p-group, metabelian, nilpotent (class 3), monomial
Aliases: D4⋊5M4(2), C42.50D4, C42.611C23, D4⋊C8⋊29C2, C4⋊C8⋊45C22, (C4×C8)⋊50C22, (C4×D4).18C4, C22.21C4≀C2, C42.64(C2×C4), (C4×M4(2))⋊15C2, (C22×D4).25C4, (C22×C4).658D4, C4.22(C2×M4(2)), C4.134(C8⋊C22), C42.6C4⋊26C2, (C4×D4).267C22, (C2×C42).167C22, C23.171(C22⋊C4), C2.16(C24.4C4), C2.5(C23.37D4), (C2×C4×D4).8C2, C2.10(C2×C4≀C2), (C2×C4⋊C4).42C4, C4⋊C4.183(C2×C4), (C2×D4).195(C2×C4), (C2×C4).1139(C2×D4), (C2×C4).80(C22⋊C4), (C2×C4).316(C22×C4), (C22×C4).189(C2×C4), C22.166(C2×C22⋊C4), 2-Sylow(CO-(4,5)), SmallGroup(128,222)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for D4⋊5M4(2)
G = < a,b,c,d | a4=b2=c8=d2=1, bab=a-1, ac=ca, ad=da, cbc-1=a-1b, bd=db, dcd=c5 >
Subgroups: 340 in 158 conjugacy classes, 52 normal (26 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C8, 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, C8⋊C4, C22⋊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), C42.6C4, C2×C4×D4, D4⋊5M4(2)
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, M4(2), C22×C4, C2×D4, C4≀C2, C2×C22⋊C4, C2×M4(2), C8⋊C22, C24.4C4, C23.37D4, C2×C4≀C2, D4⋊5M4(2)
►On 32 points
(1 15 25 19)(2 16 26 20)(3 9 27 21)(4 10 28 22)(5 11 29 23)(6 12 30 24)(7 13 31 17)(8 14 32 18)
(1 29)(2 24)(3 7)(4 14)(5 25)(6 20)(8 10)(9 17)(11 15)(12 26)(13 21)(16 30)(18 28)(19 23)(22 32)(27 31)
(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 25)(2 30)(3 27)(4 32)(5 29)(6 26)(7 31)(8 28)(9 21)(10 18)(11 23)(12 20)(13 17)(14 22)(15 19)(16 24)
G:=sub<Sym(32)| (1,15,25,19)(2,16,26,20)(3,9,27,21)(4,10,28,22)(5,11,29,23)(6,12,30,24)(7,13,31,17)(8,14,32,18), (1,29)(2,24)(3,7)(4,14)(5,25)(6,20)(8,10)(9,17)(11,15)(12,26)(13,21)(16,30)(18,28)(19,23)(22,32)(27,31), (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), (1,25)(2,30)(3,27)(4,32)(5,29)(6,26)(7,31)(8,28)(9,21)(10,18)(11,23)(12,20)(13,17)(14,22)(15,19)(16,24)>;
G:=Group( (1,15,25,19)(2,16,26,20)(3,9,27,21)(4,10,28,22)(5,11,29,23)(6,12,30,24)(7,13,31,17)(8,14,32,18), (1,29)(2,24)(3,7)(4,14)(5,25)(6,20)(8,10)(9,17)(11,15)(12,26)(13,21)(16,30)(18,28)(19,23)(22,32)(27,31), (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), (1,25)(2,30)(3,27)(4,32)(5,29)(6,26)(7,31)(8,28)(9,21)(10,18)(11,23)(12,20)(13,17)(14,22)(15,19)(16,24) );
G=PermutationGroup([[(1,15,25,19),(2,16,26,20),(3,9,27,21),(4,10,28,22),(5,11,29,23),(6,12,30,24),(7,13,31,17),(8,14,32,18)], [(1,29),(2,24),(3,7),(4,14),(5,25),(6,20),(8,10),(9,17),(11,15),(12,26),(13,21),(16,30),(18,28),(19,23),(22,32),(27,31)], [(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)], [(1,25),(2,30),(3,27),(4,32),(5,29),(6,26),(7,31),(8,28),(9,21),(10,18),(11,23),(12,20),(13,17),(14,22),(15,19),(16,24)]])
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 | 4 |
| type | + | + | + | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | D4 | D4 | M4(2) | C4≀C2 | C8⋊C22 |
| kernel | D4⋊5M4(2) | D4⋊C8 | C4×M4(2) | C42.6C4 | C2×C4×D4 | C2×C4⋊C4 | C4×D4 | C22×D4 | C42 | C22×C4 | D4 | C22 | C4 |
| # reps | 1 | 4 | 1 | 1 | 1 | 2 | 4 | 2 | 2 | 2 | 8 | 8 | 2 |
Matrix representation of D4⋊5M4(2) ►in GL4(𝔽17) generated by
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 16 |
| 0 | 0 | 1 | 0 |
| 16 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
| 4 | 0 | 0 | 0 |
| 0 | 0 | 6 | 6 |
| 0 | 0 | 11 | 6 |
| 1 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
G:=sub<GL(4,GF(17))| [16,0,0,0,0,16,0,0,0,0,0,1,0,0,16,0],[16,0,0,0,0,1,0,0,0,0,16,0,0,0,0,1],[0,4,0,0,1,0,0,0,0,0,6,11,0,0,6,6],[1,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16] >;
D4⋊5M4(2) in GAP, Magma, Sage, TeX
D_4\rtimes_5M_4(2)
% in TeX
G:=Group("D4:5M4(2)"); // GroupNames label
G:=SmallGroup(128,222);
// by ID
G=gap.SmallGroup(128,222);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,1430,387,184,1123,570,136,172]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=c^8=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=a^-1*b,b*d=d*b,d*c*d=c^5>;
// generators/relations