p-group, metabelian, nilpotent (class 2), monomial
Aliases: C8.23C42, M5(2)⋊12C4, C23.11M4(2), M5(2).18C22, (C2×C16)⋊8C4, (C4×C8).9C4, C16.3(C2×C4), C8○(C16⋊C4), C16⋊C4⋊6C2, C8.62(C22×C4), C4.44(C2×C42), (C2×C4).75C42, C42.23(C2×C4), C4.10(C8⋊C4), (C2×C8).383C23, C4.48(C2×M4(2)), (C2×C4).23M4(2), C42⋊C2.20C4, C22.9(C8⋊C4), (C2×M4(2)).27C4, (C2×M5(2)).20C2, C8⋊C4.149C22, C8○2M4(2).19C2, (C22×C8).413C22, C22.22(C2×M4(2)), C2.12(C2×C8⋊C4), (C2×C8).248(C2×C4), (C22×C4).285(C2×C4), (C2×C4).557(C22×C4), SmallGroup(128,842)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C8.23C42
G = < a,b,c | a8=c4=1, b4=a2, ab=ba, ac=ca, cbc-1=a2b >
Subgroups: 100 in 80 conjugacy classes, 66 normal (18 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C8, C8, C8, C2×C4, C2×C4, C2×C4, C23, C16, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), C22×C4, C4×C8, C8⋊C4, C2×C16, M5(2), C42⋊C2, C22×C8, C2×M4(2), C16⋊C4, C8○2M4(2), C2×M5(2), C8.23C42
Quotients: C1, C2, C4, C22, C2×C4, C23, C42, M4(2), C22×C4, C8⋊C4, C2×C42, C2×M4(2), C2×C8⋊C4, C8.23C42
►On 32 points
(1 3 5 7 9 11 13 15)(2 4 6 8 10 12 14 16)(17 27 21 31 25 19 29 23)(18 28 22 32 26 20 30 24)
(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 26)(2 23 10 31)(3 20)(4 17 12 25)(5 30)(6 27 14 19)(7 24)(8 21 16 29)(9 18)(11 28)(13 22)(15 32)
G:=sub<Sym(32)| (1,3,5,7,9,11,13,15)(2,4,6,8,10,12,14,16)(17,27,21,31,25,19,29,23)(18,28,22,32,26,20,30,24), (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), (1,26)(2,23,10,31)(3,20)(4,17,12,25)(5,30)(6,27,14,19)(7,24)(8,21,16,29)(9,18)(11,28)(13,22)(15,32)>;
G:=Group( (1,3,5,7,9,11,13,15)(2,4,6,8,10,12,14,16)(17,27,21,31,25,19,29,23)(18,28,22,32,26,20,30,24), (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), (1,26)(2,23,10,31)(3,20)(4,17,12,25)(5,30)(6,27,14,19)(7,24)(8,21,16,29)(9,18)(11,28)(13,22)(15,32) );
G=PermutationGroup([[(1,3,5,7,9,11,13,15),(2,4,6,8,10,12,14,16),(17,27,21,31,25,19,29,23),(18,28,22,32,26,20,30,24)], [(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)], [(1,26),(2,23,10,31),(3,20),(4,17,12,25),(5,30),(6,27,14,19),(7,24),(8,21,16,29),(9,18),(11,28),(13,22),(15,32)]])
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 8A | 8B | 8C | 8D | 8E | ··· | 8J | 8K | 8L | 8M | 8N | 16A | ··· | 16P |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | ··· | 8 | 8 | 8 | 8 | 8 | 16 | ··· | 16 |
| size | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 |
| type | + | + | + | + | ||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | C4 | M4(2) | M4(2) | C8.23C42 |
| kernel | C8.23C42 | C16⋊C4 | C8○2M4(2) | C2×M5(2) | C4×C8 | C2×C16 | M5(2) | C42⋊C2 | C2×M4(2) | C2×C4 | C23 | C1 |
| # reps | 1 | 4 | 1 | 2 | 4 | 8 | 8 | 2 | 2 | 6 | 2 | 4 |
Matrix representation of C8.23C42 ►in GL4(𝔽17) generated by
| 2 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 |
| 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 2 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 2 | 0 | 0 | 0 |
| 0 | 15 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 |
| 0 | 0 | 4 | 0 |
G:=sub<GL(4,GF(17))| [2,0,0,0,0,2,0,0,0,0,2,0,0,0,0,2],[0,0,2,0,0,0,0,15,1,0,0,0,0,1,0,0],[0,1,0,0,1,0,0,0,0,0,0,4,0,0,4,0] >;
C8.23C42 in GAP, Magma, Sage, TeX
C_8._{23}C_4^2 % in TeX
G:=Group("C8.23C4^2"); // GroupNames label
G:=SmallGroup(128,842);
// by ID
G=gap.SmallGroup(128,842);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,-2,56,477,120,723,1018,136,2804,124]);
// Polycyclic
G:=Group<a,b,c|a^8=c^4=1,b^4=a^2,a*b=b*a,a*c=c*a,c*b*c^-1=a^2*b>;
// generators/relations