p-group, metabelian, nilpotent (class 2), monomial
Aliases: M4(2)⋊9Q8, C42.288C23, (C8×Q8)⋊28C2, C4⋊Q8.31C4, C8.39(C2×Q8), C4.34(C4×Q8), C8⋊4Q8⋊33C2, C4.38(C8○D4), M4(2)○2(C4⋊C8), C22⋊Q8.20C4, C4.66(C22×Q8), C22.12(C4×Q8), C4⋊C8.359C22, (C2×C8).425C23, C42.213(C2×C4), (C2×C4).659C24, (C4×C8).331C22, C42.C2.16C4, (C4×Q8).58C22, (C4×M4(2)).29C2, C8⋊C4.160C22, C8○2M4(2).24C2, C2.20(Q8○M4(2)), C22.185(C23×C4), C23.144(C22×C4), (C22×C8).445C22, (C2×C42).772C22, (C22×C4).1521C23, C42⋊C2.305C22, (C2×M4(2)).362C22, C42.6C22.14C2, C23.37C23.23C2, C2.26(C2×C4×Q8), (C2×C4⋊C8).58C2, C2.22(C2×C8○D4), C4⋊C4.163(C2×C4), C4.310(C2×C4○D4), (C2×C4).314(C2×Q8), C22⋊C4.40(C2×C4), (C2×C4).76(C22×C4), (C2×Q8).119(C2×C4), (C2×C4).832(C4○D4), (C22×C4).346(C2×C4), SmallGroup(128,1694)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for M4(2)⋊9Q8
G = < a,b,c,d | a8=b2=c4=1, d2=c2, bab=dad-1=a5, ac=ca, bc=cb, dbd-1=a4b, dcd-1=c-1 >
Subgroups: 220 in 178 conjugacy classes, 142 normal (32 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C8, C8, C2×C4, C2×C4, C2×C4, Q8, C23, C42, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), M4(2), C22×C4, C2×Q8, C4×C8, C8⋊C4, C4⋊C8, C4⋊C8, C2×C42, C42⋊C2, C4×Q8, C22⋊Q8, C42.C2, C4⋊Q8, C22×C8, C2×M4(2), C4×M4(2), C8○2M4(2), C2×C4⋊C8, C42.6C22, C8×Q8, C8⋊4Q8, C23.37C23, M4(2)⋊9Q8
Quotients: C1, C2, C4, C22, C2×C4, Q8, C23, C22×C4, C2×Q8, C4○D4, C24, C4×Q8, C8○D4, C23×C4, C22×Q8, C2×C4○D4, C2×C4×Q8, C2×C8○D4, Q8○M4(2), M4(2)⋊9Q8
►On 64 points
(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)(33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56)(57 58 59 60 61 62 63 64)
(2 6)(4 8)(9 13)(11 15)(18 22)(20 24)(25 29)(27 31)(34 38)(36 40)(41 45)(43 47)(50 54)(52 56)(57 61)(59 63)
(1 28 23 35)(2 29 24 36)(3 30 17 37)(4 31 18 38)(5 32 19 39)(6 25 20 40)(7 26 21 33)(8 27 22 34)(9 43 59 50)(10 44 60 51)(11 45 61 52)(12 46 62 53)(13 47 63 54)(14 48 64 55)(15 41 57 56)(16 42 58 49)
(1 45 23 52)(2 42 24 49)(3 47 17 54)(4 44 18 51)(5 41 19 56)(6 46 20 53)(7 43 21 50)(8 48 22 55)(9 33 59 26)(10 38 60 31)(11 35 61 28)(12 40 62 25)(13 37 63 30)(14 34 64 27)(15 39 57 32)(16 36 58 29)
G:=sub<Sym(64)| (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)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64), (2,6)(4,8)(9,13)(11,15)(18,22)(20,24)(25,29)(27,31)(34,38)(36,40)(41,45)(43,47)(50,54)(52,56)(57,61)(59,63), (1,28,23,35)(2,29,24,36)(3,30,17,37)(4,31,18,38)(5,32,19,39)(6,25,20,40)(7,26,21,33)(8,27,22,34)(9,43,59,50)(10,44,60,51)(11,45,61,52)(12,46,62,53)(13,47,63,54)(14,48,64,55)(15,41,57,56)(16,42,58,49), (1,45,23,52)(2,42,24,49)(3,47,17,54)(4,44,18,51)(5,41,19,56)(6,46,20,53)(7,43,21,50)(8,48,22,55)(9,33,59,26)(10,38,60,31)(11,35,61,28)(12,40,62,25)(13,37,63,30)(14,34,64,27)(15,39,57,32)(16,36,58,29)>;
G:=Group( (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)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64), (2,6)(4,8)(9,13)(11,15)(18,22)(20,24)(25,29)(27,31)(34,38)(36,40)(41,45)(43,47)(50,54)(52,56)(57,61)(59,63), (1,28,23,35)(2,29,24,36)(3,30,17,37)(4,31,18,38)(5,32,19,39)(6,25,20,40)(7,26,21,33)(8,27,22,34)(9,43,59,50)(10,44,60,51)(11,45,61,52)(12,46,62,53)(13,47,63,54)(14,48,64,55)(15,41,57,56)(16,42,58,49), (1,45,23,52)(2,42,24,49)(3,47,17,54)(4,44,18,51)(5,41,19,56)(6,46,20,53)(7,43,21,50)(8,48,22,55)(9,33,59,26)(10,38,60,31)(11,35,61,28)(12,40,62,25)(13,37,63,30)(14,34,64,27)(15,39,57,32)(16,36,58,29) );
G=PermutationGroup([[(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),(33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56),(57,58,59,60,61,62,63,64)], [(2,6),(4,8),(9,13),(11,15),(18,22),(20,24),(25,29),(27,31),(34,38),(36,40),(41,45),(43,47),(50,54),(52,56),(57,61),(59,63)], [(1,28,23,35),(2,29,24,36),(3,30,17,37),(4,31,18,38),(5,32,19,39),(6,25,20,40),(7,26,21,33),(8,27,22,34),(9,43,59,50),(10,44,60,51),(11,45,61,52),(12,46,62,53),(13,47,63,54),(14,48,64,55),(15,41,57,56),(16,42,58,49)], [(1,45,23,52),(2,42,24,49),(3,47,17,54),(4,44,18,51),(5,41,19,56),(6,46,20,53),(7,43,21,50),(8,48,22,55),(9,33,59,26),(10,38,60,31),(11,35,61,28),(12,40,62,25),(13,37,63,30),(14,34,64,27),(15,39,57,32),(16,36,58,29)]])
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | ··· | 4J | 4K | ··· | 4T | 8A | ··· | 8P | 8Q | ··· | 8X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | - | ||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | Q8 | C4○D4 | C8○D4 | Q8○M4(2) |
| kernel | M4(2)⋊9Q8 | C4×M4(2) | C8○2M4(2) | C2×C4⋊C8 | C42.6C22 | C8×Q8 | C8⋊4Q8 | C23.37C23 | C22⋊Q8 | C42.C2 | C4⋊Q8 | M4(2) | C2×C4 | C4 | C2 |
| # reps | 1 | 1 | 2 | 1 | 2 | 2 | 6 | 1 | 8 | 4 | 4 | 4 | 4 | 8 | 2 |
Matrix representation of M4(2)⋊9Q8 ►in GL4(𝔽17) generated by
| 0 | 1 | 0 | 0 |
| 13 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 13 | 0 |
| 0 | 0 | 0 | 4 |
| 0 | 8 | 0 | 0 |
| 15 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 16 | 0 |
G:=sub<GL(4,GF(17))| [0,13,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,16,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,13,0,0,0,0,4],[0,15,0,0,8,0,0,0,0,0,0,16,0,0,1,0] >;
M4(2)⋊9Q8 in GAP, Magma, Sage, TeX
M_4(2)\rtimes_9Q_8
% in TeX
G:=Group("M4(2):9Q8"); // GroupNames label
G:=SmallGroup(128,1694);
// by ID
G=gap.SmallGroup(128,1694);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,120,268,521,124]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=c^4=1,d^2=c^2,b*a*b=d*a*d^-1=a^5,a*c=c*a,b*c=c*b,d*b*d^-1=a^4*b,d*c*d^-1=c^-1>;
// generators/relations