p-group, metabelian, nilpotent (class 3), monomial
Aliases: M4(2).44D4, C8⋊C22⋊1C4, C4.63(C4×D4), (C2×C8).27D4, C4○D4.45D4, (C2×D4).71D4, C8.C22⋊1C4, (C2×Q8).69D4, Q8○M4(2)⋊5C2, C22.35(C4×D4), C4.125C22≀C2, C4.C42⋊4C2, D4.3(C22⋊C4), C23.7(C4○D4), M4(2).4(C2×C4), Q8.3(C22⋊C4), C4.129(C4⋊D4), M4(2)⋊4C4⋊2C2, D8⋊C22.1C2, C23.24D4⋊2C2, (C22×C8).39C22, C42⋊C22⋊11C2, C22.42(C4⋊D4), (C22×C4).679C23, C42⋊C2.16C22, C2.20(C23.23D4), (C2×M4(2)).178C22, C22.21(C22.D4), C4○D4.7(C2×C4), (C2×D4).74(C2×C4), C4.15(C2×C22⋊C4), (C2×C4).9(C22×C4), (C2×Q8).65(C2×C4), (C2×C4).1000(C2×D4), (C22×C8)⋊C2⋊23C2, (C2×C4).318(C4○D4), (C2×C4○D4).16C22, SmallGroup(128,613)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for M4(2).44D4
G = < a,b,c,d | a8=b2=1, c4=a4, d2=b, bab=a5, cac-1=dad-1=a3b, bc=cb, bd=db, dcd-1=a4bc3 >
Subgroups: 300 in 155 conjugacy classes, 54 normal (36 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), D8, SD16, Q16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C4○D4, C22⋊C8, D4⋊C4, Q8⋊C4, C4≀C2, C42⋊C2, C22×C8, C2×M4(2), C2×M4(2), C8○D4, C4○D8, C8⋊C22, C8⋊C22, C8.C22, C8.C22, C2×C4○D4, C4.C42, M4(2)⋊4C4, (C22×C8)⋊C2, C23.24D4, C42⋊C22, Q8○M4(2), D8⋊C22, M4(2).44D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C4○D4, C2×C22⋊C4, C4×D4, C22≀C2, C4⋊D4, C22.D4, C23.23D4, M4(2).44D4
►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)(9 13)(11 15)(18 22)(20 24)(25 29)(27 31)
(1 14 23 26 5 10 19 30)(2 9 20 25 6 13 24 29)(3 16 17 28 7 12 21 32)(4 11 22 27 8 15 18 31)
(1 30)(2 25 6 29)(3 32)(4 27 8 31)(5 26)(7 28)(9 24 13 20)(10 23)(11 18 15 22)(12 17)(14 19)(16 21)
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)(9,13)(11,15)(18,22)(20,24)(25,29)(27,31), (1,14,23,26,5,10,19,30)(2,9,20,25,6,13,24,29)(3,16,17,28,7,12,21,32)(4,11,22,27,8,15,18,31), (1,30)(2,25,6,29)(3,32)(4,27,8,31)(5,26)(7,28)(9,24,13,20)(10,23)(11,18,15,22)(12,17)(14,19)(16,21)>;
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)(9,13)(11,15)(18,22)(20,24)(25,29)(27,31), (1,14,23,26,5,10,19,30)(2,9,20,25,6,13,24,29)(3,16,17,28,7,12,21,32)(4,11,22,27,8,15,18,31), (1,30)(2,25,6,29)(3,32)(4,27,8,31)(5,26)(7,28)(9,24,13,20)(10,23)(11,18,15,22)(12,17)(14,19)(16,21) );
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),(9,13),(11,15),(18,22),(20,24),(25,29),(27,31)], [(1,14,23,26,5,10,19,30),(2,9,20,25,6,13,24,29),(3,16,17,28,7,12,21,32),(4,11,22,27,8,15,18,31)], [(1,30),(2,25,6,29),(3,32),(4,27,8,31),(5,26),(7,28),(9,24,13,20),(10,23),(11,18,15,22),(12,17),(14,19),(16,21)]])
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 8A | ··· | 8L | 8M | 8N |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 8 | 8 |
| size | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 8 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 8 | 8 | 8 | 4 | ··· | 4 | 8 | 8 |
32 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 | D4 | D4 | C4○D4 | C4○D4 | M4(2).44D4 |
| kernel | M4(2).44D4 | C4.C42 | M4(2)⋊4C4 | (C22×C8)⋊C2 | C23.24D4 | C42⋊C22 | Q8○M4(2) | D8⋊C22 | C8⋊C22 | C8.C22 | C2×C8 | M4(2) | C2×D4 | C2×Q8 | C4○D4 | C2×C4 | C23 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 4 |
Matrix representation of M4(2).44D4 ►in GL4(𝔽17) generated by
| 0 | 16 | 13 | 10 |
| 0 | 16 | 0 | 10 |
| 16 | 1 | 0 | 0 |
| 0 | 2 | 0 | 1 |
| 1 | 0 | 0 | 1 |
| 0 | 1 | 0 | 1 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 14 | 14 | 12 | 14 |
| 3 | 14 | 12 | 0 |
| 0 | 0 | 10 | 12 |
| 0 | 0 | 10 | 0 |
| 14 | 3 | 12 | 5 |
| 3 | 3 | 12 | 8 |
| 0 | 0 | 10 | 12 |
| 0 | 0 | 10 | 7 |
G:=sub<GL(4,GF(17))| [0,0,16,0,16,16,1,2,13,0,0,0,10,10,0,1],[1,0,0,0,0,1,0,0,0,0,16,0,1,1,0,16],[14,3,0,0,14,14,0,0,12,12,10,10,14,0,12,0],[14,3,0,0,3,3,0,0,12,12,10,10,5,8,12,7] >;
M4(2).44D4 in GAP, Magma, Sage, TeX
M_4(2)._{44}D_4 % in TeX
G:=Group("M4(2).44D4"); // GroupNames label
G:=SmallGroup(128,613);
// by ID
G=gap.SmallGroup(128,613);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,288,422,1018,248,2804,718,172,2028]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=1,c^4=a^4,d^2=b,b*a*b=a^5,c*a*c^-1=d*a*d^-1=a^3*b,b*c=c*b,b*d=d*b,d*c*d^-1=a^4*b*c^3>;
// generators/relations