p-group, metabelian, nilpotent (class 3), monomial
Aliases: M4(2)⋊13D4, C42.434D4, C4⋊2C4≀C2, C4⋊Q8⋊20C4, C4.94(C4×D4), C4⋊1D4⋊16C4, C4.4D4⋊18C4, (C4×M4(2))⋊26C2, C42.158(C2×C4), C23.572(C2×D4), (C22×C4).301D4, C4.208(C4⋊D4), C22.6(C4⋊1D4), (C2×C42).326C22, (C22×C4).1416C23, C22.35(C4.4D4), C2.44(C42⋊C22), (C2×M4(2)).326C22, C22.26C24.22C2, C2.32(C24.3C22), (C4×C4⋊C4)⋊8C2, (C2×C4≀C2)⋊22C2, C2.44(C2×C4≀C2), (C2×D4).118(C2×C4), (C2×C4).1545(C2×D4), (C2×Q8).103(C2×C4), (C2×C4).868(C4○D4), (C2×C4).430(C22×C4), (C2×C4○D4).42C22, (C2×C4).203(C22⋊C4), C22.290(C2×C22⋊C4), SmallGroup(128,712)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for M4(2)⋊13D4
G = < a,b,c,d | a8=b2=c4=d2=1, bab=a5, ac=ca, dad=a-1b, bc=cb, dbd=a4b, dcd=c-1 >
Subgroups: 348 in 168 conjugacy classes, 58 normal (28 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), M4(2), C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C2.C42, C4×C8, C8⋊C4, C4≀C2, C2×C42, C2×C42, C2×C4⋊C4, C4×D4, C4⋊D4, C4.4D4, C4⋊1D4, C4⋊Q8, C2×M4(2), C2×C4○D4, C4×C4⋊C4, C4×M4(2), C2×C4≀C2, C22.26C24, M4(2)⋊13D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C4○D4, C4≀C2, C2×C22⋊C4, C4×D4, C4⋊D4, C4.4D4, C4⋊1D4, C24.3C22, C2×C4≀C2, C42⋊C22, M4(2)⋊13D4
►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)
(2 6)(4 8)(10 14)(12 16)(17 21)(19 23)(26 30)(28 32)
(1 24 27 9)(2 17 28 10)(3 18 29 11)(4 19 30 12)(5 20 31 13)(6 21 32 14)(7 22 25 15)(8 23 26 16)
(1 28)(2 27)(3 30)(4 29)(5 32)(6 31)(7 26)(8 25)(9 10)(11 12)(13 14)(15 16)(17 24)(18 19)(20 21)(22 23)
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), (2,6)(4,8)(10,14)(12,16)(17,21)(19,23)(26,30)(28,32), (1,24,27,9)(2,17,28,10)(3,18,29,11)(4,19,30,12)(5,20,31,13)(6,21,32,14)(7,22,25,15)(8,23,26,16), (1,28)(2,27)(3,30)(4,29)(5,32)(6,31)(7,26)(8,25)(9,10)(11,12)(13,14)(15,16)(17,24)(18,19)(20,21)(22,23)>;
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), (2,6)(4,8)(10,14)(12,16)(17,21)(19,23)(26,30)(28,32), (1,24,27,9)(2,17,28,10)(3,18,29,11)(4,19,30,12)(5,20,31,13)(6,21,32,14)(7,22,25,15)(8,23,26,16), (1,28)(2,27)(3,30)(4,29)(5,32)(6,31)(7,26)(8,25)(9,10)(11,12)(13,14)(15,16)(17,24)(18,19)(20,21)(22,23) );
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)], [(2,6),(4,8),(10,14),(12,16),(17,21),(19,23),(26,30),(28,32)], [(1,24,27,9),(2,17,28,10),(3,18,29,11),(4,19,30,12),(5,20,31,13),(6,21,32,14),(7,22,25,15),(8,23,26,16)], [(1,28),(2,27),(3,30),(4,29),(5,32),(6,31),(7,26),(8,25),(9,10),(11,12),(13,14),(15,16),(17,24),(18,19),(20,21),(22,23)]])
38 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | ··· | 4J | 4K | ··· | 4T | 4U | 4V | 8A | ··· | 8H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | 4 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 8 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 | 4 | ··· | 4 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | ||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | D4 | D4 | D4 | C4○D4 | C4≀C2 | C42⋊C22 |
| kernel | M4(2)⋊13D4 | C4×C4⋊C4 | C4×M4(2) | C2×C4≀C2 | C22.26C24 | C4.4D4 | C4⋊1D4 | C4⋊Q8 | C42 | M4(2) | C22×C4 | C2×C4 | C4 | C2 |
| # reps | 1 | 1 | 1 | 4 | 1 | 4 | 2 | 2 | 2 | 4 | 2 | 4 | 8 | 2 |
Matrix representation of M4(2)⋊13D4 ►in GL4(𝔽17) generated by
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 4 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 16 |
| 16 | 16 | 0 | 0 |
| 2 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 15 | 16 | 0 | 0 |
| 0 | 0 | 0 | 13 |
| 0 | 0 | 4 | 0 |
G:=sub<GL(4,GF(17))| [16,0,0,0,0,16,0,0,0,0,0,4,0,0,1,0],[1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,16],[16,2,0,0,16,1,0,0,0,0,1,0,0,0,0,1],[1,15,0,0,0,16,0,0,0,0,0,4,0,0,13,0] >;
M4(2)⋊13D4 in GAP, Magma, Sage, TeX
M_4(2)\rtimes_{13}D_4 % in TeX
G:=Group("M4(2):13D4"); // GroupNames label
G:=SmallGroup(128,712);
// by ID
G=gap.SmallGroup(128,712);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,288,422,100,1018,248,2028]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=c^4=d^2=1,b*a*b=a^5,a*c=c*a,d*a*d=a^-1*b,b*c=c*b,d*b*d=a^4*b,d*c*d=c^-1>;
// generators/relations