metabelian, supersoluble, monomial
Aliases: C33⋊10M4(2), C12.94S32, C32⋊4C8⋊16S3, (C3×C12).169D6, C6.18(S3×Dic3), C32⋊8(C8⋊S3), C3⋊Dic3.6Dic3, C3⋊2(D6.Dic3), C3⋊2(C12.31D6), C6.19(C6.D6), C4.7(C32⋊4D6), C32⋊6(C4.Dic3), (C32×C12).71C22, (C4×C3⋊S3).8S3, (C6×C3⋊S3).8C4, (C12×C3⋊S3).2C2, (C3×C6).54(C4×S3), (C2×C3⋊S3).6Dic3, (C3×C3⋊Dic3).6C4, C2.3(C33⋊9(C2×C4)), (C3×C32⋊4C8)⋊14C2, (C32×C6).45(C2×C4), (C3×C6).41(C2×Dic3), SmallGroup(432,456)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C33⋊10M4(2)
G = < a,b,c,d,e | a3=b3=c3=d8=e2=1, ab=ba, ac=ca, ad=da, eae=a-1, bc=cb, dbd-1=b-1, be=eb, dcd-1=ece=c-1, ede=d5 >
Subgroups: 456 in 118 conjugacy classes, 35 normal (19 characteristic)
C1, C2, C2, C3, C3, C3, C4, C4, C22, S3, C6, C6, C6, C8, C2×C4, C32, C32, C32, Dic3, C12, C12, C12, D6, C2×C6, M4(2), C3×S3, C3⋊S3, C3×C6, C3×C6, C3×C6, C3⋊C8, C24, C4×S3, C2×C12, C33, C3×Dic3, C3⋊Dic3, C3×C12, C3×C12, C3×C12, S3×C6, C2×C3⋊S3, C8⋊S3, C4.Dic3, C3×C3⋊S3, C32×C6, C3×C3⋊C8, C32⋊4C8, S3×C12, C4×C3⋊S3, C3×C3⋊Dic3, C32×C12, C6×C3⋊S3, D6.Dic3, C12.31D6, C3×C32⋊4C8, C12×C3⋊S3, C33⋊10M4(2)
Quotients: C1, C2, C4, C22, S3, C2×C4, Dic3, D6, M4(2), C4×S3, C2×Dic3, S32, C8⋊S3, C4.Dic3, S3×Dic3, C6.D6, C32⋊4D6, D6.Dic3, C12.31D6, C33⋊9(C2×C4), C33⋊10M4(2)
►On 48 points
(1 38 45)(2 39 46)(3 40 47)(4 33 48)(5 34 41)(6 35 42)(7 36 43)(8 37 44)(9 23 28)(10 24 29)(11 17 30)(12 18 31)(13 19 32)(14 20 25)(15 21 26)(16 22 27)
(1 38 45)(2 46 39)(3 40 47)(4 48 33)(5 34 41)(6 42 35)(7 36 43)(8 44 37)(9 23 28)(10 29 24)(11 17 30)(12 31 18)(13 19 32)(14 25 20)(15 21 26)(16 27 22)
(1 45 38)(2 39 46)(3 47 40)(4 33 48)(5 41 34)(6 35 42)(7 43 36)(8 37 44)(9 23 28)(10 29 24)(11 17 30)(12 31 18)(13 19 32)(14 25 20)(15 21 26)(16 27 22)
(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 22)(2 19)(3 24)(4 21)(5 18)(6 23)(7 20)(8 17)(9 35)(10 40)(11 37)(12 34)(13 39)(14 36)(15 33)(16 38)(25 43)(26 48)(27 45)(28 42)(29 47)(30 44)(31 41)(32 46)
G:=sub<Sym(48)| (1,38,45)(2,39,46)(3,40,47)(4,33,48)(5,34,41)(6,35,42)(7,36,43)(8,37,44)(9,23,28)(10,24,29)(11,17,30)(12,18,31)(13,19,32)(14,20,25)(15,21,26)(16,22,27), (1,38,45)(2,46,39)(3,40,47)(4,48,33)(5,34,41)(6,42,35)(7,36,43)(8,44,37)(9,23,28)(10,29,24)(11,17,30)(12,31,18)(13,19,32)(14,25,20)(15,21,26)(16,27,22), (1,45,38)(2,39,46)(3,47,40)(4,33,48)(5,41,34)(6,35,42)(7,43,36)(8,37,44)(9,23,28)(10,29,24)(11,17,30)(12,31,18)(13,19,32)(14,25,20)(15,21,26)(16,27,22), (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), (1,22)(2,19)(3,24)(4,21)(5,18)(6,23)(7,20)(8,17)(9,35)(10,40)(11,37)(12,34)(13,39)(14,36)(15,33)(16,38)(25,43)(26,48)(27,45)(28,42)(29,47)(30,44)(31,41)(32,46)>;
G:=Group( (1,38,45)(2,39,46)(3,40,47)(4,33,48)(5,34,41)(6,35,42)(7,36,43)(8,37,44)(9,23,28)(10,24,29)(11,17,30)(12,18,31)(13,19,32)(14,20,25)(15,21,26)(16,22,27), (1,38,45)(2,46,39)(3,40,47)(4,48,33)(5,34,41)(6,42,35)(7,36,43)(8,44,37)(9,23,28)(10,29,24)(11,17,30)(12,31,18)(13,19,32)(14,25,20)(15,21,26)(16,27,22), (1,45,38)(2,39,46)(3,47,40)(4,33,48)(5,41,34)(6,35,42)(7,43,36)(8,37,44)(9,23,28)(10,29,24)(11,17,30)(12,31,18)(13,19,32)(14,25,20)(15,21,26)(16,27,22), (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), (1,22)(2,19)(3,24)(4,21)(5,18)(6,23)(7,20)(8,17)(9,35)(10,40)(11,37)(12,34)(13,39)(14,36)(15,33)(16,38)(25,43)(26,48)(27,45)(28,42)(29,47)(30,44)(31,41)(32,46) );
G=PermutationGroup([[(1,38,45),(2,39,46),(3,40,47),(4,33,48),(5,34,41),(6,35,42),(7,36,43),(8,37,44),(9,23,28),(10,24,29),(11,17,30),(12,18,31),(13,19,32),(14,20,25),(15,21,26),(16,22,27)], [(1,38,45),(2,46,39),(3,40,47),(4,48,33),(5,34,41),(6,42,35),(7,36,43),(8,44,37),(9,23,28),(10,29,24),(11,17,30),(12,31,18),(13,19,32),(14,25,20),(15,21,26),(16,27,22)], [(1,45,38),(2,39,46),(3,47,40),(4,33,48),(5,41,34),(6,35,42),(7,43,36),(8,37,44),(9,23,28),(10,29,24),(11,17,30),(12,31,18),(13,19,32),(14,25,20),(15,21,26),(16,27,22)], [(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)], [(1,22),(2,19),(3,24),(4,21),(5,18),(6,23),(7,20),(8,17),(9,35),(10,40),(11,37),(12,34),(13,39),(14,36),(15,33),(16,38),(25,43),(26,48),(27,45),(28,42),(29,47),(30,44),(31,41),(32,46)]])
54 conjugacy classes
| class | 1 | 2A | 2B | 3A | 3B | 3C | 3D | ··· | 3H | 4A | 4B | 4C | 6A | 6B | 6C | 6D | ··· | 6H | 6I | 6J | 8A | 8B | 8C | 8D | 12A | ··· | 12F | 12G | ··· | 12P | 12Q | 12R | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 3 | 3 | 3 | 3 | ··· | 3 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | ··· | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | ··· | 12 | 12 | ··· | 12 | 12 | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 18 | 2 | 2 | 2 | 4 | ··· | 4 | 1 | 1 | 18 | 2 | 2 | 2 | 4 | ··· | 4 | 18 | 18 | 18 | 18 | 18 | 18 | 2 | ··· | 2 | 4 | ··· | 4 | 18 | 18 | 18 | ··· | 18 |
54 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | - | + | - | + | - | + | |||||||||||
| image | C1 | C2 | C2 | C4 | C4 | S3 | S3 | Dic3 | D6 | Dic3 | M4(2) | C4×S3 | C8⋊S3 | C4.Dic3 | S32 | S3×Dic3 | C6.D6 | C32⋊4D6 | D6.Dic3 | C12.31D6 | C33⋊9(C2×C4) | C33⋊10M4(2) |
| kernel | C33⋊10M4(2) | C3×C32⋊4C8 | C12×C3⋊S3 | C3×C3⋊Dic3 | C6×C3⋊S3 | C32⋊4C8 | C4×C3⋊S3 | C3⋊Dic3 | C3×C12 | C2×C3⋊S3 | C33 | C3×C6 | C32 | C32 | C12 | C6 | C6 | C4 | C3 | C3 | C2 | C1 |
| # reps | 1 | 2 | 1 | 2 | 2 | 2 | 1 | 1 | 3 | 1 | 2 | 4 | 8 | 4 | 3 | 2 | 1 | 2 | 4 | 2 | 2 | 4 |
Matrix representation of C33⋊10M4(2) ►in GL6(𝔽73)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 1 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 1 | 72 | 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 | 72 | 72 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 46 |
| 0 | 0 | 0 | 0 | 45 | 72 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 71 | 60 |
| 0 | 0 | 0 | 0 | 62 | 2 |
G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,72,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,72,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,1,0,0,0,0,72,0],[0,72,0,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,45,0,0,0,0,46,72],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,71,62,0,0,0,0,60,2] >;
C33⋊10M4(2) in GAP, Magma, Sage, TeX
C_3^3\rtimes_{10}M_4(2) % in TeX
G:=Group("C3^3:10M4(2)"); // GroupNames label
G:=SmallGroup(432,456);
// by ID
G=gap.SmallGroup(432,456);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,141,36,58,1124,571,2028,14118]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^3=c^3=d^8=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,e*a*e=a^-1,b*c=c*b,d*b*d^-1=b^-1,b*e=e*b,d*c*d^-1=e*c*e=c^-1,e*d*e=d^5>;
// generators/relations