p-group, metabelian, nilpotent (class 3), monomial
Aliases: M4(2).24D4, C4.70(C4×D4), D4⋊C4⋊8C4, Q8⋊C4⋊8C4, (C2×C8).328D4, C2.16(C8○D8), C42⋊6C4⋊27C2, C22.166(C4×D4), C2.16(C8.26D4), C4.193(C4⋊D4), C8○2M4(2)⋊24C2, C4.40(C42⋊C2), C23.206(C4○D4), (C2×C42).300C22, (C22×C8).394C22, C23.24D4.6C2, C22.6(C4.4D4), C22.7C42⋊25C2, (C22×C4).1383C23, C42⋊C2.272C22, C22.15(C42⋊2C2), C4.142(C22.D4), (C2×M4(2)).321C22, C2.14(C24.C22), (C2×C8).43(C2×C4), (C2×C4≀C2).10C2, (C2×C8.C4)⋊9C2, C4⋊C4.153(C2×C4), (C2×D4).96(C2×C4), (C2×Q8).81(C2×C4), (C2×C4).1537(C2×D4), (C2×C4).761(C4○D4), (C2×C4).401(C22×C4), (C2×C4○D4).33C22, (C22×C8)⋊C2.15C2, SmallGroup(128,661)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for M4(2).24D4
G = < a,b,c,d | a8=b2=d2=1, c4=a4, bab=a5, ac=ca, dad=a-1b, bc=cb, dbd=a4b, dcd=a6c3 >
Subgroups: 212 in 112 conjugacy classes, 48 normal (46 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), M4(2), C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C4×C8, C8⋊C4, C22⋊C8, D4⋊C4, Q8⋊C4, C4≀C2, C8.C4, C2×C42, C42⋊C2, C22×C8, C2×M4(2), C2×C4○D4, C22.7C42, C42⋊6C4, C8○2M4(2), (C22×C8)⋊C2, C23.24D4, C2×C4≀C2, C2×C8.C4, M4(2).24D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22×C4, C2×D4, C4○D4, C42⋊C2, C4×D4, C4⋊D4, C22.D4, C4.4D4, C42⋊2C2, C24.C22, C8○D8, C8.26D4, M4(2).24D4
►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 5)(3 7)(10 14)(12 16)(18 22)(20 24)(25 29)(27 31)
(1 12 27 22 5 16 31 18)(2 13 28 23 6 9 32 19)(3 14 29 24 7 10 25 20)(4 15 30 17 8 11 26 21)
(1 32)(2 27)(3 26)(4 29)(5 28)(6 31)(7 30)(8 25)(9 10)(11 12)(13 14)(15 16)(17 18)(19 20)(21 22)(23 24)
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), (1,5)(3,7)(10,14)(12,16)(18,22)(20,24)(25,29)(27,31), (1,12,27,22,5,16,31,18)(2,13,28,23,6,9,32,19)(3,14,29,24,7,10,25,20)(4,15,30,17,8,11,26,21), (1,32)(2,27)(3,26)(4,29)(5,28)(6,31)(7,30)(8,25)(9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)>;
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), (1,5)(3,7)(10,14)(12,16)(18,22)(20,24)(25,29)(27,31), (1,12,27,22,5,16,31,18)(2,13,28,23,6,9,32,19)(3,14,29,24,7,10,25,20)(4,15,30,17,8,11,26,21), (1,32)(2,27)(3,26)(4,29)(5,28)(6,31)(7,30)(8,25)(9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24) );
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)], [(1,5),(3,7),(10,14),(12,16),(18,22),(20,24),(25,29),(27,31)], [(1,12,27,22,5,16,31,18),(2,13,28,23,6,9,32,19),(3,14,29,24,7,10,25,20),(4,15,30,17,8,11,26,21)], [(1,32),(2,27),(3,26),(4,29),(5,28),(6,31),(7,30),(8,25),(9,10),(11,12),(13,14),(15,16),(17,18),(19,20),(21,22),(23,24)]])
38 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | ··· | 4N | 4O | 8A | 8B | 8C | 8D | 8E | ··· | 8N | 8O | 8P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 8 | 8 | 8 | 8 | 8 | ··· | 8 | 8 | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | ··· | 4 | 8 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | 8 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | ||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | D4 | D4 | C4○D4 | C4○D4 | C8○D8 | C8.26D4 |
| kernel | M4(2).24D4 | C22.7C42 | C42⋊6C4 | C8○2M4(2) | (C22×C8)⋊C2 | C23.24D4 | C2×C4≀C2 | C2×C8.C4 | D4⋊C4 | Q8⋊C4 | C2×C8 | M4(2) | C2×C4 | C23 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 | 2 | 2 | 6 | 2 | 8 | 2 |
Matrix representation of M4(2).24D4 ►in GL4(𝔽17) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 15 |
| 0 | 0 | 2 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
| 16 | 0 | 0 | 0 |
| 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 8 |
| 1 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 15 |
| 0 | 0 | 8 | 0 |
G:=sub<GL(4,GF(17))| [1,0,0,0,0,1,0,0,0,0,0,2,0,0,15,0],[1,0,0,0,0,1,0,0,0,0,16,0,0,0,0,1],[0,16,0,0,1,0,0,0,0,0,8,0,0,0,0,8],[1,0,0,0,0,16,0,0,0,0,0,8,0,0,15,0] >;
M4(2).24D4 in GAP, Magma, Sage, TeX
M_4(2)._{24}D_4 % in TeX
G:=Group("M4(2).24D4"); // GroupNames label
G:=SmallGroup(128,661);
// by ID
G=gap.SmallGroup(128,661);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,288,422,58,2804,718,172,124]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=d^2=1,c^4=a^4,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=a^6*c^3>;
// generators/relations