p-group, metabelian, nilpotent (class 3), monomial
Aliases: C8.5M4(2), C23.18M4(2), M5(2).10C22, (C4xC8).10C4, C16:C4:4C2, C8:C4.8C4, C8.C8:11C2, C8.84(C4oD4), C42.28(C2xC4), (C2xC42).26C4, C23.C8.6C2, (C2xC8).386C23, (C4xC8).158C22, (C4xM4(2)).3C2, (C2xC4).32M4(2), C4.51(C2xM4(2)), (C2xM4(2)).15C4, C8:C4.150C22, C4.87(C42:C2), C22.25(C2xM4(2)), C2.10(C42.6C4), (C2xM4(2)).330C22, (C2xC8).15(C2xC4), (C22xC4).85(C2xC4), (C2xC4).562(C22xC4), SmallGroup(128,897)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C8.5M4(2)
G = < a,b,c | a8=1, b8=c2=a4, bab-1=a3, cac-1=a5, cbc-1=a6b5 >
Subgroups: 100 in 64 conjugacy classes, 38 normal (24 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C8, C2xC4, C2xC4, C23, C16, C42, C42, C2xC8, M4(2), C22xC4, C22xC4, C4xC8, C8:C4, M5(2), C2xC42, C2xM4(2), C16:C4, C23.C8, C8.C8, C4xM4(2), C8.5M4(2)
Quotients: C1, C2, C4, C22, C2xC4, C23, M4(2), C22xC4, C4oD4, C42:C2, C2xM4(2), C42.6C4, C8.5M4(2)
►On 16 points - transitive group 16T220
(1 7 13 3 9 15 5 11)(2 4 6 8 10 12 14 16)
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16)
(1 13 9 5)(2 6 10 14)(3 7 11 15)(4 16 12 8)
G:=sub<Sym(16)| (1,7,13,3,9,15,5,11)(2,4,6,8,10,12,14,16), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16), (1,13,9,5)(2,6,10,14)(3,7,11,15)(4,16,12,8)>;
G:=Group( (1,7,13,3,9,15,5,11)(2,4,6,8,10,12,14,16), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16), (1,13,9,5)(2,6,10,14)(3,7,11,15)(4,16,12,8) );
G=PermutationGroup([[(1,7,13,3,9,15,5,11),(2,4,6,8,10,12,14,16)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)], [(1,13,9,5),(2,6,10,14),(3,7,11,15),(4,16,12,8)]])
G:=TransitiveGroup(16,220);
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | ··· | 4G | 4H | 4I | 4J | 8A | 8B | 8C | 8D | 8E | ··· | 8J | 16A | ··· | 16H |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | ··· | 8 | 16 | ··· | 16 |
| size | 1 | 1 | 2 | 4 | 1 | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | ··· | 8 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | |||||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | M4(2) | C4oD4 | M4(2) | M4(2) | C8.5M4(2) |
| kernel | C8.5M4(2) | C16:C4 | C23.C8 | C8.C8 | C4xM4(2) | C4xC8 | C8:C4 | C2xC42 | C2xM4(2) | C8 | C8 | C2xC4 | C23 | C1 |
| # reps | 1 | 2 | 2 | 2 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 2 | 2 | 4 |
Matrix representation of C8.5M4(2) ►in GL4(F5) generated by
| 3 | 4 | 1 | 0 |
| 3 | 4 | 0 | 2 |
| 3 | 0 | 1 | 4 |
| 0 | 4 | 3 | 2 |
| 3 | 4 | 3 | 1 |
| 0 | 4 | 1 | 1 |
| 3 | 4 | 2 | 3 |
| 4 | 0 | 1 | 1 |
| 3 | 3 | 1 | 0 |
| 0 | 1 | 1 | 0 |
| 0 | 3 | 4 | 0 |
| 0 | 4 | 1 | 2 |
G:=sub<GL(4,GF(5))| [3,3,3,0,4,4,0,4,1,0,1,3,0,2,4,2],[3,0,3,4,4,4,4,0,3,1,2,1,1,1,3,1],[3,0,0,0,3,1,3,4,1,1,4,1,0,0,0,2] >;
C8.5M4(2) in GAP, Magma, Sage, TeX
C_8._5M_4(2)
% in TeX
G:=Group("C8.5M4(2)"); // GroupNames label
G:=SmallGroup(128,897);
// by ID
G=gap.SmallGroup(128,897);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,-2,112,141,758,58,2019,1411,718,102,124]);
// Polycyclic
G:=Group<a,b,c|a^8=1,b^8=c^2=a^4,b*a*b^-1=a^3,c*a*c^-1=a^5,c*b*c^-1=a^6*b^5>;
// generators/relations