p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.7C8, M5(2)⋊4C4, C23.26M4(2), C4.3(C4×C8), (C2×C8).10C8, C4.24(C4⋊C8), C8.33(C4⋊C4), (C2×C8).56Q8, (C2×C8).373D4, (C2×C4).53C42, (C2×C42).42C4, (C22×C8).24C4, C2.3(C8.C8), C22.19(C4⋊C8), C4.20(C22⋊C8), C8.55(C22⋊C4), (C2×M5(2)).7C2, (C2×C4).69M4(2), C22.3(C8⋊C4), C22.23(C22⋊C8), (C22×C8).571C22, C4.26(C2.C42), C2.14(C22.7C42), (C2×C4×C8).57C2, (C2×C4).74(C2×C8), (C2×C8).188(C2×C4), (C2×C4).159(C4⋊C4), (C22×C4).467(C2×C4), (C2×C4).347(C22⋊C4), SmallGroup(128,108)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.7C8
G = < a,b,c | a4=b4=1, c8=b2, cac-1=ab=ba, cbc-1=b-1 >
Subgroups: 104 in 74 conjugacy classes, 44 normal (24 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, C23, C16, C42, C42, C2×C8, C2×C8, C2×C8, C22×C4, C22×C4, C4×C8, C2×C16, M5(2), M5(2), C2×C42, C22×C8, C2×C4×C8, C2×M5(2), C42.7C8
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, Q8, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), C2.C42, C4×C8, C8⋊C4, C22⋊C8, C4⋊C8, C22.7C42, C8.C8, C42.7C8
►On 32 points
(1 13 9 5)(2 25)(3 15 11 7)(4 27)(6 29)(8 31)(10 17)(12 19)(14 21)(16 23)(18 30 26 22)(20 32 28 24)
(1 28 9 20)(2 21 10 29)(3 30 11 22)(4 23 12 31)(5 32 13 24)(6 25 14 17)(7 18 15 26)(8 27 16 19)
(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)
G:=sub<Sym(32)| (1,13,9,5)(2,25)(3,15,11,7)(4,27)(6,29)(8,31)(10,17)(12,19)(14,21)(16,23)(18,30,26,22)(20,32,28,24), (1,28,9,20)(2,21,10,29)(3,30,11,22)(4,23,12,31)(5,32,13,24)(6,25,14,17)(7,18,15,26)(8,27,16,19), (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)>;
G:=Group( (1,13,9,5)(2,25)(3,15,11,7)(4,27)(6,29)(8,31)(10,17)(12,19)(14,21)(16,23)(18,30,26,22)(20,32,28,24), (1,28,9,20)(2,21,10,29)(3,30,11,22)(4,23,12,31)(5,32,13,24)(6,25,14,17)(7,18,15,26)(8,27,16,19), (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) );
G=PermutationGroup([[(1,13,9,5),(2,25),(3,15,11,7),(4,27),(6,29),(8,31),(10,17),(12,19),(14,21),(16,23),(18,30,26,22),(20,32,28,24)], [(1,28,9,20),(2,21,10,29),(3,30,11,22),(4,23,12,31),(5,32,13,24),(6,25,14,17),(7,18,15,26),(8,27,16,19)], [(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)]])
56 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | ··· | 4N | 8A | ··· | 8H | 8I | ··· | 8T | 16A | ··· | 16P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 | 8 | ··· | 8 | 16 | ··· | 16 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | ||||||||
| image | C1 | C2 | C2 | C4 | C4 | C4 | C8 | C8 | D4 | Q8 | M4(2) | M4(2) | C8.C8 |
| kernel | C42.7C8 | C2×C4×C8 | C2×M5(2) | M5(2) | C2×C42 | C22×C8 | C42 | C2×C8 | C2×C8 | C2×C8 | C2×C4 | C23 | C2 |
| # reps | 1 | 1 | 2 | 8 | 2 | 2 | 8 | 8 | 3 | 1 | 2 | 2 | 16 |
Matrix representation of C42.7C8 ►in GL3(𝔽17) generated by
| 1 | 0 | 0 |
| 0 | 13 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 0 | 4 | 0 |
| 0 | 0 | 13 |
| 15 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 15 | 0 |
G:=sub<GL(3,GF(17))| [1,0,0,0,13,0,0,0,1],[1,0,0,0,4,0,0,0,13],[15,0,0,0,0,15,0,1,0] >;
C42.7C8 in GAP, Magma, Sage, TeX
C_4^2._7C_8
% in TeX
G:=Group("C4^2.7C8"); // GroupNames label
G:=SmallGroup(128,108);
// by ID
G=gap.SmallGroup(128,108);
# by ID
G:=PCGroup([7,-2,2,-2,2,2,-2,-2,56,85,120,1430,352,136,2804,124]);
// Polycyclic
G:=Group<a,b,c|a^4=b^4=1,c^8=b^2,c*a*c^-1=a*b=b*a,c*b*c^-1=b^-1>;
// generators/relations