p-group, metabelian, nilpotent (class 3), monomial
Aliases: M4(2).45D4, C4.94C22≀C2, C24.59(C2×C4), C22.52(C4×D4), (C22×C4).295D4, C4.134(C4⋊D4), C22.C42⋊18C2, C22⋊1(C4.10D4), C24.4C4.22C2, (C22×C4).689C23, (C23×C4).264C22, C23.195(C22×C4), (C22×Q8).20C22, C23.126(C22⋊C4), C4.89(C22.D4), (C22×M4(2)).22C2, C2.40(C23.23D4), (C2×M4(2)).188C22, C2.27(M4(2).8C22), (C2×C4⋊C4).23C4, (C2×C4).1337(C2×D4), (C2×C22⋊C4).12C4, (C2×C4⋊C4).65C22, (C22×C4).21(C2×C4), (C2×C22⋊Q8).10C2, (C2×C4).323(C4○D4), (C2×C4.10D4)⋊17C2, C2.25(C2×C4.10D4), (C2×C4).198(C22⋊C4), C22.276(C2×C22⋊C4), SmallGroup(128,633)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for M4(2).45D4
G = < a,b,c,d | a8=b2=1, c4=a4, d2=a2b, bab=a5, cac-1=dad-1=ab, cbc-1=dbd-1=a4b, dcd-1=a6bc3 >
Subgroups: 316 in 169 conjugacy classes, 58 normal (28 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, Q8, C23, C23, C22⋊C4, C4⋊C4, C2×C8, M4(2), M4(2), C22×C4, C22×C4, C22×C4, C2×Q8, C24, C22⋊C8, C4.10D4, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C22⋊Q8, C22×C8, C2×M4(2), C2×M4(2), C23×C4, C22×Q8, C22.C42, C24.4C4, C2×C4.10D4, C2×C22⋊Q8, C22×M4(2), M4(2).45D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C4○D4, C4.10D4, C2×C22⋊C4, C4×D4, C22≀C2, C4⋊D4, C22.D4, C23.23D4, C2×C4.10D4, M4(2).8C22, M4(2).45D4
►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 22)(2 19)(3 24)(4 21)(5 18)(6 23)(7 20)(8 17)(9 30)(10 27)(11 32)(12 29)(13 26)(14 31)(15 28)(16 25)
(1 13 3 11 5 9 7 15)(2 27 8 29 6 31 4 25)(10 21 12 19 14 17 16 23)(18 26 20 32 22 30 24 28)
(1 26 24 15 5 30 20 11)(2 14 21 25 6 10 17 29)(3 32 18 13 7 28 22 9)(4 12 23 31 8 16 19 27)
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), (1,22)(2,19)(3,24)(4,21)(5,18)(6,23)(7,20)(8,17)(9,30)(10,27)(11,32)(12,29)(13,26)(14,31)(15,28)(16,25), (1,13,3,11,5,9,7,15)(2,27,8,29,6,31,4,25)(10,21,12,19,14,17,16,23)(18,26,20,32,22,30,24,28), (1,26,24,15,5,30,20,11)(2,14,21,25,6,10,17,29)(3,32,18,13,7,28,22,9)(4,12,23,31,8,16,19,27)>;
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), (1,22)(2,19)(3,24)(4,21)(5,18)(6,23)(7,20)(8,17)(9,30)(10,27)(11,32)(12,29)(13,26)(14,31)(15,28)(16,25), (1,13,3,11,5,9,7,15)(2,27,8,29,6,31,4,25)(10,21,12,19,14,17,16,23)(18,26,20,32,22,30,24,28), (1,26,24,15,5,30,20,11)(2,14,21,25,6,10,17,29)(3,32,18,13,7,28,22,9)(4,12,23,31,8,16,19,27) );
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)], [(1,22),(2,19),(3,24),(4,21),(5,18),(6,23),(7,20),(8,17),(9,30),(10,27),(11,32),(12,29),(13,26),(14,31),(15,28),(16,25)], [(1,13,3,11,5,9,7,15),(2,27,8,29,6,31,4,25),(10,21,12,19,14,17,16,23),(18,26,20,32,22,30,24,28)], [(1,26,24,15,5,30,20,11),(2,14,21,25,6,10,17,29),(3,32,18,13,7,28,22,9),(4,12,23,31,8,16,19,27)]])
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 4K | 8A | ··· | 8H | 8I | 8J | 8K | 8L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 8 | 8 | 8 | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 2 | ··· | 2 | 4 | 8 | 8 | 8 | 8 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | D4 | D4 | C4○D4 | C4.10D4 | M4(2).8C22 |
| kernel | M4(2).45D4 | C22.C42 | C24.4C4 | C2×C4.10D4 | C2×C22⋊Q8 | C22×M4(2) | C2×C22⋊C4 | C2×C4⋊C4 | M4(2) | C22×C4 | C2×C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 2 | 1 | 1 | 4 | 4 | 4 | 4 | 4 | 2 | 2 |
Matrix representation of M4(2).45D4 ►in GL6(𝔽17)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 16 | 15 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 11 | 6 | 0 | 0 | 0 | 0 |
| 8 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 7 |
| 0 | 0 | 0 | 0 | 5 | 10 |
| 0 | 0 | 0 | 7 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 6 | 15 | 0 | 0 | 0 | 0 |
| 9 | 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 10 | 10 |
| 0 | 0 | 0 | 0 | 12 | 7 |
| 0 | 0 | 0 | 10 | 0 | 0 |
| 0 | 0 | 5 | 0 | 0 | 0 |
G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,1,0,0,0,0,15,1,0,0,1,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[11,8,0,0,0,0,6,6,0,0,0,0,0,0,0,0,0,12,0,0,0,0,7,0,0,0,7,5,0,0,0,0,7,10,0,0],[6,9,0,0,0,0,15,11,0,0,0,0,0,0,0,0,0,5,0,0,0,0,10,0,0,0,10,12,0,0,0,0,10,7,0,0] >;
M4(2).45D4 in GAP, Magma, Sage, TeX
M_4(2)._{45}D_4 % in TeX
G:=Group("M4(2).45D4"); // GroupNames label
G:=SmallGroup(128,633);
// by ID
G=gap.SmallGroup(128,633);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,736,422,2019,1018,2028,124]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=1,c^4=a^4,d^2=a^2*b,b*a*b=a^5,c*a*c^-1=d*a*d^-1=a*b,c*b*c^-1=d*b*d^-1=a^4*b,d*c*d^-1=a^6*b*c^3>;
// generators/relations