p-group, metabelian, nilpotent (class 2), monomial
Aliases: M4(2)⋊9Q8, C42.288C23, (C8×Q8)⋊28C2, C4⋊Q8.31C4, C4.34(C4×Q8), C8.39(C2×Q8), C8⋊4Q8⋊33C2, C4.38(C8○D4), M4(2)○2(C4⋊C8), C22⋊Q8.20C4, C22.12(C4×Q8), C4.66(C22×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
Subgroups: 220 in 178 conjugacy classes, 142 normal (32 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×4], C4 [×11], C22, C22 [×2], C22 [×2], C8 [×4], C8 [×6], C2×C4 [×6], C2×C4 [×12], C2×C4 [×3], Q8 [×6], C23, C42 [×4], C42 [×4], C22⋊C4 [×4], C4⋊C4 [×16], C2×C8 [×8], C2×C8 [×4], M4(2) [×4], M4(2) [×2], C22×C4 [×3], C2×Q8 [×4], C4×C8 [×6], C8⋊C4 [×6], C4⋊C8 [×2], C4⋊C8 [×10], C2×C42, C42⋊C2 [×2], C4×Q8 [×4], C22⋊Q8 [×4], C42.C2 [×2], C4⋊Q8 [×2], C22×C8 [×2], C2×M4(2) [×2], C4×M4(2), C8○2M4(2) [×2], C2×C4⋊C8, C42.6C22 [×2], C8×Q8 [×2], C8⋊4Q8 [×6], C23.37C23, M4(2)⋊9Q8
Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], Q8 [×4], C23 [×15], C22×C4 [×14], C2×Q8 [×6], C4○D4 [×2], C24, C4×Q8 [×4], C8○D4 [×2], C23×C4, C22×Q8, C2×C4○D4, C2×C4×Q8, C2×C8○D4, Q8○M4(2), M4(2)⋊9Q8
Generators and relations
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 >
►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 26 19 35)(2 27 20 36)(3 28 21 37)(4 29 22 38)(5 30 23 39)(6 31 24 40)(7 32 17 33)(8 25 18 34)(9 41 57 50)(10 42 58 51)(11 43 59 52)(12 44 60 53)(13 45 61 54)(14 46 62 55)(15 47 63 56)(16 48 64 49)
(1 43 19 52)(2 48 20 49)(3 45 21 54)(4 42 22 51)(5 47 23 56)(6 44 24 53)(7 41 17 50)(8 46 18 55)(9 33 57 32)(10 38 58 29)(11 35 59 26)(12 40 60 31)(13 37 61 28)(14 34 62 25)(15 39 63 30)(16 36 64 27)
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), (1,26,19,35)(2,27,20,36)(3,28,21,37)(4,29,22,38)(5,30,23,39)(6,31,24,40)(7,32,17,33)(8,25,18,34)(9,41,57,50)(10,42,58,51)(11,43,59,52)(12,44,60,53)(13,45,61,54)(14,46,62,55)(15,47,63,56)(16,48,64,49), (1,43,19,52)(2,48,20,49)(3,45,21,54)(4,42,22,51)(5,47,23,56)(6,44,24,53)(7,41,17,50)(8,46,18,55)(9,33,57,32)(10,38,58,29)(11,35,59,26)(12,40,60,31)(13,37,61,28)(14,34,62,25)(15,39,63,30)(16,36,64,27)>;
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), (1,26,19,35)(2,27,20,36)(3,28,21,37)(4,29,22,38)(5,30,23,39)(6,31,24,40)(7,32,17,33)(8,25,18,34)(9,41,57,50)(10,42,58,51)(11,43,59,52)(12,44,60,53)(13,45,61,54)(14,46,62,55)(15,47,63,56)(16,48,64,49), (1,43,19,52)(2,48,20,49)(3,45,21,54)(4,42,22,51)(5,47,23,56)(6,44,24,53)(7,41,17,50)(8,46,18,55)(9,33,57,32)(10,38,58,29)(11,35,59,26)(12,40,60,31)(13,37,61,28)(14,34,62,25)(15,39,63,30)(16,36,64,27) );
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)], [(1,26,19,35),(2,27,20,36),(3,28,21,37),(4,29,22,38),(5,30,23,39),(6,31,24,40),(7,32,17,33),(8,25,18,34),(9,41,57,50),(10,42,58,51),(11,43,59,52),(12,44,60,53),(13,45,61,54),(14,46,62,55),(15,47,63,56),(16,48,64,49)], [(1,43,19,52),(2,48,20,49),(3,45,21,54),(4,42,22,51),(5,47,23,56),(6,44,24,53),(7,41,17,50),(8,46,18,55),(9,33,57,32),(10,38,58,29),(11,35,59,26),(12,40,60,31),(13,37,61,28),(14,34,62,25),(15,39,63,30),(16,36,64,27)])
Matrix representation ►G ⊆ 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] >;
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 |
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