p-group, metabelian, nilpotent (class 3), monomial
Aliases: M4(2).1Q8, (C2×C8).58D4, C4⋊C4.113D4, C4.38(C4⋊Q8), C4.14(C22⋊Q8), C4.C42.4C2, C2.23(D4.5D4), C2.23(D4.4D4), C23.284(C4○D4), C22.C42.5C2, M4(2)⋊C4.8C2, (C22×C8).122C22, (C22×C4).740C23, C22.262(C4⋊D4), C42⋊C2.71C22, (C2×M4(2)).33C22, C42.6C22.6C2, C22.11(C42.C2), C4.118(C22.D4), C2.11(C23.81C23), (C2×C4).22(C2×Q8), (C2×C2.D8).19C2, (C2×C4).1382(C2×D4), (C2×C4).356(C4○D4), (C2×C4⋊C4).160C22, SmallGroup(128,821)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for M4(2).Q8
G = < a,b,c,d | a8=b2=1, c4=a4, d2=a6bc2, bab=a5, cac-1=ab, dad-1=a3, cbc-1=dbd-1=a4b, dcd-1=a2bc3 >
Subgroups: 168 in 88 conjugacy classes, 42 normal (22 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, C23, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), M4(2), C22×C4, C22×C4, C4⋊C8, C4.Q8, C2.D8, C2×C4⋊C4, C42⋊C2, C22×C8, C2×M4(2), C2×M4(2), C4.C42, C22.C42, C42.6C22, C2×C2.D8, M4(2)⋊C4, M4(2).Q8
Quotients: C1, C2, C22, D4, Q8, C23, C2×D4, C2×Q8, C4○D4, C4⋊D4, C22⋊Q8, C22.D4, C42.C2, C4⋊Q8, C23.81C23, D4.4D4, D4.5D4, M4(2).Q8
Character table of M4(2).Q8
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 8I | 8J | |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 8 | 8 | 8 | 8 | 8 | 8 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 8 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | linear of order 2 |
| ρ3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | -1 | 1 | 1 | linear of order 2 |
| ρ4 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ5 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | -1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | -1 | -1 | 1 | linear of order 2 |
| ρ6 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | -1 | -1 | 1 | -1 | linear of order 2 |
| ρ7 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | -1 | -1 | -1 | -1 | 1 | -1 | 1 | 1 | -1 | 1 | linear of order 2 |
| ρ8 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | -1 | 1 | 1 | -1 | linear of order 2 |
| ρ9 | 2 | 2 | 2 | 2 | -2 | -2 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | 2 | 2 | 2 | -2 | -2 | -2 | -2 | 2 | 2 | 2 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ11 | 2 | 2 | 2 | 2 | -2 | -2 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ12 | 2 | 2 | 2 | 2 | -2 | -2 | -2 | -2 | 2 | 2 | -2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ13 | 2 | -2 | -2 | 2 | -2 | 2 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | -2 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ14 | 2 | -2 | -2 | 2 | 2 | -2 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | -2 | symplectic lifted from Q8, Schur index 2 |
| ρ15 | 2 | -2 | -2 | 2 | -2 | 2 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 2 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ16 | 2 | -2 | -2 | 2 | 2 | -2 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 2 | symplectic lifted from Q8, Schur index 2 |
| ρ17 | 2 | -2 | -2 | 2 | 2 | -2 | -2 | 2 | 2 | -2 | 0 | -2i | 0 | 0 | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ18 | 2 | -2 | -2 | 2 | -2 | 2 | -2 | 2 | -2 | 2 | 0 | 0 | -2i | 0 | 0 | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ19 | 2 | -2 | -2 | 2 | -2 | 2 | -2 | 2 | -2 | 2 | 0 | 0 | 2i | 0 | 0 | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ20 | 2 | 2 | 2 | 2 | 2 | 2 | -2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2i | 0 | 0 | -2i | 0 | 0 | complex lifted from C4○D4 |
| ρ21 | 2 | -2 | -2 | 2 | 2 | -2 | -2 | 2 | 2 | -2 | 0 | 2i | 0 | 0 | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ22 | 2 | 2 | 2 | 2 | 2 | 2 | -2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2i | 0 | 0 | 2i | 0 | 0 | complex lifted from C4○D4 |
| ρ23 | 4 | -4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2√2 | 2√2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4.4D4 |
| ρ24 | 4 | -4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2√2 | -2√2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4.4D4 |
| ρ25 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2√2 | -2√2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from D4.5D4, Schur index 2 |
| ρ26 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2√2 | 2√2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from D4.5D4, Schur index 2 |
►On 64 points
Generators in S64
(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)(10 14)(12 16)(17 21)(19 23)(26 30)(28 32)(34 38)(36 40)(41 45)(43 47)(50 54)(52 56)(58 62)(60 64)
(1 28 57 52 5 32 61 56)(2 25 62 53 6 29 58 49)(3 26 59 50 7 30 63 54)(4 31 64 51 8 27 60 55)(9 17 33 45 13 21 37 41)(10 22 38 46 14 18 34 42)(11 23 35 43 15 19 39 47)(12 20 40 44 16 24 36 48)
(1 10 63 40)(2 13 64 35)(3 16 57 38)(4 11 58 33)(5 14 59 36)(6 9 60 39)(7 12 61 34)(8 15 62 37)(17 25 47 51)(18 28 48 54)(19 31 41 49)(20 26 42 52)(21 29 43 55)(22 32 44 50)(23 27 45 53)(24 30 46 56)
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)(10,14)(12,16)(17,21)(19,23)(26,30)(28,32)(34,38)(36,40)(41,45)(43,47)(50,54)(52,56)(58,62)(60,64), (1,28,57,52,5,32,61,56)(2,25,62,53,6,29,58,49)(3,26,59,50,7,30,63,54)(4,31,64,51,8,27,60,55)(9,17,33,45,13,21,37,41)(10,22,38,46,14,18,34,42)(11,23,35,43,15,19,39,47)(12,20,40,44,16,24,36,48), (1,10,63,40)(2,13,64,35)(3,16,57,38)(4,11,58,33)(5,14,59,36)(6,9,60,39)(7,12,61,34)(8,15,62,37)(17,25,47,51)(18,28,48,54)(19,31,41,49)(20,26,42,52)(21,29,43,55)(22,32,44,50)(23,27,45,53)(24,30,46,56)>;
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)(10,14)(12,16)(17,21)(19,23)(26,30)(28,32)(34,38)(36,40)(41,45)(43,47)(50,54)(52,56)(58,62)(60,64), (1,28,57,52,5,32,61,56)(2,25,62,53,6,29,58,49)(3,26,59,50,7,30,63,54)(4,31,64,51,8,27,60,55)(9,17,33,45,13,21,37,41)(10,22,38,46,14,18,34,42)(11,23,35,43,15,19,39,47)(12,20,40,44,16,24,36,48), (1,10,63,40)(2,13,64,35)(3,16,57,38)(4,11,58,33)(5,14,59,36)(6,9,60,39)(7,12,61,34)(8,15,62,37)(17,25,47,51)(18,28,48,54)(19,31,41,49)(20,26,42,52)(21,29,43,55)(22,32,44,50)(23,27,45,53)(24,30,46,56) );
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),(10,14),(12,16),(17,21),(19,23),(26,30),(28,32),(34,38),(36,40),(41,45),(43,47),(50,54),(52,56),(58,62),(60,64)], [(1,28,57,52,5,32,61,56),(2,25,62,53,6,29,58,49),(3,26,59,50,7,30,63,54),(4,31,64,51,8,27,60,55),(9,17,33,45,13,21,37,41),(10,22,38,46,14,18,34,42),(11,23,35,43,15,19,39,47),(12,20,40,44,16,24,36,48)], [(1,10,63,40),(2,13,64,35),(3,16,57,38),(4,11,58,33),(5,14,59,36),(6,9,60,39),(7,12,61,34),(8,15,62,37),(17,25,47,51),(18,28,48,54),(19,31,41,49),(20,26,42,52),(21,29,43,55),(22,32,44,50),(23,27,45,53),(24,30,46,56)]])
Matrix representation of M4(2).Q8 ►in GL6(𝔽17)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 11 | 2 | 3 |
| 0 | 0 | 10 | 4 | 3 | 15 |
| 0 | 0 | 15 | 5 | 1 | 10 |
| 0 | 0 | 7 | 11 | 9 | 9 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 16 | 0 |
| 0 | 0 | 3 | 4 | 0 | 16 |
| 0 | 5 | 0 | 0 | 0 | 0 |
| 7 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 5 | 6 | 6 |
| 0 | 0 | 11 | 5 | 11 | 6 |
| 0 | 0 | 4 | 4 | 3 | 3 |
| 0 | 0 | 11 | 2 | 14 | 4 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 4 | 0 | 15 |
| 0 | 0 | 16 | 0 | 2 | 0 |
| 0 | 0 | 10 | 11 | 0 | 16 |
| 0 | 0 | 2 | 6 | 4 | 14 |
G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,16,0,0,0,0,0,0,3,10,15,7,0,0,11,4,5,11,0,0,2,3,1,9,0,0,3,15,10,9],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,1,3,0,0,0,1,0,4,0,0,0,0,16,0,0,0,0,0,0,16],[0,7,0,0,0,0,5,0,0,0,0,0,0,0,5,11,4,11,0,0,5,5,4,2,0,0,6,11,3,14,0,0,6,6,3,4],[4,0,0,0,0,0,0,13,0,0,0,0,0,0,3,16,10,2,0,0,4,0,11,6,0,0,0,2,0,4,0,0,15,0,16,14] >;
M4(2).Q8 in GAP, Magma, Sage, TeX
M_4(2).Q_8
% in TeX
G:=Group("M4(2).Q8"); // GroupNames label
G:=SmallGroup(128,821);
// by ID
G=gap.SmallGroup(128,821);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,112,141,64,422,387,58,718,172,2028,1027,124]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=1,c^4=a^4,d^2=a^6*b*c^2,b*a*b=a^5,c*a*c^-1=a*b,d*a*d^-1=a^3,c*b*c^-1=d*b*d^-1=a^4*b,d*c*d^-1=a^2*b*c^3>;
// generators/relations
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