p-group, metabelian, nilpotent (class 3), monomial
Aliases: M4(2).29C23, C8○D4.1C4, C4○D4.59D4, D4○(C8.C4), (C2×Q8).24Q8, (C2×D4).33Q8, Q8○(C8.C4), D4.11(C4⋊C4), Q8.11(C4⋊C4), C8.17(C22×C4), C4.54(C23×C4), C23.48(C2×Q8), (C2×C8).583C23, (C2×C4).192C24, C4.190(C22×D4), Q8○M4(2).7C2, C22.3(C22×Q8), M4(2).26(C2×C4), M4(2).C4⋊11C2, C8.C4.24C22, (C22×C4).909C23, (C22×C8).247C22, (C2×M4(2)).243C22, C4.23(C2×C4⋊C4), (C2×C8).95(C2×C4), (C2×C8○D4).8C2, C22.5(C2×C4⋊C4), (C2×C4).98(C2×Q8), C4○D4.34(C2×C4), C4○D4○(C8.C4), C2.31(C22×C4⋊C4), (C2×C8.C4)⋊22C2, (C2×C4).1083(C2×D4), (C2×C4).254(C22×C4), (C2×C4○D4).279C22, SmallGroup(128,1648)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for M4(2).29C23
G = < a,b,c,d,e | a8=b2=d2=e2=1, c2=a2, bab=a5, cac-1=a-1b, ad=da, ae=ea, cbc-1=a4b, bd=db, be=eb, cd=dc, ce=ec, ede=a4d >
Subgroups: 276 in 220 conjugacy classes, 170 normal (13 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C8, C8, C8, C2×C4, C2×C4, D4, Q8, C23, C2×C8, C2×C8, C2×C8, M4(2), M4(2), C22×C4, C2×D4, C2×Q8, C4○D4, C8.C4, C22×C8, C2×M4(2), C8○D4, C8○D4, C2×C4○D4, C2×C8.C4, M4(2).C4, C2×C8○D4, Q8○M4(2), M4(2).29C23
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C4⋊C4, C22×C4, C2×D4, C2×Q8, C24, C2×C4⋊C4, C23×C4, C22×D4, C22×Q8, C22×C4⋊C4, M4(2).29C23
►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)(17 21)(19 23)(26 30)(28 32)
(1 9 3 11 5 13 7 15)(2 12 4 14 6 16 8 10)(17 31 19 25 21 27 23 29)(18 26 20 28 22 30 24 32)
(1 30)(2 31)(3 32)(4 25)(5 26)(6 27)(7 28)(8 29)(9 24)(10 17)(11 18)(12 19)(13 20)(14 21)(15 22)(16 23)
(17 21)(18 22)(19 23)(20 24)(25 29)(26 30)(27 31)(28 32)
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)(17,21)(19,23)(26,30)(28,32), (1,9,3,11,5,13,7,15)(2,12,4,14,6,16,8,10)(17,31,19,25,21,27,23,29)(18,26,20,28,22,30,24,32), (1,30)(2,31)(3,32)(4,25)(5,26)(6,27)(7,28)(8,29)(9,24)(10,17)(11,18)(12,19)(13,20)(14,21)(15,22)(16,23), (17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32)>;
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)(17,21)(19,23)(26,30)(28,32), (1,9,3,11,5,13,7,15)(2,12,4,14,6,16,8,10)(17,31,19,25,21,27,23,29)(18,26,20,28,22,30,24,32), (1,30)(2,31)(3,32)(4,25)(5,26)(6,27)(7,28)(8,29)(9,24)(10,17)(11,18)(12,19)(13,20)(14,21)(15,22)(16,23), (17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32) );
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),(17,21),(19,23),(26,30),(28,32)], [(1,9,3,11,5,13,7,15),(2,12,4,14,6,16,8,10),(17,31,19,25,21,27,23,29),(18,26,20,28,22,30,24,32)], [(1,30),(2,31),(3,32),(4,25),(5,26),(6,27),(7,28),(8,29),(9,24),(10,17),(11,18),(12,19),(13,20),(14,21),(15,22),(16,23)], [(17,21),(18,22),(19,23),(20,24),(25,29),(26,30),(27,31),(28,32)]])
44 conjugacy classes
| class | 1 | 2A | 2B | ··· | 2H | 4A | 4B | 4C | ··· | 4I | 8A | 8B | 8C | 8D | 8E | ··· | 8Z |
| order | 1 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | ··· | 4 | 8 | 8 | 8 | 8 | 8 | ··· | 8 |
| size | 1 | 1 | 2 | ··· | 2 | 1 | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | - | - | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C4 | Q8 | Q8 | D4 | M4(2).29C23 |
| kernel | M4(2).29C23 | C2×C8.C4 | M4(2).C4 | C2×C8○D4 | Q8○M4(2) | C8○D4 | C2×D4 | C2×Q8 | C4○D4 | C1 |
| # reps | 1 | 6 | 6 | 1 | 2 | 16 | 3 | 1 | 4 | 4 |
Matrix representation of M4(2).29C23 ►in GL4(𝔽17) generated by
| 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 8 |
| 9 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 |
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 16 |
G:=sub<GL(4,GF(17))| [0,0,9,0,0,0,0,9,8,0,0,0,0,8,0,0],[16,0,0,0,0,16,0,0,0,0,1,0,0,0,0,1],[0,0,4,0,0,0,0,4,1,0,0,0,0,1,0,0],[0,1,0,0,1,0,0,0,0,0,0,1,0,0,1,0],[1,0,0,0,0,16,0,0,0,0,1,0,0,0,0,16] >;
M4(2).29C23 in GAP, Magma, Sage, TeX
M_4(2)._{29}C_2^3 % in TeX
G:=Group("M4(2).29C2^3"); // GroupNames label
G:=SmallGroup(128,1648);
// by ID
G=gap.SmallGroup(128,1648);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,120,521,2804,172,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^8=b^2=d^2=e^2=1,c^2=a^2,b*a*b=a^5,c*a*c^-1=a^-1*b,a*d=d*a,a*e=e*a,c*b*c^-1=a^4*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=a^4*d>;
// generators/relations