p-group, metabelian, nilpotent (class 3), monomial
Aliases: M4(2).42D4, C4.57(C4×D4), C4○D4.6Q8, D4⋊C4⋊6C4, D4.7(C4⋊C4), Q8⋊C4⋊4C4, C4○D4.44D4, (C2×C8).320D4, Q8.7(C4⋊C4), C2.13(C8○D8), C42⋊6C4⋊19C2, C4.119C22≀C2, C22.148(C4×D4), C2.13(C8.26D4), C4.113(C22⋊Q8), C23.203(C4○D4), (C22×C8).387C22, (C2×C42).272C22, C23.24D4.5C2, C22.16(C22⋊Q8), (C22×C4).1357C23, C42⋊C2.13C22, C42.6C22⋊20C2, C22.7C42⋊23C2, C2.22(C23.8Q8), (C2×M4(2)).316C22, C22.4(C22.D4), C4.16(C2×C4⋊C4), (C2×C4≀C2).8C2, C4⋊C4.67(C2×C4), (C2×C8).36(C2×C4), (C2×C8.C4)⋊8C2, (C2×C8○D4).14C2, (C2×C4).992(C2×D4), (C2×C4).268(C2×Q8), (C2×D4).163(C2×C4), (C2×Q8).145(C2×C4), (C2×C4).753(C4○D4), (C2×C4).375(C22×C4), (C2×C4○D4).262C22, SmallGroup(128,598)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for M4(2).42D4
G = < a,b,c,d | a8=b2=c4=1, d2=a6, bab=a5, cac-1=a3b, dad-1=a-1b, bc=cb, dbd-1=a4b, dcd-1=a6c-1 >
Subgroups: 236 in 135 conjugacy classes, 56 normal (46 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), M4(2), C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C4○D4, D4⋊C4, Q8⋊C4, C4≀C2, C4⋊C8, C8.C4, C2×C42, C42⋊C2, C22×C8, C22×C8, C2×M4(2), C2×M4(2), C8○D4, C2×C4○D4, C22.7C42, C42⋊6C4, C23.24D4, C2×C4≀C2, C42.6C22, C2×C8.C4, C2×C8○D4, M4(2).42D4
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, C22≀C2, C22⋊Q8, C22.D4, C23.8Q8, C8○D8, C8.26D4, M4(2).42D4
►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 12)(2 9)(3 14)(4 11)(5 16)(6 13)(7 10)(8 15)(17 25)(18 30)(19 27)(20 32)(21 29)(22 26)(23 31)(24 28)
(1 5)(2 11 6 15)(3 7)(4 13 8 9)(10 14)(12 16)(17 19 21 23)(18 30)(20 32)(22 26)(24 28)(25 27 29 31)
(1 19 7 17 5 23 3 21)(2 30 8 28 6 26 4 32)(9 22 15 20 13 18 11 24)(10 29 16 27 14 25 12 31)
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), (1,12)(2,9)(3,14)(4,11)(5,16)(6,13)(7,10)(8,15)(17,25)(18,30)(19,27)(20,32)(21,29)(22,26)(23,31)(24,28), (1,5)(2,11,6,15)(3,7)(4,13,8,9)(10,14)(12,16)(17,19,21,23)(18,30)(20,32)(22,26)(24,28)(25,27,29,31), (1,19,7,17,5,23,3,21)(2,30,8,28,6,26,4,32)(9,22,15,20,13,18,11,24)(10,29,16,27,14,25,12,31)>;
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), (1,12)(2,9)(3,14)(4,11)(5,16)(6,13)(7,10)(8,15)(17,25)(18,30)(19,27)(20,32)(21,29)(22,26)(23,31)(24,28), (1,5)(2,11,6,15)(3,7)(4,13,8,9)(10,14)(12,16)(17,19,21,23)(18,30)(20,32)(22,26)(24,28)(25,27,29,31), (1,19,7,17,5,23,3,21)(2,30,8,28,6,26,4,32)(9,22,15,20,13,18,11,24)(10,29,16,27,14,25,12,31) );
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)], [(1,12),(2,9),(3,14),(4,11),(5,16),(6,13),(7,10),(8,15),(17,25),(18,30),(19,27),(20,32),(21,29),(22,26),(23,31),(24,28)], [(1,5),(2,11,6,15),(3,7),(4,13,8,9),(10,14),(12,16),(17,19,21,23),(18,30),(20,32),(22,26),(24,28),(25,27,29,31)], [(1,19,7,17,5,23,3,21),(2,30,8,28,6,26,4,32),(9,22,15,20,13,18,11,24),(10,29,16,27,14,25,12,31)]])
38 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | ··· | 4L | 4M | 4N | 8A | 8B | 8C | 8D | 8E | ··· | 8N | 8O | 8P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | ··· | 8 | 8 | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | ··· | 4 | 8 | 8 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | 8 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | - | ||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | D4 | D4 | D4 | Q8 | C4○D4 | C4○D4 | C8○D8 | C8.26D4 |
| kernel | M4(2).42D4 | C22.7C42 | C42⋊6C4 | C23.24D4 | C2×C4≀C2 | C42.6C22 | C2×C8.C4 | C2×C8○D4 | D4⋊C4 | Q8⋊C4 | C2×C8 | M4(2) | C4○D4 | C4○D4 | C2×C4 | C23 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 | 2 | 2 | 2 | 2 | 2 | 2 | 8 | 2 |
Matrix representation of M4(2).42D4 ►in GL4(𝔽17) generated by
| 0 | 16 | 0 | 0 |
| 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 8 |
| 0 | 0 | 9 | 0 |
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 4 |
| 1 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 4 |
| 0 | 0 | 16 | 0 |
G:=sub<GL(4,GF(17))| [0,16,0,0,16,0,0,0,0,0,0,9,0,0,8,0],[16,0,0,0,0,16,0,0,0,0,16,0,0,0,0,1],[1,0,0,0,0,16,0,0,0,0,16,0,0,0,0,4],[1,0,0,0,0,16,0,0,0,0,0,16,0,0,4,0] >;
M4(2).42D4 in GAP, Magma, Sage, TeX
M_4(2)._{42}D_4 % in TeX
G:=Group("M4(2).42D4"); // GroupNames label
G:=SmallGroup(128,598);
// by ID
G=gap.SmallGroup(128,598);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,232,422,2019,1018,521,248,2804,1411,718,172,124]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=c^4=1,d^2=a^6,b*a*b=a^5,c*a*c^-1=a^3*b,d*a*d^-1=a^-1*b,b*c=c*b,d*b*d^-1=a^4*b,d*c*d^-1=a^6*c^-1>;
// generators/relations