p-group, metabelian, nilpotent (class 3), monomial
Aliases: M4(2).27D4, C8.9(C4⋊C4), (C2×C8).17Q8, C8.C4⋊9C4, (C2×C8).357D4, C4.149(C4×D4), C22⋊C4.5Q8, C23.5(C2×Q8), C22.7(C4⋊Q8), C22.10(C4×Q8), C4.202(C4⋊D4), C8○2M4(2).4C2, M4(2).12(C2×C4), C4.18(C42.C2), M4(2).C4.1C2, M4(2)⋊4C4.1C2, (C22×C8).224C22, (C22×C4).698C23, C22.27(C22⋊Q8), C42⋊C2.277C22, (C2×M4(2)).205C22, C2.20(C23.65C23), C4.47(C2×C4⋊C4), (C2×C8).70(C2×C4), (C2×C4).118(C2×Q8), (C2×C4).63(C4○D4), (C2×C4).1018(C2×D4), (C2×C8.C4).18C2, (C2×C4).199(C22×C4), SmallGroup(128,685)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for M4(2).27D4
G = < a,b,c,d | a8=b2=1, c4=a4, d2=a2, bab=a5, ac=ca, dad-1=a-1b, bc=cb, dbd-1=a4b, dcd-1=c3 >
Subgroups: 140 in 92 conjugacy classes, 54 normal (34 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, C23, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), M4(2), C22×C4, C4×C8, C8⋊C4, C8.C4, C8.C4, C42⋊C2, C22×C8, C2×M4(2), C2×M4(2), M4(2)⋊4C4, C8○2M4(2), C2×C8.C4, M4(2).C4, M4(2).27D4
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C4⋊C4, C22×C4, C2×D4, C2×Q8, C4○D4, C2×C4⋊C4, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C42.C2, C4⋊Q8, C23.65C23, M4(2).27D4
►On 32 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)
(1 17)(2 22)(3 19)(4 24)(5 21)(6 18)(7 23)(8 20)(9 25)(10 30)(11 27)(12 32)(13 29)(14 26)(15 31)(16 28)
(1 24 3 18 5 20 7 22)(2 17 4 19 6 21 8 23)(9 12 15 10 13 16 11 14)(25 32 31 30 29 28 27 26)
(1 31 3 25 5 27 7 29)(2 10 4 12 6 14 8 16)(9 17 11 19 13 21 15 23)(18 30 20 32 22 26 24 28)
G:=sub<Sym(32)| (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), (1,17)(2,22)(3,19)(4,24)(5,21)(6,18)(7,23)(8,20)(9,25)(10,30)(11,27)(12,32)(13,29)(14,26)(15,31)(16,28), (1,24,3,18,5,20,7,22)(2,17,4,19,6,21,8,23)(9,12,15,10,13,16,11,14)(25,32,31,30,29,28,27,26), (1,31,3,25,5,27,7,29)(2,10,4,12,6,14,8,16)(9,17,11,19,13,21,15,23)(18,30,20,32,22,26,24,28)>;
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), (1,17)(2,22)(3,19)(4,24)(5,21)(6,18)(7,23)(8,20)(9,25)(10,30)(11,27)(12,32)(13,29)(14,26)(15,31)(16,28), (1,24,3,18,5,20,7,22)(2,17,4,19,6,21,8,23)(9,12,15,10,13,16,11,14)(25,32,31,30,29,28,27,26), (1,31,3,25,5,27,7,29)(2,10,4,12,6,14,8,16)(9,17,11,19,13,21,15,23)(18,30,20,32,22,26,24,28) );
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)], [(1,17),(2,22),(3,19),(4,24),(5,21),(6,18),(7,23),(8,20),(9,25),(10,30),(11,27),(12,32),(13,29),(14,26),(15,31),(16,28)], [(1,24,3,18,5,20,7,22),(2,17,4,19,6,21,8,23),(9,12,15,10,13,16,11,14),(25,32,31,30,29,28,27,26)], [(1,31,3,25,5,27,7,29),(2,10,4,12,6,14,8,16),(9,17,11,19,13,21,15,23),(18,30,20,32,22,26,24,28)]])
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 8A | 8B | 8C | 8D | 8E | ··· | 8J | 8K | ··· | 8R |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | ··· | 8 | 8 | ··· | 8 |
| size | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | ··· | 8 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | - | + | - | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C4 | Q8 | D4 | Q8 | D4 | C4○D4 | M4(2).27D4 |
| kernel | M4(2).27D4 | M4(2)⋊4C4 | C8○2M4(2) | C2×C8.C4 | M4(2).C4 | C8.C4 | C22⋊C4 | C2×C8 | C2×C8 | M4(2) | C2×C4 | C1 |
| # reps | 1 | 2 | 1 | 2 | 2 | 8 | 2 | 2 | 2 | 2 | 4 | 4 |
Matrix representation of M4(2).27D4 ►in GL4(𝔽17) generated by
| 9 | 0 | 0 | 0 |
| 16 | 8 | 0 | 0 |
| 0 | 0 | 8 | 8 |
| 0 | 0 | 0 | 9 |
| 13 | 13 | 0 | 0 |
| 8 | 4 | 0 | 0 |
| 0 | 0 | 4 | 4 |
| 0 | 0 | 9 | 13 |
| 9 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 |
| 0 | 0 | 15 | 0 |
| 0 | 0 | 0 | 15 |
| 0 | 0 | 15 | 0 |
| 0 | 0 | 0 | 15 |
| 2 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 |
G:=sub<GL(4,GF(17))| [9,16,0,0,0,8,0,0,0,0,8,0,0,0,8,9],[13,8,0,0,13,4,0,0,0,0,4,9,0,0,4,13],[9,0,0,0,0,9,0,0,0,0,15,0,0,0,0,15],[0,0,2,0,0,0,0,2,15,0,0,0,0,15,0,0] >;
M4(2).27D4 in GAP, Magma, Sage, TeX
M_4(2)._{27}D_4 % in TeX
G:=Group("M4(2).27D4"); // GroupNames label
G:=SmallGroup(128,685);
// by ID
G=gap.SmallGroup(128,685);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,232,422,100,2019,1018,248,2804,172,124]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=1,c^4=a^4,d^2=a^2,b*a*b=a^5,a*c=c*a,d*a*d^-1=a^-1*b,b*c=c*b,d*b*d^-1=a^4*b,d*c*d^-1=c^3>;
// generators/relations