p-group, metabelian, nilpotent (class 2), monomial
Aliases: M4(2)○2M4(2), C23.18C42, C42.588C23, C4⋊C4○M4(2), (C4×C8)⋊51C22, C22⋊C4○M4(2), C4.59(C23×C4), (C2×C4).27C42, C4.32(C2×C42), C24.71(C2×C4), C8.47(C22×C4), C8⋊C4⋊69C22, M4(2)○(C2×M4(2)), (C2×M4(2))⋊20C4, (C4×M4(2))⋊27C2, M4(2)⋊23(C2×C4), (C2×C8).611C23, (C2×C4).623C24, C42.198(C2×C4), C8○2M4(2)⋊26C2, C42⋊C2.25C4, C2.1(Q8○M4(2)), C22.18(C2×C42), C2.15(C22×C42), C22.34(C23×C4), M4(2)○(C42⋊C2), (C22×C8).421C22, (C2×C42).749C22, (C23×C4).504C22, C23.135(C22×C4), (C22×C4).1488C23, (C22×M4(2)).24C2, C42⋊C2.346C22, (C2×M4(2)).380C22, (C2×C8)⋊25(C2×C4), (C2×C4⋊C4).64C4, C8⋊C4○(C8⋊C4), C4⋊C4.243(C2×C4), C22⋊C4.86(C2×C4), (C2×C22⋊C4).32C4, (C2×C4).569(C22×C4), (C22×C4).134(C2×C4), C42⋊C2○(C2×M4(2)), (C2×C42⋊C2).49C2, SmallGroup(128,1605)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for M4(2)○2M4(2)
G = < a,b,c,d | a8=b2=d2=1, c4=a4, bab=a5, ac=ca, ad=da, bc=cb, bd=db, dcd=a4c >
Subgroups: 332 in 282 conjugacy classes, 252 normal (12 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, C23, C23, C23, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, C22×C4, C24, C4×C8, C8⋊C4, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C22×C8, C2×M4(2), C23×C4, C4×M4(2), C8○2M4(2), C2×C42⋊C2, C22×M4(2), M4(2)○2M4(2)
Quotients: C1, C2, C4, C22, C2×C4, C23, C42, C22×C4, C24, C2×C42, C23×C4, C22×C42, Q8○M4(2), M4(2)○2M4(2)
►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)
(2 6)(4 8)(10 14)(12 16)(17 21)(19 23)(25 29)(27 31)
(1 18 26 9 5 22 30 13)(2 19 27 10 6 23 31 14)(3 20 28 11 7 24 32 15)(4 21 29 12 8 17 25 16)
(1 28)(2 29)(3 30)(4 31)(5 32)(6 25)(7 26)(8 27)(9 20)(10 21)(11 22)(12 23)(13 24)(14 17)(15 18)(16 19)
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), (2,6)(4,8)(10,14)(12,16)(17,21)(19,23)(25,29)(27,31), (1,18,26,9,5,22,30,13)(2,19,27,10,6,23,31,14)(3,20,28,11,7,24,32,15)(4,21,29,12,8,17,25,16), (1,28)(2,29)(3,30)(4,31)(5,32)(6,25)(7,26)(8,27)(9,20)(10,21)(11,22)(12,23)(13,24)(14,17)(15,18)(16,19)>;
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), (2,6)(4,8)(10,14)(12,16)(17,21)(19,23)(25,29)(27,31), (1,18,26,9,5,22,30,13)(2,19,27,10,6,23,31,14)(3,20,28,11,7,24,32,15)(4,21,29,12,8,17,25,16), (1,28)(2,29)(3,30)(4,31)(5,32)(6,25)(7,26)(8,27)(9,20)(10,21)(11,22)(12,23)(13,24)(14,17)(15,18)(16,19) );
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)], [(2,6),(4,8),(10,14),(12,16),(17,21),(19,23),(25,29),(27,31)], [(1,18,26,9,5,22,30,13),(2,19,27,10,6,23,31,14),(3,20,28,11,7,24,32,15),(4,21,29,12,8,17,25,16)], [(1,28),(2,29),(3,30),(4,31),(5,32),(6,25),(7,26),(8,27),(9,20),(10,21),(11,22),(12,23),(13,24),(14,17),(15,18),(16,19)]])
68 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | ··· | 2I | 4A | 4B | 4C | 4D | 4E | ··· | 4Z | 8A | ··· | 8AF |
| order | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
68 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 |
| type | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | Q8○M4(2) |
| kernel | M4(2)○2M4(2) | C4×M4(2) | C8○2M4(2) | C2×C42⋊C2 | C22×M4(2) | C2×C22⋊C4 | C2×C4⋊C4 | C42⋊C2 | C2×M4(2) | C2 |
| # reps | 1 | 4 | 8 | 1 | 2 | 4 | 4 | 8 | 32 | 4 |
Matrix representation of M4(2)○2M4(2) ►in GL5(𝔽17)
| 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 13 | 0 |
| 0 | 0 | 0 | 15 | 4 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 9 | 16 | 0 | 0 |
| 16 | 0 | 0 | 0 | 0 |
| 0 | 1 | 13 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 1 | 13 |
| 0 | 0 | 0 | 0 | 16 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 1 |
| 0 | 0 | 0 | 0 | 13 |
| 0 | 1 | 13 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 |
| 16 | 0 | 0 | 0 | 0 |
| 0 | 1 | 13 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 16 | 4 |
| 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(5,GF(17))| [13,0,0,0,0,0,0,0,1,9,0,0,0,0,16,0,13,15,0,0,0,0,4,0,0],[16,0,0,0,0,0,1,0,0,0,0,13,16,0,0,0,0,0,1,0,0,0,0,13,16],[1,0,0,0,0,0,0,0,1,0,0,0,0,13,16,0,4,0,0,0,0,1,13,0,0],[16,0,0,0,0,0,1,0,0,0,0,13,16,0,0,0,0,0,16,0,0,0,0,4,1] >;
M4(2)○2M4(2) in GAP, Magma, Sage, TeX
M_4(2)\circ_2M_4(2)
% in TeX
G:=Group("M4(2)o2M4(2)"); // GroupNames label
G:=SmallGroup(128,1605);
// by ID
G=gap.SmallGroup(128,1605);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,112,232,723,2019,172]);
// 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,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=a^4*c>;
// generators/relations