p-group, metabelian, nilpotent (class 3), monomial
Aliases: M4(2)⋊14D4, C24.111D4, C8⋊6(C2×D4), C8⋊2D4⋊2C2, C8⋊D4⋊1C2, C8⋊8D4⋊11C2, C8⋊7D4⋊25C2, C4.Q8⋊3C22, (C2×D8)⋊18C22, C4⋊C4.24C23, C2.D8⋊14C22, C4⋊D4⋊55C22, C22⋊1(C8⋊C22), (C2×C8).251C23, (C2×C4).259C24, (C22×C8)⋊20C22, (C2×D4).62C23, C23.384(C2×D4), (C22×C4).429D4, C4.153(C22×D4), C22⋊Q8⋊67C22, (C2×Q8).50C23, C4.172(C4⋊D4), D4⋊C4⋊50C22, C22.19C24⋊8C2, Q8⋊C4⋊51C22, (C2×SD16)⋊11C22, (C22×M4(2))⋊4C2, M4(2)⋊C4⋊12C2, C23.36D4⋊40C2, C23.37D4⋊33C2, C22.84(C4⋊D4), (C2×M4(2))⋊52C22, (C23×C4).551C22, (C22×C4).981C23, C22.519(C22×D4), C2.15(D8⋊C22), (C22×D4).348C22, C42⋊C2.108C22, C4.26(C2×C4○D4), (C2×C4⋊D4)⋊48C2, (C2×C8⋊C22)⋊18C2, (C2×C4).475(C2×D4), C2.77(C2×C4⋊D4), C2.19(C2×C8⋊C22), (C2×C4).477(C4○D4), (C2×C4⋊C4).591C22, (C2×C4○D4).125C22, SmallGroup(128,1787)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 588 in 274 conjugacy classes, 102 normal (44 characteristic)
C1, C2 [×3], C2 [×8], C4 [×2], C4 [×2], C4 [×7], C22, C22 [×4], C22 [×24], C8 [×4], C8 [×2], C2×C4 [×4], C2×C4 [×4], C2×C4 [×19], D4 [×22], Q8 [×2], C23 [×3], C23 [×14], C42, C22⋊C4 [×9], C4⋊C4 [×2], C4⋊C4 [×2], C4⋊C4 [×3], C2×C8 [×4], C2×C8 [×4], M4(2) [×4], M4(2) [×6], D8 [×4], SD16 [×4], C22×C4 [×6], C22×C4 [×6], C2×D4, C2×D4 [×2], C2×D4 [×16], C2×Q8, C4○D4 [×4], C24, C24, D4⋊C4 [×6], Q8⋊C4 [×2], C4.Q8 [×2], C2.D8 [×2], C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C4×D4 [×2], C22≀C2, C4⋊D4 [×2], C4⋊D4 [×4], C4⋊D4 [×2], C22⋊Q8 [×2], C22.D4, C22×C8 [×2], C2×M4(2) [×2], C2×M4(2) [×2], C2×M4(2) [×4], C2×D8 [×2], C2×SD16 [×2], C8⋊C22 [×4], C23×C4, C22×D4, C22×D4, C2×C4○D4, C23.36D4, C23.37D4, M4(2)⋊C4, C8⋊8D4 [×2], C8⋊7D4 [×2], C8⋊D4 [×2], C8⋊2D4 [×2], C2×C4⋊D4, C22.19C24, C22×M4(2), C2×C8⋊C22, M4(2)⋊14D4
Quotients:
C1, C2 [×15], C22 [×35], D4 [×8], C23 [×15], C2×D4 [×12], C4○D4 [×2], C24, C4⋊D4 [×4], C8⋊C22 [×2], C22×D4 [×2], C2×C4○D4, C2×C4⋊D4, C2×C8⋊C22, D8⋊C22, M4(2)⋊14D4
Generators and relations
G = < a,b,c,d | a8=b2=c4=d2=1, bab=a5, cac-1=dad=a-1, bc=cb, bd=db, dcd=c-1 >
►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 29)(2 26)(3 31)(4 28)(5 25)(6 30)(7 27)(8 32)(9 23)(10 20)(11 17)(12 22)(13 19)(14 24)(15 21)(16 18)
(1 18 29 16)(2 17 30 15)(3 24 31 14)(4 23 32 13)(5 22 25 12)(6 21 26 11)(7 20 27 10)(8 19 28 9)
(2 8)(3 7)(4 6)(9 17)(10 24)(11 23)(12 22)(13 21)(14 20)(15 19)(16 18)(26 32)(27 31)(28 30)
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), (1,29)(2,26)(3,31)(4,28)(5,25)(6,30)(7,27)(8,32)(9,23)(10,20)(11,17)(12,22)(13,19)(14,24)(15,21)(16,18), (1,18,29,16)(2,17,30,15)(3,24,31,14)(4,23,32,13)(5,22,25,12)(6,21,26,11)(7,20,27,10)(8,19,28,9), (2,8)(3,7)(4,6)(9,17)(10,24)(11,23)(12,22)(13,21)(14,20)(15,19)(16,18)(26,32)(27,31)(28,30)>;
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), (1,29)(2,26)(3,31)(4,28)(5,25)(6,30)(7,27)(8,32)(9,23)(10,20)(11,17)(12,22)(13,19)(14,24)(15,21)(16,18), (1,18,29,16)(2,17,30,15)(3,24,31,14)(4,23,32,13)(5,22,25,12)(6,21,26,11)(7,20,27,10)(8,19,28,9), (2,8)(3,7)(4,6)(9,17)(10,24)(11,23)(12,22)(13,21)(14,20)(15,19)(16,18)(26,32)(27,31)(28,30) );
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)], [(1,29),(2,26),(3,31),(4,28),(5,25),(6,30),(7,27),(8,32),(9,23),(10,20),(11,17),(12,22),(13,19),(14,24),(15,21),(16,18)], [(1,18,29,16),(2,17,30,15),(3,24,31,14),(4,23,32,13),(5,22,25,12),(6,21,26,11),(7,20,27,10),(8,19,28,9)], [(2,8),(3,7),(4,6),(9,17),(10,24),(11,23),(12,22),(13,21),(14,20),(15,19),(16,18),(26,32),(27,31),(28,30)])
Matrix representation ►G ⊆ GL6(𝔽17)
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 16 | 0 | 0 |
| 0 | 0 | 2 | 15 | 16 | 0 |
| 0 | 0 | 0 | 1 | 1 | 16 |
| 0 | 0 | 2 | 16 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 16 | 1 |
| 0 | 0 | 0 | 16 | 15 | 2 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 15 | 0 | 0 | 0 | 0 |
| 1 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 13 |
| 0 | 0 | 8 | 13 | 13 | 13 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 13 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 1 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 16 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,2,0,2,0,0,16,15,1,16,0,0,0,16,1,0,0,0,0,0,16,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,16,15,1,0,0,0,1,2,0,1],[1,1,0,0,0,0,15,16,0,0,0,0,0,0,4,8,0,0,0,0,0,13,0,0,0,0,0,13,4,0,0,0,13,13,0,13],[1,1,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,16,16,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 4A | ··· | 4F | 4G | 4H | ··· | 4L | 8A | ··· | 8H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 8 | 8 | 8 | 2 | ··· | 2 | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | C4○D4 | C8⋊C22 | D8⋊C22 |
| kernel | M4(2)⋊14D4 | C23.36D4 | C23.37D4 | M4(2)⋊C4 | C8⋊8D4 | C8⋊7D4 | C8⋊D4 | C8⋊2D4 | C2×C4⋊D4 | C22.19C24 | C22×M4(2) | C2×C8⋊C22 | M4(2) | C22×C4 | C24 | C2×C4 | C22 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 4 | 3 | 1 | 4 | 2 | 2 |
In GAP, Magma, Sage, TeX
M_{4(2)}\rtimes_{14}D_4 % in TeX
G:=Group("M4(2):14D4"); // GroupNames label
G:=SmallGroup(128,1787);
// by ID
G=gap.SmallGroup(128,1787);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,120,758,1018,2804,172]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=c^4=d^2=1,b*a*b=a^5,c*a*c^-1=d*a*d=a^-1,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations