p-group, metabelian, nilpotent (class 3), monomial
Aliases: (C4×C8)⋊5C4, C8⋊C4⋊8C4, C8.4(C4⋊C4), (C2×C8).9Q8, (C2×C8).107D4, C42.21(C2×C4), C4.9C42.3C2, C23.5(C4○D4), C2.6(C42⋊8C4), C4.47(C4.4D4), C8○2M4(2).1C2, C4.10(C42.C2), C4.86(C42⋊C2), M4(2)⋊4C4.3C2, (C22×C4).670C23, (C22×C8).219C22, C22.6(C42.C2), C23.25D4.12C2, C22.17(C4.4D4), C42⋊C2.269C22, C22.32(C42⋊C2), (C2×M4(2)).166C22, C4.32(C2×C4⋊C4), (C2×C8).13(C2×C4), (C2×C4).237(C2×D4), (C2×C4).115(C2×Q8), (C2×C4).49(C4○D4), (C2×C8.C4).10C2, (C2×C4).535(C22×C4), SmallGroup(128,565)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C8.(C4⋊C4)
G = < a,b,c | a8=c4=1, b4=a4, bab-1=a5, cac-1=a-1, cbc-1=a6b3 >
Subgroups: 148 in 86 conjugacy classes, 50 normal (34 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, C23, C42, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, C2×C8, M4(2), C22×C4, C4×C8, C8⋊C4, C4.Q8, C2.D8, C8.C4, C42⋊C2, C42⋊C2, C22×C8, C2×M4(2), C2×M4(2), C4.9C42, M4(2)⋊4C4, C8○2M4(2), C23.25D4, C2×C8.C4, C8.(C4⋊C4)
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C4⋊C4, C22×C4, C2×D4, C2×Q8, C4○D4, C2×C4⋊C4, C42⋊C2, C4.4D4, C42.C2, C42⋊8C4, C8.(C4⋊C4)
►On 32 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)
(1 12 25 22 5 16 29 18)(2 9 26 19 6 13 30 23)(3 14 27 24 7 10 31 20)(4 11 28 21 8 15 32 17)
(2 8)(3 7)(4 6)(9 19 13 23)(10 18 14 22)(11 17 15 21)(12 24 16 20)(26 32)(27 31)(28 30)
G:=sub<Sym(32)| (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), (1,12,25,22,5,16,29,18)(2,9,26,19,6,13,30,23)(3,14,27,24,7,10,31,20)(4,11,28,21,8,15,32,17), (2,8)(3,7)(4,6)(9,19,13,23)(10,18,14,22)(11,17,15,21)(12,24,16,20)(26,32)(27,31)(28,30)>;
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), (1,12,25,22,5,16,29,18)(2,9,26,19,6,13,30,23)(3,14,27,24,7,10,31,20)(4,11,28,21,8,15,32,17), (2,8)(3,7)(4,6)(9,19,13,23)(10,18,14,22)(11,17,15,21)(12,24,16,20)(26,32)(27,31)(28,30) );
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)], [(1,12,25,22,5,16,29,18),(2,9,26,19,6,13,30,23),(3,14,27,24,7,10,31,20),(4,11,28,21,8,15,32,17)], [(2,8),(3,7),(4,6),(9,19,13,23),(10,18,14,22),(11,17,15,21),(12,24,16,20),(26,32),(27,31),(28,30)]])
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 8A | 8B | 8C | 8D | 8E | ··· | 8J | 8K | 8L | 8M | 8N |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | ··· | 8 | 8 | 8 | 8 | 8 |
| size | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | - | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | D4 | Q8 | C4○D4 | C4○D4 | C8.(C4⋊C4) |
| kernel | C8.(C4⋊C4) | C4.9C42 | M4(2)⋊4C4 | C8○2M4(2) | C23.25D4 | C2×C8.C4 | C4×C8 | C8⋊C4 | C2×C8 | C2×C8 | C2×C4 | C23 | C1 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 4 | 4 | 2 | 2 | 6 | 2 | 4 |
Matrix representation of C8.(C4⋊C4) ►in GL4(𝔽17) generated by
| 14 | 12 | 0 | 0 |
| 12 | 14 | 0 | 0 |
| 0 | 0 | 3 | 5 |
| 0 | 0 | 5 | 3 |
| 0 | 0 | 14 | 5 |
| 0 | 0 | 5 | 14 |
| 5 | 14 | 0 | 0 |
| 14 | 5 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 |
| 0 | 0 | 13 | 0 |
G:=sub<GL(4,GF(17))| [14,12,0,0,12,14,0,0,0,0,3,5,0,0,5,3],[0,0,5,14,0,0,14,5,14,5,0,0,5,14,0,0],[0,1,0,0,16,0,0,0,0,0,0,13,0,0,4,0] >;
C8.(C4⋊C4) in GAP, Magma, Sage, TeX
C_8.(C_4\rtimes C_4)
% in TeX
G:=Group("C8.(C4:C4)"); // GroupNames label
G:=SmallGroup(128,565);
// by ID
G=gap.SmallGroup(128,565);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,120,422,58,2019,172,2028,1027]);
// Polycyclic
G:=Group<a,b,c|a^8=c^4=1,b^4=a^4,b*a*b^-1=a^5,c*a*c^-1=a^-1,c*b*c^-1=a^6*b^3>;
// generators/relations