p-group, metabelian, nilpotent (class 3), monomial
Aliases: C8.19M4(2), C23.4M4(2), M5(2).11C22, (C4×C8).5C4, C16⋊C4⋊5C2, C8.C8⋊12C2, C8⋊C4.19C4, C8.85(C4○D4), C42.29(C2×C4), (C22×C8).11C4, C23.C8.7C2, (C2×C4).5M4(2), (C4×C8).159C22, (C2×C8).387C23, C4.52(C2×M4(2)), C42⋊C2.24C4, C8○2M4(2).7C2, C8⋊C4.151C22, C4.88(C42⋊C2), C22.26(C2×M4(2)), C2.11(C42.6C4), (C2×M4(2)).331C22, (C2×C8).16(C2×C4), (C22×C4).86(C2×C4), (C2×C4).563(C22×C4), SmallGroup(128,898)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C8.19M4(2)
G = < a,b,c | a8=1, b8=c2=a4, bab-1=a3, ac=ca, cbc-1=a2b5 >
Subgroups: 92 in 61 conjugacy classes, 38 normal (26 characteristic)
C1, C2, C2 [×2], C4 [×2], C4 [×3], C22, C22 [×2], C8 [×2], C8 [×2], C8 [×2], C2×C4 [×2], C2×C4 [×4], C23, C16 [×4], C42 [×2], C22⋊C4, C4⋊C4, C2×C8 [×4], C2×C8 [×2], M4(2) [×2], C22×C4, C4×C8 [×2], C8⋊C4 [×2], M5(2) [×4], C42⋊C2, C22×C8, C2×M4(2), C16⋊C4 [×2], C23.C8 [×2], C8.C8 [×2], C8○2M4(2), C8.19M4(2)
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], C23, M4(2) [×4], C22×C4, C4○D4 [×2], C42⋊C2, C2×M4(2) [×2], C42.6C4, C8.19M4(2)
►On 32 points
(1 22 5 26 9 30 13 18)(2 27 14 23 10 19 6 31)(3 24 7 28 11 32 15 20)(4 29 16 25 12 21 8 17)
(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 20 9 28)(2 29 10 21)(3 30 11 22)(4 23 12 31)(5 24 13 32)(6 17 14 25)(7 18 15 26)(8 27 16 19)
G:=sub<Sym(32)| (1,22,5,26,9,30,13,18)(2,27,14,23,10,19,6,31)(3,24,7,28,11,32,15,20)(4,29,16,25,12,21,8,17), (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), (1,20,9,28)(2,29,10,21)(3,30,11,22)(4,23,12,31)(5,24,13,32)(6,17,14,25)(7,18,15,26)(8,27,16,19)>;
G:=Group( (1,22,5,26,9,30,13,18)(2,27,14,23,10,19,6,31)(3,24,7,28,11,32,15,20)(4,29,16,25,12,21,8,17), (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), (1,20,9,28)(2,29,10,21)(3,30,11,22)(4,23,12,31)(5,24,13,32)(6,17,14,25)(7,18,15,26)(8,27,16,19) );
G=PermutationGroup([(1,22,5,26,9,30,13,18),(2,27,14,23,10,19,6,31),(3,24,7,28,11,32,15,20),(4,29,16,25,12,21,8,17)], [(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)], [(1,20,9,28),(2,29,10,21),(3,30,11,22),(4,23,12,31),(5,24,13,32),(6,17,14,25),(7,18,15,26),(8,27,16,19)])
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | ··· | 4H | 8A | ··· | 8H | 8I | 8J | 8K | 8L | 16A | ··· | 16H |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 | 8 | 8 | 8 | 8 | 16 | ··· | 16 |
| size | 1 | 1 | 2 | 4 | 1 | 1 | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | 4 | 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) | C4○D4 | M4(2) | M4(2) | C8.19M4(2) |
| kernel | C8.19M4(2) | C16⋊C4 | C23.C8 | C8.C8 | C8○2M4(2) | C4×C8 | C8⋊C4 | C42⋊C2 | C22×C8 | C8 | C8 | C2×C4 | C23 | C1 |
| # reps | 1 | 2 | 2 | 2 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 2 | 2 | 4 |
Matrix representation of C8.19M4(2) ►in GL4(𝔽17) generated by
| 8 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 |
| 0 | 0 | 2 | 0 |
| 12 | 0 | 0 | 2 |
| 9 | 0 | 9 | 4 |
| 0 | 0 | 9 | 0 |
| 8 | 0 | 0 | 0 |
| 0 | 4 | 1 | 8 |
| 0 | 15 | 0 | 0 |
| 9 | 0 | 0 | 0 |
| 9 | 0 | 9 | 4 |
| 1 | 13 | 5 | 8 |
G:=sub<GL(4,GF(17))| [8,0,0,12,0,8,0,0,0,0,2,0,0,0,0,2],[9,0,8,0,0,0,0,4,9,9,0,1,4,0,0,8],[0,9,9,1,15,0,0,13,0,0,9,5,0,0,4,8] >;
C8.19M4(2) in GAP, Magma, Sage, TeX
C_8._{19}M_4(2) % in TeX
G:=Group("C8.19M4(2)"); // GroupNames label
G:=SmallGroup(128,898);
// by ID
G=gap.SmallGroup(128,898);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,-2,112,141,758,723,58,2019,1411,718,102,124]);
// Polycyclic
G:=Group<a,b,c|a^8=1,b^8=c^2=a^4,b*a*b^-1=a^3,a*c=c*a,c*b*c^-1=a^2*b^5>;
// generators/relations