p-group, metabelian, nilpotent (class 3), monomial
Aliases: M4(2)⋊15D4, C24.112D4, C8⋊D4⋊2C2, C8.22(C2×D4), C8⋊8D4⋊12C2, C8.D4⋊2C2, C4.Q8⋊4C22, C4⋊C4.25C23, C2.D8⋊15C22, C8.18D4⋊25C2, (C2×C8).252C23, (C2×C4).260C24, (C2×Q16)⋊18C22, (C2×D4).63C23, C4.154(C22×D4), C23.385(C2×D4), (C22×C4).430D4, (C2×Q8).51C23, C4.173(C4⋊D4), D4⋊C4⋊51C22, Q8⋊C4⋊52C22, (C2×SD16)⋊12C22, C22⋊1(C8.C22), (C22×M4(2))⋊5C2, M4(2)⋊C4⋊13C2, C4⋊D4.150C22, C23.38D4⋊33C2, C23.36D4⋊41C2, C22.85(C4⋊D4), (C22×C8).259C22, (C23×C4).552C22, (C22×C4).982C23, C22.520(C22×D4), C22⋊Q8.155C22, C2.16(D8⋊C22), C22.19C24.16C2, (C22×Q8).281C22, C42⋊C2.109C22, (C2×M4(2)).264C22, C4.27(C2×C4○D4), (C2×C4).476(C2×D4), C2.78(C2×C4⋊D4), (C2×C22⋊Q8)⋊56C2, (C2×C8.C22)⋊18C2, C2.19(C2×C8.C22), (C2×C4).478(C4○D4), (C2×C4⋊C4).592C22, (C2×C4○D4).126C22, SmallGroup(128,1788)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 460 in 248 conjugacy classes, 102 normal (44 characteristic)
C1, C2 [×3], C2 [×6], C4 [×2], C4 [×2], C4 [×9], C22, C22 [×4], C22 [×14], C8 [×4], C8 [×2], C2×C4 [×4], C2×C4 [×4], C2×C4 [×23], D4 [×8], Q8 [×8], C23 [×3], C23 [×6], C42, C22⋊C4 [×9], C4⋊C4 [×2], C4⋊C4 [×2], C4⋊C4 [×7], C2×C8 [×4], C2×C8 [×4], M4(2) [×4], M4(2) [×6], SD16 [×4], Q16 [×4], C22×C4 [×6], C22×C4 [×7], C2×D4, C2×D4 [×3], C2×Q8, C2×Q8 [×2], C2×Q8 [×5], C4○D4 [×4], C24, D4⋊C4 [×2], Q8⋊C4 [×6], C4.Q8 [×2], C2.D8 [×2], C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C42⋊C2, C4×D4 [×2], C22≀C2, C4⋊D4 [×2], C22⋊Q8 [×2], C22⋊Q8 [×4], C22⋊Q8 [×2], C22.D4, C22×C8 [×2], C2×M4(2) [×2], C2×M4(2) [×2], C2×M4(2) [×4], C2×SD16 [×2], C2×Q16 [×2], C8.C22 [×4], C23×C4, C22×Q8, C2×C4○D4, C23.36D4, C23.38D4, M4(2)⋊C4, C8⋊8D4 [×2], C8.18D4 [×2], C8⋊D4 [×2], C8.D4 [×2], C2×C22⋊Q8, C22.19C24, C22×M4(2), C2×C8.C22, M4(2)⋊15D4
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)⋊15D4
Generators and relations
G = < a,b,c,d | a8=b2=c4=d2=1, bab=a5, cac-1=dad=a3, cbc-1=dbd=a4b, 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 19)(2 24)(3 21)(4 18)(5 23)(6 20)(7 17)(8 22)(9 31)(10 28)(11 25)(12 30)(13 27)(14 32)(15 29)(16 26)
(1 13 19 31)(2 16 20 26)(3 11 21 29)(4 14 22 32)(5 9 23 27)(6 12 24 30)(7 15 17 25)(8 10 18 28)
(1 31)(2 26)(3 29)(4 32)(5 27)(6 30)(7 25)(8 28)(9 23)(10 18)(11 21)(12 24)(13 19)(14 22)(15 17)(16 20)
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), (1,19)(2,24)(3,21)(4,18)(5,23)(6,20)(7,17)(8,22)(9,31)(10,28)(11,25)(12,30)(13,27)(14,32)(15,29)(16,26), (1,13,19,31)(2,16,20,26)(3,11,21,29)(4,14,22,32)(5,9,23,27)(6,12,24,30)(7,15,17,25)(8,10,18,28), (1,31)(2,26)(3,29)(4,32)(5,27)(6,30)(7,25)(8,28)(9,23)(10,18)(11,21)(12,24)(13,19)(14,22)(15,17)(16,20)>;
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), (1,19)(2,24)(3,21)(4,18)(5,23)(6,20)(7,17)(8,22)(9,31)(10,28)(11,25)(12,30)(13,27)(14,32)(15,29)(16,26), (1,13,19,31)(2,16,20,26)(3,11,21,29)(4,14,22,32)(5,9,23,27)(6,12,24,30)(7,15,17,25)(8,10,18,28), (1,31)(2,26)(3,29)(4,32)(5,27)(6,30)(7,25)(8,28)(9,23)(10,18)(11,21)(12,24)(13,19)(14,22)(15,17)(16,20) );
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)], [(1,19),(2,24),(3,21),(4,18),(5,23),(6,20),(7,17),(8,22),(9,31),(10,28),(11,25),(12,30),(13,27),(14,32),(15,29),(16,26)], [(1,13,19,31),(2,16,20,26),(3,11,21,29),(4,14,22,32),(5,9,23,27),(6,12,24,30),(7,15,17,25),(8,10,18,28)], [(1,31),(2,26),(3,29),(4,32),(5,27),(6,30),(7,25),(8,28),(9,23),(10,18),(11,21),(12,24),(13,19),(14,22),(15,17),(16,20)])
Matrix representation ►G ⊆ GL6(𝔽17)
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 16 | 2 | 0 | 0 |
| 0 | 0 | 16 | 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 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 7 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 10 | 0 | 0 |
| 0 | 0 | 5 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 7 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 10 | 0 | 0 |
| 0 | 0 | 5 | 0 | 0 | 0 |
G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,16,0,0,0,0,2,1,0,0,1,0,0,0,0,0,0,1,0,0],[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],[0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,10,0,0,0,0,12,0,0,0,0,7,0,0,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,10,0,0,0,0,12,0,0,0,0,7,0,0,0] >;
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | ··· | 4F | 4G | 4H | ··· | 4N | 8A | ··· | 8H |
| order | 1 | 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 | 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)⋊15D4 | C23.36D4 | C23.38D4 | M4(2)⋊C4 | C8⋊8D4 | C8.18D4 | C8⋊D4 | C8.D4 | C2×C22⋊Q8 | 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_{15}D_4 % in TeX
G:=Group("M4(2):15D4"); // GroupNames label
G:=SmallGroup(128,1788);
// by ID
G=gap.SmallGroup(128,1788);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,120,758,1018,2804,172]);
// 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^3,c*b*c^-1=d*b*d=a^4*b,d*c*d=c^-1>;
// generators/relations