p-group, metabelian, nilpotent (class 3), monomial
Aliases: M4(2)⋊1C4, C23.40D4, C8⋊1(C2×C4), C4.Q8⋊2C2, (C2×C4).6Q8, C4.4(C2×Q8), C2.D8⋊10C2, C4.15(C4⋊C4), (C2×C4).128D4, C4⋊C4.49C22, C2.3(C8⋊C22), C4.27(C22×C4), (C2×C4).70C23, (C2×C8).13C22, C22.50(C2×D4), C22.10(C4⋊C4), C42⋊C2.7C2, C2.3(C8.C22), (C2×M4(2)).1C2, (C22×C4).38C22, C2.14(C2×C4⋊C4), (C2×C4⋊C4).15C2, (C2×C4).26(C2×C4), SmallGroup(64,109)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for M4(2)⋊C4
G = < a,b,c | a8=b2=c4=1, bab=a5, cac-1=a-1, bc=cb >
Subgroups: 89 in 59 conjugacy classes, 41 normal (17 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, C23, C42, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, C22×C4, C4.Q8, C2.D8, C2×C4⋊C4, C42⋊C2, C2×M4(2), M4(2)⋊C4
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C4⋊C4, C22×C4, C2×D4, C2×Q8, C2×C4⋊C4, C8⋊C22, C8.C22, M4(2)⋊C4
Character table of M4(2)⋊C4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 8A | 8B | 8C | 8D | |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | linear of order 2 |
| ρ3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ4 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | linear of order 2 |
| ρ5 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | 1 | -1 | linear of order 2 |
| ρ6 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ7 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | linear of order 2 |
| ρ8 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ9 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | -i | -i | -i | -i | i | i | i | i | -1 | -1 | 1 | 1 | linear of order 4 |
| ρ10 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | -i | -i | i | i | i | i | -i | -i | 1 | 1 | -1 | -1 | linear of order 4 |
| ρ11 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | -i | i | -i | i | i | -i | i | -i | -1 | 1 | 1 | -1 | linear of order 4 |
| ρ12 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | -i | i | i | -i | i | -i | -i | i | 1 | -1 | -1 | 1 | linear of order 4 |
| ρ13 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | i | -i | i | -i | -i | i | -i | i | -1 | 1 | 1 | -1 | linear of order 4 |
| ρ14 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | i | -i | -i | i | -i | i | i | -i | 1 | -1 | -1 | 1 | linear of order 4 |
| ρ15 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | i | i | i | i | -i | -i | -i | -i | -1 | -1 | 1 | 1 | linear of order 4 |
| ρ16 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | i | i | -i | -i | -i | -i | i | i | 1 | 1 | -1 | -1 | linear of order 4 |
| ρ17 | 2 | 2 | 2 | 2 | 2 | 2 | -2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ18 | 2 | 2 | 2 | 2 | -2 | -2 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ19 | 2 | -2 | 2 | -2 | 2 | -2 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ20 | 2 | -2 | 2 | -2 | -2 | 2 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ21 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C8⋊C22 |
| ρ22 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from C8.C22, Schur index 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)
(1 23)(2 20)(3 17)(4 22)(5 19)(6 24)(7 21)(8 18)(9 27)(10 32)(11 29)(12 26)(13 31)(14 28)(15 25)(16 30)
(1 10 23 32)(2 9 24 31)(3 16 17 30)(4 15 18 29)(5 14 19 28)(6 13 20 27)(7 12 21 26)(8 11 22 25)
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), (1,23)(2,20)(3,17)(4,22)(5,19)(6,24)(7,21)(8,18)(9,27)(10,32)(11,29)(12,26)(13,31)(14,28)(15,25)(16,30), (1,10,23,32)(2,9,24,31)(3,16,17,30)(4,15,18,29)(5,14,19,28)(6,13,20,27)(7,12,21,26)(8,11,22,25)>;
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), (1,23)(2,20)(3,17)(4,22)(5,19)(6,24)(7,21)(8,18)(9,27)(10,32)(11,29)(12,26)(13,31)(14,28)(15,25)(16,30), (1,10,23,32)(2,9,24,31)(3,16,17,30)(4,15,18,29)(5,14,19,28)(6,13,20,27)(7,12,21,26)(8,11,22,25) );
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)], [(1,23),(2,20),(3,17),(4,22),(5,19),(6,24),(7,21),(8,18),(9,27),(10,32),(11,29),(12,26),(13,31),(14,28),(15,25),(16,30)], [(1,10,23,32),(2,9,24,31),(3,16,17,30),(4,15,18,29),(5,14,19,28),(6,13,20,27),(7,12,21,26),(8,11,22,25)]])
M4(2)⋊C4 is a maximal subgroup of
C24.21D4 C4.10D4⋊2C4 C4≀C2⋊C4 C42⋊9(C2×C4) M4(2)⋊6D4 M4(2).7D4 M4(2)⋊Q8 C42⋊3Q8 C24.100D4 C4○D4.7Q8 C4○D4.8Q8 C4×C8⋊C22 C4×C8.C22 C42.275C23 C42.276C23 M4(2)⋊14D4 M4(2)⋊15D4 M4(2)⋊16D4 M4(2)⋊17D4 C42.219D4 C42.220D4 C42.448D4 C42.449D4 C42.20C23 C42.21C23 C42.22C23 C42.23C23 C24.183D4 C24.116D4 C24.117D4 C24.118D4 (C2×D4).301D4 (C2×D4).302D4 (C2×D4).303D4 (C2×D4).304D4 M4(2)⋊3Q8 M4(2)⋊4Q8 M4(2)⋊5Q8 M4(2)⋊6Q8 C42.57C23 C42.58C23 C42.59C23 C42.60C23 C42.61C23 C42.62C23 C42.63C23 C42.64C23 C42.492C23 C42.493C23 C42.494C23 C42.495C23 C42.496C23 C42.497C23 C42.498C23
C4p.(C4⋊C4): M4(2).5Q8 M4(2).6Q8 C4⋊C4.225D6 C4⋊C4.232D6 C23.52D12 C20.47(C4⋊C4) C20.64(C4⋊C4) C23.47D20 ...
(C2×C8).D2p: C4.10D4⋊3C4 C4.D4⋊3C4 M4(2).12D4 M4(2).13D4 M4(2).Q8 M4(2).2Q8 C8⋊(C4×S3) C8⋊S3⋊C4 ...
M4(2)⋊C4 is a maximal quotient of
M4(2)⋊1C8 C8⋊1M4(2) C42.92D4 C42.21Q8 C24.152D4 C8⋊C42 C24.67D4 C42.24Q8 C42.26Q8 M4(2)⋊1F5
C23.D4p: C23.37D8 C23.52D12 C23.47D20 C23.47D28 ...
C4⋊C4.D2p: C24.159D4 C24.71D4 C42.29Q8 C42.30Q8 C42.31Q8 C8⋊(C4×S3) C8⋊S3⋊C4 C4⋊C4.225D6 ...
Matrix representation of M4(2)⋊C4 ►in GL6(𝔽17)
| 0 | 16 | 0 | 0 | 0 | 0 |
| 1 | 0 | 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 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 5 | 5 | 0 | 0 | 0 | 0 |
| 5 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 14 | 0 | 0 |
| 0 | 0 | 1 | 6 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 | 15 |
| 0 | 0 | 0 | 0 | 7 | 9 |
G:=sub<GL(6,GF(17))| [0,1,0,0,0,0,16,0,0,0,0,0,0,0,0,0,16,16,0,0,0,0,2,1,0,0,1,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[5,5,0,0,0,0,5,12,0,0,0,0,0,0,11,1,0,0,0,0,14,6,0,0,0,0,0,0,8,7,0,0,0,0,15,9] >;
M4(2)⋊C4 in GAP, Magma, Sage, TeX
M_4(2)\rtimes C_4
% in TeX
G:=Group("M4(2):C4"); // GroupNames label
G:=SmallGroup(64,109);
// by ID
G=gap.SmallGroup(64,109);
# by ID
G:=PCGroup([6,-2,2,2,-2,2,-2,96,121,55,332,963,117]);
// Polycyclic
G:=Group<a,b,c|a^8=b^2=c^4=1,b*a*b=a^5,c*a*c^-1=a^-1,b*c=c*b>;
// generators/relations
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