p-group, metabelian, nilpotent (class 3), monomial
Aliases: M4(2)⋊6Q8, C42.254D4, C42.720C23, C8⋊Q8⋊7C2, C8.5(C2×Q8), C4.43(C4⋊Q8), C8.5Q8⋊16C2, C4.17(C22×Q8), C4⋊C4.105C23, (C2×C4).364C24, (C4×C8).180C22, (C2×C8).275C23, C23.687(C2×D4), (C22×C4).474D4, (C4×M4(2)).5C2, C4⋊Q8.288C22, C22.19(C4⋊Q8), C4.Q8.25C22, C2.D8.95C22, C8⋊C4.125C22, (C2×C42).861C22, C22.624(C22×D4), C2.41(D8⋊C22), M4(2)⋊C4.15C2, (C22×C4).1048C23, C42.C2.119C22, C42⋊C2.149C22, (C2×M4(2)).278C22, C23.37C23.34C2, C2.34(C2×C4⋊Q8), (C2×C4).697(C2×D4), (C2×C4).108(C2×Q8), (C2×C4⋊C4).635C22, (C2×C42.C2).35C2, SmallGroup(128,1898)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 276 in 174 conjugacy classes, 108 normal (14 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×4], C4 [×12], C22, C22 [×2], C22 [×2], C8 [×8], C2×C4 [×2], C2×C4 [×8], C2×C4 [×14], Q8 [×4], C23, C42 [×2], C42 [×2], C42 [×2], C22⋊C4 [×2], C4⋊C4 [×8], C4⋊C4 [×16], C2×C8 [×4], M4(2) [×8], C22×C4, C22×C4 [×2], C22×C4 [×2], C2×Q8 [×2], C4×C8 [×2], C8⋊C4 [×2], C4.Q8 [×8], C2.D8 [×8], C2×C42, C2×C4⋊C4 [×2], C2×C4⋊C4 [×2], C42⋊C2 [×2], C4×Q8 [×2], C22⋊Q8 [×2], C42.C2 [×6], C42.C2 [×2], C4⋊Q8 [×2], C2×M4(2) [×2], C4×M4(2), M4(2)⋊C4 [×4], C8.5Q8 [×4], C8⋊Q8 [×4], C2×C42.C2, C23.37C23, M4(2)⋊6Q8
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], Q8 [×8], C23 [×15], C2×D4 [×6], C2×Q8 [×12], C24, C4⋊Q8 [×4], C22×D4, C22×Q8 [×2], C2×C4⋊Q8, D8⋊C22 [×2], M4(2)⋊6Q8
Generators and relations
G = < a,b,c,d | a8=b2=c4=1, d2=c2, bab=cac-1=a5, dad-1=a-1, 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)
(1 41)(2 46)(3 43)(4 48)(5 45)(6 42)(7 47)(8 44)(9 19)(10 24)(11 21)(12 18)(13 23)(14 20)(15 17)(16 22)(25 53)(26 50)(27 55)(28 52)(29 49)(30 54)(31 51)(32 56)(33 57)(34 62)(35 59)(36 64)(37 61)(38 58)(39 63)(40 60)
(1 20 45 10)(2 17 46 15)(3 22 47 12)(4 19 48 9)(5 24 41 14)(6 21 42 11)(7 18 43 16)(8 23 44 13)(25 33 53 57)(26 38 54 62)(27 35 55 59)(28 40 56 64)(29 37 49 61)(30 34 50 58)(31 39 51 63)(32 36 52 60)
(1 29 45 49)(2 28 46 56)(3 27 47 55)(4 26 48 54)(5 25 41 53)(6 32 42 52)(7 31 43 51)(8 30 44 50)(9 38 19 62)(10 37 20 61)(11 36 21 60)(12 35 22 59)(13 34 23 58)(14 33 24 57)(15 40 17 64)(16 39 18 63)
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), (1,41)(2,46)(3,43)(4,48)(5,45)(6,42)(7,47)(8,44)(9,19)(10,24)(11,21)(12,18)(13,23)(14,20)(15,17)(16,22)(25,53)(26,50)(27,55)(28,52)(29,49)(30,54)(31,51)(32,56)(33,57)(34,62)(35,59)(36,64)(37,61)(38,58)(39,63)(40,60), (1,20,45,10)(2,17,46,15)(3,22,47,12)(4,19,48,9)(5,24,41,14)(6,21,42,11)(7,18,43,16)(8,23,44,13)(25,33,53,57)(26,38,54,62)(27,35,55,59)(28,40,56,64)(29,37,49,61)(30,34,50,58)(31,39,51,63)(32,36,52,60), (1,29,45,49)(2,28,46,56)(3,27,47,55)(4,26,48,54)(5,25,41,53)(6,32,42,52)(7,31,43,51)(8,30,44,50)(9,38,19,62)(10,37,20,61)(11,36,21,60)(12,35,22,59)(13,34,23,58)(14,33,24,57)(15,40,17,64)(16,39,18,63)>;
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), (1,41)(2,46)(3,43)(4,48)(5,45)(6,42)(7,47)(8,44)(9,19)(10,24)(11,21)(12,18)(13,23)(14,20)(15,17)(16,22)(25,53)(26,50)(27,55)(28,52)(29,49)(30,54)(31,51)(32,56)(33,57)(34,62)(35,59)(36,64)(37,61)(38,58)(39,63)(40,60), (1,20,45,10)(2,17,46,15)(3,22,47,12)(4,19,48,9)(5,24,41,14)(6,21,42,11)(7,18,43,16)(8,23,44,13)(25,33,53,57)(26,38,54,62)(27,35,55,59)(28,40,56,64)(29,37,49,61)(30,34,50,58)(31,39,51,63)(32,36,52,60), (1,29,45,49)(2,28,46,56)(3,27,47,55)(4,26,48,54)(5,25,41,53)(6,32,42,52)(7,31,43,51)(8,30,44,50)(9,38,19,62)(10,37,20,61)(11,36,21,60)(12,35,22,59)(13,34,23,58)(14,33,24,57)(15,40,17,64)(16,39,18,63) );
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)], [(1,41),(2,46),(3,43),(4,48),(5,45),(6,42),(7,47),(8,44),(9,19),(10,24),(11,21),(12,18),(13,23),(14,20),(15,17),(16,22),(25,53),(26,50),(27,55),(28,52),(29,49),(30,54),(31,51),(32,56),(33,57),(34,62),(35,59),(36,64),(37,61),(38,58),(39,63),(40,60)], [(1,20,45,10),(2,17,46,15),(3,22,47,12),(4,19,48,9),(5,24,41,14),(6,21,42,11),(7,18,43,16),(8,23,44,13),(25,33,53,57),(26,38,54,62),(27,35,55,59),(28,40,56,64),(29,37,49,61),(30,34,50,58),(31,39,51,63),(32,36,52,60)], [(1,29,45,49),(2,28,46,56),(3,27,47,55),(4,26,48,54),(5,25,41,53),(6,32,42,52),(7,31,43,51),(8,30,44,50),(9,38,19,62),(10,37,20,61),(11,36,21,60),(12,35,22,59),(13,34,23,58),(14,33,24,57),(15,40,17,64),(16,39,18,63)])
Matrix representation ►G ⊆ GL6(𝔽17)
| 1 | 15 | 0 | 0 | 0 | 0 |
| 1 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 4 | 4 | 13 | 9 |
| 0 | 0 | 0 | 13 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 13 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 1 | 1 | 0 | 16 |
| 16 | 2 | 0 | 0 | 0 | 0 |
| 16 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 13 | 0 |
| 0 | 0 | 4 | 4 | 0 | 13 |
| 8 | 3 | 0 | 0 | 0 | 0 |
| 1 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 13 | 13 | 15 | 8 |
| 0 | 0 | 6 | 6 | 7 | 5 |
| 0 | 0 | 6 | 4 | 0 | 0 |
| 0 | 0 | 13 | 2 | 11 | 15 |
G:=sub<GL(6,GF(17))| [1,1,0,0,0,0,15,16,0,0,0,0,0,0,0,4,0,0,0,0,0,4,13,4,0,0,4,13,0,0,0,0,0,9,0,13],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,1,0,1,0,0,0,0,16,0,0,0,0,0,0,16],[16,16,0,0,0,0,2,1,0,0,0,0,0,0,4,0,0,4,0,0,0,4,0,4,0,0,0,0,13,0,0,0,0,0,0,13],[8,1,0,0,0,0,3,9,0,0,0,0,0,0,13,6,6,13,0,0,13,6,4,2,0,0,15,7,0,11,0,0,8,5,0,15] >;
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | ··· | 4H | 4I | 4J | 4K | ··· | 4R | 8A | ··· | 8H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | - | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | Q8 | D4 | D8⋊C22 |
| kernel | M4(2)⋊6Q8 | C4×M4(2) | M4(2)⋊C4 | C8.5Q8 | C8⋊Q8 | C2×C42.C2 | C23.37C23 | C42 | M4(2) | C22×C4 | C2 |
| # reps | 1 | 1 | 4 | 4 | 4 | 1 | 1 | 2 | 8 | 2 | 4 |
In GAP, Magma, Sage, TeX
M_{4(2)}\rtimes_6Q_8 % in TeX
G:=Group("M4(2):6Q8"); // GroupNames label
G:=SmallGroup(128,1898);
// by ID
G=gap.SmallGroup(128,1898);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,112,253,568,758,184,521,4037,124]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=c^4=1,d^2=c^2,b*a*b=c*a*c^-1=a^5,d*a*d^-1=a^-1,b*c=c*b,d*b*d^-1=a^4*b,d*c*d^-1=c^-1>;
// generators/relations