C6, C2×C4, C32, C32, Dic3, C12, D6, C2×C6, C4⋊C4, C3×S3, C3⋊S3, C3×C6, C3×C6, C4×S3, C2×Dic3, C2×C12, C33, C3×Dic3, C3⋊Dic3, C3×C12, C32⋊C4, S3×C6, C2×C3⋊S3, Dic3⋊C4, C3×C3⋊S3, C32×C6, S3×Dic3, C6.D6, C6.D6, S3×C12, C2×C32⋊C4, C32×Dic3, C3×C3⋊Dic3, C33⋊C4, C6×C3⋊S3, C3⋊S3.Q8, C3×C6.D6, C33⋊9(C2×C4), C2×C33⋊C4, C33⋊C4⋊C4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, D6, C4⋊C4, Dic6, C4×S3, C3⋊D4, Dic3⋊C4, S3≀C2, C3⋊S3.Q8, C33⋊D4, C33⋊C4⋊C4
►On 48 points
(1 6 45)(2 7 46)(3 47 8)(4 48 5)(9 29 34)(10 30 35)(11 36 31)(12 33 32)(13 25 40)(14 37 26)(15 38 27)(16 28 39)(17 43 21)(18 22 44)(19 23 41)(20 42 24)
(2 46 7)(4 5 48)(10 35 30)(12 32 33)(13 40 25)(15 27 38)(17 21 43)(19 41 23)
(1 45 6)(2 7 46)(3 47 8)(4 5 48)(9 34 29)(10 30 35)(11 36 31)(12 32 33)(13 25 40)(14 37 26)(15 27 38)(16 39 28)(17 43 21)(18 22 44)(19 41 23)(20 24 42)
(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 33 28 23)(2 36 25 22)(3 35 26 21)(4 34 27 24)(5 29 38 42)(6 32 39 41)(7 31 40 44)(8 30 37 43)(9 15 20 48)(10 14 17 47)(11 13 18 46)(12 16 19 45)
G:=sub<Sym(48)| (1,6,45)(2,7,46)(3,47,8)(4,48,5)(9,29,34)(10,30,35)(11,36,31)(12,33,32)(13,25,40)(14,37,26)(15,38,27)(16,28,39)(17,43,21)(18,22,44)(19,23,41)(20,42,24), (2,46,7)(4,5,48)(10,35,30)(12,32,33)(13,40,25)(15,27,38)(17,21,43)(19,41,23), (1,45,6)(2,7,46)(3,47,8)(4,5,48)(9,34,29)(10,30,35)(11,36,31)(12,32,33)(13,25,40)(14,37,26)(15,27,38)(16,39,28)(17,43,21)(18,22,44)(19,41,23)(20,24,42), (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), (1,33,28,23)(2,36,25,22)(3,35,26,21)(4,34,27,24)(5,29,38,42)(6,32,39,41)(7,31,40,44)(8,30,37,43)(9,15,20,48)(10,14,17,47)(11,13,18,46)(12,16,19,45)>;
G:=Group( (1,6,45)(2,7,46)(3,47,8)(4,48,5)(9,29,34)(10,30,35)(11,36,31)(12,33,32)(13,25,40)(14,37,26)(15,38,27)(16,28,39)(17,43,21)(18,22,44)(19,23,41)(20,42,24), (2,46,7)(4,5,48)(10,35,30)(12,32,33)(13,40,25)(15,27,38)(17,21,43)(19,41,23), (1,45,6)(2,7,46)(3,47,8)(4,5,48)(9,34,29)(10,30,35)(11,36,31)(12,32,33)(13,25,40)(14,37,26)(15,27,38)(16,39,28)(17,43,21)(18,22,44)(19,41,23)(20,24,42), (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), (1,33,28,23)(2,36,25,22)(3,35,26,21)(4,34,27,24)(5,29,38,42)(6,32,39,41)(7,31,40,44)(8,30,37,43)(9,15,20,48)(10,14,17,47)(11,13,18,46)(12,16,19,45) );
G=PermutationGroup([[(1,6,45),(2,7,46),(3,47,8),(4,48,5),(9,29,34),(10,30,35),(11,36,31),(12,33,32),(13,25,40),(14,37,26),(15,38,27),(16,28,39),(17,43,21),(18,22,44),(19,23,41),(20,42,24)], [(2,46,7),(4,5,48),(10,35,30),(12,32,33),(13,40,25),(15,27,38),(17,21,43),(19,41,23)], [(1,45,6),(2,7,46),(3,47,8),(4,5,48),(9,34,29),(10,30,35),(11,36,31),(12,32,33),(13,25,40),(14,37,26),(15,27,38),(16,39,28),(17,43,21),(18,22,44),(19,41,23),(20,24,42)], [(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)], [(1,33,28,23),(2,36,25,22),(3,35,26,21),(4,34,27,24),(5,29,38,42),(6,32,39,41),(7,31,40,44),(8,30,37,43),(9,15,20,48),(10,14,17,47),(11,13,18,46),(12,16,19,45)]])
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 3E | 3F | 4A | 4B | 4C | 4D | 4E | 4F | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 12A | 12B | 12C | 12D | 12E | ··· | 12J | 12K | 12L |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 12 | 12 |
| size | 1 | 1 | 9 | 9 | 2 | 4 | 4 | 4 | 4 | 8 | 6 | 6 | 18 | 18 | 54 | 54 | 2 | 4 | 4 | 4 | 4 | 8 | 18 | 18 | 6 | 6 | 6 | 6 | 12 | ··· | 12 | 36 | 36 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 8 |
| type | + | + | + | + | + | - | + | + | - | + | + | - | ||||||
| image | C1 | C2 | C2 | C2 | C4 | S3 | Q8 | D4 | D6 | Dic6 | C4×S3 | C3⋊D4 | S3≀C2 | C3⋊S3.Q8 | C33⋊D4 | C33⋊C4⋊C4 | C33⋊D4 | C33⋊C4⋊C4 |
| kernel | C33⋊C4⋊C4 | C3×C6.D6 | C33⋊9(C2×C4) | C2×C33⋊C4 | C33⋊C4 | C6.D6 | C3×C3⋊S3 | C32×C6 | C2×C3⋊S3 | C3⋊S3 | C3⋊S3 | C3×C6 | C6 | C3 | C2 | C1 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 1 | 1 |
Matrix representation of C33⋊C4⋊C4 ►in GL6(𝔽13)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 12 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 12 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 | 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 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 9 | 2 | 0 | 0 | 0 | 0 |
| 11 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 0 | 0 | 1 | 0 | 1 | 0 |
| 2 | 9 | 0 | 0 | 0 | 0 |
| 4 | 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 5 | 0 | 0 |
| 0 | 0 | 5 | 0 | 0 | 0 |
| 0 | 0 | 0 | 8 | 0 | 8 |
| 0 | 0 | 8 | 0 | 8 | 0 |
G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,1,0,0,0,0,0,0,12,0,0,12,0,0,0,0,0,0,1,0,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,1,0,0,0,0,1,0,0,0,0,12,0,0],[0,1,0,0,0,0,12,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[9,11,0,0,0,0,2,4,0,0,0,0,0,0,0,12,0,1,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,12,0],[2,4,0,0,0,0,9,11,0,0,0,0,0,0,0,5,0,8,0,0,5,0,8,0,0,0,0,0,0,8,0,0,0,0,8,0] >;
C33⋊C4⋊C4 in GAP, Magma, Sage, TeX
C_3^3\rtimes C_4\rtimes C_4
% in TeX
G:=Group("C3^3:C4:C4"); // GroupNames label
G:=SmallGroup(432,581);
// by ID
G=gap.SmallGroup(432,581);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,3,-3,56,85,64,1684,571,298,677,1027,14118]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^3=c^3=d^4=e^4=1,a*b=b*a,a*c=c*a,d*a*d^-1=a*b^-1,a*e=e*a,b*c=c*b,d*b*d^-1=e*b*e^-1=a^-1*b^-1,d*c*d^-1=c^-1,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations