p-group, metabelian, nilpotent (class 3), monomial
Aliases: D4⋊3M4(2), C42.48D4, C42.607C23, D4⋊C8⋊26C2, (C4×C8)⋊2C22, C4.80(C2×D8), C4⋊C8⋊44C22, (C4×D4).15C4, (C2×C4).127D8, C42.60(C2×C4), C4.93(C2×SD16), (C2×C4).98SD16, (C22×D4).24C4, (C22×C4).730D4, C4.18(C2×M4(2)), C4.34(D4⋊C4), C4⋊M4(2)⋊15C2, (C4×D4).264C22, C42.12C4⋊11C2, (C2×C42).163C22, C22.24(D4⋊C4), C23.169(C22⋊C4), C2.12(C24.4C4), C2.9(C42⋊C22), (C2×C4×D4).7C2, (C2×C4⋊C4).40C4, C4⋊C4.179(C2×C4), C2.5(C2×D4⋊C4), (C2×D4).192(C2×C4), (C2×C4).1449(C2×D4), (C2×C4).78(C22⋊C4), (C2×C4).312(C22×C4), (C22×C4).185(C2×C4), C22.162(C2×C22⋊C4), SmallGroup(128,218)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for D4⋊M4(2)
G = < a,b,c,d | a4=b2=c8=d2=1, bab=cac-1=a-1, ad=da, cbc-1=ab, bd=db, dcd=c5 >
Subgroups: 340 in 156 conjugacy classes, 58 normal (28 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, D4, C23, C23, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, C22×C4, C2×D4, C2×D4, C24, C4×C8, C22⋊C8, C4⋊C8, C4⋊C8, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C4×D4, C4×D4, C2×M4(2), C23×C4, C22×D4, D4⋊C8, C4⋊M4(2), C42.12C4, C2×C4×D4, D4⋊M4(2)
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, M4(2), D8, SD16, C22×C4, C2×D4, D4⋊C4, C2×C22⋊C4, C2×M4(2), C2×D8, C2×SD16, C24.4C4, C2×D4⋊C4, C42⋊C22, D4⋊M4(2)
►On 32 points
(1 25 14 19)(2 20 15 26)(3 27 16 21)(4 22 9 28)(5 29 10 23)(6 24 11 30)(7 31 12 17)(8 18 13 32)
(1 25)(3 27)(5 29)(7 31)(10 23)(12 17)(14 19)(16 21)(18 32)(20 26)(22 28)(24 30)
(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 10)(2 15)(3 12)(4 9)(5 14)(6 11)(7 16)(8 13)(17 27)(18 32)(19 29)(20 26)(21 31)(22 28)(23 25)(24 30)
G:=sub<Sym(32)| (1,25,14,19)(2,20,15,26)(3,27,16,21)(4,22,9,28)(5,29,10,23)(6,24,11,30)(7,31,12,17)(8,18,13,32), (1,25)(3,27)(5,29)(7,31)(10,23)(12,17)(14,19)(16,21)(18,32)(20,26)(22,28)(24,30), (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), (1,10)(2,15)(3,12)(4,9)(5,14)(6,11)(7,16)(8,13)(17,27)(18,32)(19,29)(20,26)(21,31)(22,28)(23,25)(24,30)>;
G:=Group( (1,25,14,19)(2,20,15,26)(3,27,16,21)(4,22,9,28)(5,29,10,23)(6,24,11,30)(7,31,12,17)(8,18,13,32), (1,25)(3,27)(5,29)(7,31)(10,23)(12,17)(14,19)(16,21)(18,32)(20,26)(22,28)(24,30), (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), (1,10)(2,15)(3,12)(4,9)(5,14)(6,11)(7,16)(8,13)(17,27)(18,32)(19,29)(20,26)(21,31)(22,28)(23,25)(24,30) );
G=PermutationGroup([[(1,25,14,19),(2,20,15,26),(3,27,16,21),(4,22,9,28),(5,29,10,23),(6,24,11,30),(7,31,12,17),(8,18,13,32)], [(1,25),(3,27),(5,29),(7,31),(10,23),(12,17),(14,19),(16,21),(18,32),(20,26),(22,28),(24,30)], [(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)], [(1,10),(2,15),(3,12),(4,9),(5,14),(6,11),(7,16),(8,13),(17,27),(18,32),(19,29),(20,26),(21,31),(22,28),(23,25),(24,30)]])
38 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | ··· | 4J | 4K | ··· | 4P | 8A | ··· | 8H | 8I | 8J | 8K | 8L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 | 8 | 8 | 8 | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
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 | D8 | SD16 | M4(2) | C42⋊C22 |
| kernel | D4⋊M4(2) | D4⋊C8 | C4⋊M4(2) | C42.12C4 | C2×C4×D4 | C2×C4⋊C4 | C4×D4 | C22×D4 | C42 | C22×C4 | C2×C4 | C2×C4 | D4 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 2 | 4 | 2 | 2 | 2 | 4 | 4 | 8 | 2 |
Matrix representation of D4⋊M4(2) ►in GL4(𝔽17) generated by
| 0 | 1 | 0 | 0 |
| 16 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 0 | 16 | 0 | 0 |
| 16 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 1 |
| 14 | 3 | 0 | 0 |
| 3 | 3 | 0 | 0 |
| 0 | 0 | 0 | 2 |
| 0 | 0 | 2 | 0 |
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(17))| [0,16,0,0,1,0,0,0,0,0,16,0,0,0,0,16],[0,16,0,0,16,0,0,0,0,0,16,0,0,0,0,1],[14,3,0,0,3,3,0,0,0,0,0,2,0,0,2,0],[16,0,0,0,0,16,0,0,0,0,16,0,0,0,0,1] >;
D4⋊M4(2) in GAP, Magma, Sage, TeX
D_4\rtimes M_4(2)
% in TeX
G:=Group("D4:M4(2)"); // GroupNames label
G:=SmallGroup(128,218);
// by ID
G=gap.SmallGroup(128,218);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,1430,1123,570,136,172]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=c^8=d^2=1,b*a*b=c*a*c^-1=a^-1,a*d=d*a,c*b*c^-1=a*b,b*d=d*b,d*c*d=c^5>;
// generators/relations