non-abelian, soluble, monomial
Aliases: C32⋊C4⋊2C8, C4.18S3≀C2, C32⋊2(C4⋊C8), (C3×C12).18D4, C3⋊Dic3.1Q8, C3⋊S3.3M4(2), C12.29D6.3C2, C3⋊S3.3(C2×C8), (C3×C6).1(C4⋊C4), (C2×C32⋊C4).1C4, (C4×C32⋊C4).3C2, C2.1(C3⋊S3.Q8), (C4×C3⋊S3).52C22, (C2×C3⋊S3).8(C2×C4), SmallGroup(288,380)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C32⋊C4⋊C8
G = < a,b,c,d | a3=b3=c4=d8=1, cbc-1=ab=ba, cac-1=a-1b, ad=da, dbd-1=a-1b-1, dcd-1=c-1 >
Subgroups: 272 in 60 conjugacy classes, 19 normal (15 characteristic)
C1, C2, C2, C3, C4, C4, C22, S3, C6, C8, C2×C4, C32, Dic3, C12, D6, C42, C2×C8, C3⋊S3, C3×C6, C3⋊C8, C24, C4×S3, C4⋊C8, C3⋊Dic3, C3×C12, C32⋊C4, C32⋊C4, C2×C3⋊S3, S3×C8, C3×C3⋊C8, C4×C3⋊S3, C2×C32⋊C4, C12.29D6, C4×C32⋊C4, C32⋊C4⋊C8
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, Q8, C4⋊C4, C2×C8, M4(2), C4⋊C8, S3≀C2, C3⋊S3.Q8, C32⋊C4⋊C8
►On 48 points
(1 38 31)(2 39 32)(3 40 25)(4 33 26)(5 34 27)(6 35 28)(7 36 29)(8 37 30)(9 23 48)(10 24 41)(11 17 42)(12 18 43)(13 19 44)(14 20 45)(15 21 46)(16 22 47)
(1 31 38)(3 25 40)(5 27 34)(7 29 36)(9 48 23)(11 42 17)(13 44 19)(15 46 21)
(1 12 5 16)(2 9 6 13)(3 14 7 10)(4 11 8 15)(17 37 46 26)(18 27 47 38)(19 39 48 28)(20 29 41 40)(21 33 42 30)(22 31 43 34)(23 35 44 32)(24 25 45 36)
(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)
G:=sub<Sym(48)| (1,38,31)(2,39,32)(3,40,25)(4,33,26)(5,34,27)(6,35,28)(7,36,29)(8,37,30)(9,23,48)(10,24,41)(11,17,42)(12,18,43)(13,19,44)(14,20,45)(15,21,46)(16,22,47), (1,31,38)(3,25,40)(5,27,34)(7,29,36)(9,48,23)(11,42,17)(13,44,19)(15,46,21), (1,12,5,16)(2,9,6,13)(3,14,7,10)(4,11,8,15)(17,37,46,26)(18,27,47,38)(19,39,48,28)(20,29,41,40)(21,33,42,30)(22,31,43,34)(23,35,44,32)(24,25,45,36), (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)>;
G:=Group( (1,38,31)(2,39,32)(3,40,25)(4,33,26)(5,34,27)(6,35,28)(7,36,29)(8,37,30)(9,23,48)(10,24,41)(11,17,42)(12,18,43)(13,19,44)(14,20,45)(15,21,46)(16,22,47), (1,31,38)(3,25,40)(5,27,34)(7,29,36)(9,48,23)(11,42,17)(13,44,19)(15,46,21), (1,12,5,16)(2,9,6,13)(3,14,7,10)(4,11,8,15)(17,37,46,26)(18,27,47,38)(19,39,48,28)(20,29,41,40)(21,33,42,30)(22,31,43,34)(23,35,44,32)(24,25,45,36), (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) );
G=PermutationGroup([[(1,38,31),(2,39,32),(3,40,25),(4,33,26),(5,34,27),(6,35,28),(7,36,29),(8,37,30),(9,23,48),(10,24,41),(11,17,42),(12,18,43),(13,19,44),(14,20,45),(15,21,46),(16,22,47)], [(1,31,38),(3,25,40),(5,27,34),(7,29,36),(9,48,23),(11,42,17),(13,44,19),(15,46,21)], [(1,12,5,16),(2,9,6,13),(3,14,7,10),(4,11,8,15),(17,37,46,26),(18,27,47,38),(19,39,48,28),(20,29,41,40),(21,33,42,30),(22,31,43,34),(23,35,44,32),(24,25,45,36)], [(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)]])
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | 6B | 8A | ··· | 8H | 12A | 12B | 12C | 12D | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 8 | ··· | 8 | 12 | 12 | 12 | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 9 | 9 | 4 | 4 | 1 | 1 | 9 | 9 | 18 | 18 | 18 | 18 | 4 | 4 | 6 | ··· | 6 | 4 | 4 | 4 | 4 | 12 | ··· | 12 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | - | + | + | |||||
| image | C1 | C2 | C2 | C4 | C8 | Q8 | D4 | M4(2) | S3≀C2 | C3⋊S3.Q8 | C32⋊C4⋊C8 |
| kernel | C32⋊C4⋊C8 | C12.29D6 | C4×C32⋊C4 | C2×C32⋊C4 | C32⋊C4 | C3⋊Dic3 | C3×C12 | C3⋊S3 | C4 | C2 | C1 |
| # reps | 1 | 2 | 1 | 4 | 8 | 1 | 1 | 2 | 4 | 4 | 8 |
Matrix representation of C32⋊C4⋊C8 ►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 | 0 | 72 |
| 0 | 0 | 0 | 0 | 1 | 72 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 1 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 28 | 21 | 0 | 0 | 0 | 0 |
| 53 | 45 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 46 |
| 0 | 0 | 0 | 0 | 46 | 0 |
| 0 | 0 | 27 | 0 | 0 | 0 |
| 0 | 0 | 0 | 27 | 0 | 0 |
| 0 | 48 | 0 | 0 | 0 | 0 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,72,72,0,0,0,0,0,0,0,1,0,0,0,0,72,72],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[28,53,0,0,0,0,21,45,0,0,0,0,0,0,0,0,27,0,0,0,0,0,0,27,0,0,0,46,0,0,0,0,46,0,0,0],[0,4,0,0,0,0,48,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,1,0,0,0,0,0,0,1,0,0] >;
C32⋊C4⋊C8 in GAP, Magma, Sage, TeX
C_3^2\rtimes C_4\rtimes C_8
% in TeX
G:=Group("C3^2:C4:C8"); // GroupNames label
G:=SmallGroup(288,380);
// by ID
G=gap.SmallGroup(288,380);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,56,85,64,80,2693,2028,691,797,2372]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^3=c^4=d^8=1,c*b*c^-1=a*b=b*a,c*a*c^-1=a^-1*b,a*d=d*a,d*b*d^-1=a^-1*b^-1,d*c*d^-1=c^-1>;
// generators/relations