p-group, metabelian, nilpotent (class 3), monomial
Aliases: D4.2C42, Q8.2C42, C8○2C4≀C2, C4≀C2⋊6C4, C8○D4⋊6C4, C2.4(C8○D8), C4.163(C4×D4), (C2×C8).384D4, C4.7(C2×C42), C8○3(D4⋊C4), D4⋊C4⋊13C4, C8○(C42⋊6C4), C8○3(Q8⋊C4), Q8⋊C4⋊13C4, C22.28(C4×D4), C42⋊6C4⋊35C2, C8.61(C22⋊C4), C8○(C4.C42), C42.262(C2×C4), C8○2M4(2)⋊22C2, C4.C42⋊23C2, C8○(C23.24D4), M4(2).17(C2×C4), C23.199(C4○D4), (C22×C8).474C22, (C22×C4).1311C23, (C2×C42).1048C22, C22.1(C42⋊C2), C23.24D4.13C2, C42⋊C2.263C22, (C2×M4(2)).308C22, C8○(C2×C4≀C2), (C2×C8)○C4≀C2, (C2×C4×C8)⋊38C2, (C2×C4≀C2).17C2, (C2×C8)○(D4⋊C4), C4⋊C4.142(C2×C4), (C2×C8).191(C2×C4), (C2×C8)○(Q8⋊C4), C4○D4.25(C2×C4), (C2×C8○D4).12C2, C2.22(C4×C22⋊C4), (C2×D4).159(C2×C4), (C2×C4).1306(C2×D4), C4.110(C2×C22⋊C4), (C2×Q8).141(C2×C4), (C2×C4).542(C4○D4), (C2×C8)○(C4.C42), (C2×C4).354(C22×C4), (C2×C4○D4).252C22, (C2×C8)○(C23.24D4), SmallGroup(128,496)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for Q8.C42
G = < a,b,c,d | a4=c4=1, b2=d4=a2, bab-1=dad-1=a-1, dcd-1=ac=ca, cbc-1=a-1b, bd=db >
Subgroups: 236 in 150 conjugacy classes, 76 normal (36 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, C42, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, C2×C8, M4(2), M4(2), C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C4○D4, C4×C8, C8⋊C4, D4⋊C4, Q8⋊C4, C4≀C2, C2×C42, C42⋊C2, C22×C8, C22×C8, C2×M4(2), C2×M4(2), C8○D4, C8○D4, C2×C4○D4, C42⋊6C4, C4.C42, C2×C4×C8, C8○2M4(2), C23.24D4, C2×C4≀C2, C2×C8○D4, Q8.C42
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C42, C22⋊C4, C22×C4, C2×D4, C4○D4, C2×C42, C2×C22⋊C4, C42⋊C2, C4×D4, C4×C22⋊C4, C8○D8, Q8.C42
►On 32 points
(1 27 5 31)(2 32 6 28)(3 29 7 25)(4 26 8 30)(9 24 13 20)(10 21 14 17)(11 18 15 22)(12 23 16 19)
(1 9 5 13)(2 10 6 14)(3 11 7 15)(4 12 8 16)(17 28 21 32)(18 29 22 25)(19 30 23 26)(20 31 24 27)
(1 7 5 3)(2 26)(4 28)(6 30)(8 32)(9 18)(10 16 14 12)(11 20)(13 22)(15 24)(17 23 21 19)(25 31 29 27)
(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)| (1,27,5,31)(2,32,6,28)(3,29,7,25)(4,26,8,30)(9,24,13,20)(10,21,14,17)(11,18,15,22)(12,23,16,19), (1,9,5,13)(2,10,6,14)(3,11,7,15)(4,12,8,16)(17,28,21,32)(18,29,22,25)(19,30,23,26)(20,31,24,27), (1,7,5,3)(2,26)(4,28)(6,30)(8,32)(9,18)(10,16,14,12)(11,20)(13,22)(15,24)(17,23,21,19)(25,31,29,27), (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( (1,27,5,31)(2,32,6,28)(3,29,7,25)(4,26,8,30)(9,24,13,20)(10,21,14,17)(11,18,15,22)(12,23,16,19), (1,9,5,13)(2,10,6,14)(3,11,7,15)(4,12,8,16)(17,28,21,32)(18,29,22,25)(19,30,23,26)(20,31,24,27), (1,7,5,3)(2,26)(4,28)(6,30)(8,32)(9,18)(10,16,14,12)(11,20)(13,22)(15,24)(17,23,21,19)(25,31,29,27), (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([[(1,27,5,31),(2,32,6,28),(3,29,7,25),(4,26,8,30),(9,24,13,20),(10,21,14,17),(11,18,15,22),(12,23,16,19)], [(1,9,5,13),(2,10,6,14),(3,11,7,15),(4,12,8,16),(17,28,21,32),(18,29,22,25),(19,30,23,26),(20,31,24,27)], [(1,7,5,3),(2,26),(4,28),(6,30),(8,32),(9,18),(10,16,14,12),(11,20),(13,22),(15,24),(17,23,21,19),(25,31,29,27)], [(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 | 2F | 2G | 4A | 4B | 4C | 4D | 4E | ··· | 4N | 4O | ··· | 4T | 8A | ··· | 8H | 8I | ··· | 8T | 8U | ··· | 8AB |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 | 8 | ··· | 8 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | D4 | C4○D4 | C4○D4 | C8○D8 |
| kernel | Q8.C42 | C42⋊6C4 | C4.C42 | C2×C4×C8 | C8○2M4(2) | C23.24D4 | C2×C4≀C2 | C2×C8○D4 | D4⋊C4 | Q8⋊C4 | C4≀C2 | C8○D4 | C2×C8 | C2×C4 | C23 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 | 8 | 8 | 4 | 2 | 2 | 16 |
Matrix representation of Q8.C42 ►in GL4(𝔽17) generated by
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 13 | 13 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 8 | 16 |
| 0 | 0 | 14 | 9 |
| 1 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 6 | 16 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 2 |
| 0 | 0 | 6 | 16 |
G:=sub<GL(4,GF(17))| [16,0,0,0,0,16,0,0,0,0,4,13,0,0,0,13],[0,1,0,0,1,0,0,0,0,0,8,14,0,0,16,9],[1,0,0,0,0,16,0,0,0,0,4,6,0,0,0,16],[0,1,0,0,1,0,0,0,0,0,1,6,0,0,2,16] >;
Q8.C42 in GAP, Magma, Sage, TeX
Q_8.C_4^2
% in TeX
G:=Group("Q8.C4^2"); // GroupNames label
G:=SmallGroup(128,496);
// by ID
G=gap.SmallGroup(128,496);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,112,141,232,100,2019,248,172,4037]);
// Polycyclic
G:=Group<a,b,c,d|a^4=c^4=1,b^2=d^4=a^2,b*a*b^-1=d*a*d^-1=a^-1,d*c*d^-1=a*c=c*a,c*b*c^-1=a^-1*b,b*d=d*b>;
// generators/relations