p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.6D4, C4.4(C4×D4), (C2×Q16)⋊4C4, (C2×SD16)⋊4C4, (C2×Q8).73D4, C4.9C42⋊3C2, C22.56(C4×D4), C22⋊C4.121D4, C23.130(C2×D4), Q8.7(C22⋊C4), C4.138(C4⋊D4), M4(2)⋊4C4⋊6C2, C22.32C22≀C2, (C22×C4).32C23, C23.38D4⋊24C2, C22.52(C4⋊D4), C42⋊C22.5C2, (C22×Q8).22C22, C42⋊C2.30C22, C23.32C23⋊2C2, C4.11(C22.D4), (C2×M4(2)).10C22, C23.C23.6C2, C2.44(C23.23D4), (C2×C8).7(C2×C4), (C2×D4).86(C2×C4), (C2×C4).241(C2×D4), C4.21(C2×C22⋊C4), (C2×Q8).74(C2×C4), (C2×C8.C22).3C2, (C2×C4).327(C4○D4), (C2×C4).192(C22×C4), (C2×C4○D4).26C22, SmallGroup(128,637)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.6D4
G = < a,b,c,d | a4=b4=c4=1, d2=b, ab=ba, cac-1=dad-1=a-1b-1, cbc-1=b-1, bd=db, dcd-1=b-1c-1 >
Subgroups: 300 in 155 conjugacy classes, 54 normal (34 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, D4, Q8, Q8, C23, C23, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), SD16, Q16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C2×Q8, C4○D4, C23⋊C4, Q8⋊C4, C4≀C2, C42⋊C2, C42⋊C2, C4×Q8, C2×M4(2), C2×SD16, C2×Q16, C8.C22, C22×Q8, C2×C4○D4, C4.9C42, M4(2)⋊4C4, C23.C23, C23.38D4, C42⋊C22, C23.32C23, C2×C8.C22, C42.6D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C4○D4, C2×C22⋊C4, C4×D4, C22≀C2, C4⋊D4, C22.D4, C23.23D4, C42.6D4
Character table of C42.6D4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | 4Q | 4R | 4S | 8A | 8B | 8C | 8D | |
| size | 1 | 1 | 2 | 2 | 2 | 8 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 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 | -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 | 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 | -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 | 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 | -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 | 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 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ9 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | -1 | -i | i | 1 | -1 | 1 | i | -i | i | -i | i | -i | -1 | 1 | i | -i | 1 | i | -1 | -i | linear of order 4 |
| ρ10 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | -1 | i | -i | 1 | -1 | 1 | -i | i | -i | i | -i | i | -1 | 1 | -i | i | 1 | -i | -1 | i | linear of order 4 |
| ρ11 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | -1 | i | i | -1 | 1 | -1 | -i | i | -i | -i | i | -i | 1 | 1 | i | -i | -1 | -i | 1 | i | linear of order 4 |
| ρ12 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | -1 | -i | -i | -1 | 1 | -1 | i | -i | i | i | -i | i | 1 | 1 | -i | i | -1 | i | 1 | -i | linear of order 4 |
| ρ13 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | i | i | -1 | 1 | -1 | -i | i | -i | -i | i | -i | 1 | -1 | -i | i | 1 | i | -1 | -i | linear of order 4 |
| ρ14 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | -i | -i | -1 | 1 | -1 | i | -i | i | i | -i | i | 1 | -1 | i | -i | 1 | -i | -1 | i | linear of order 4 |
| ρ15 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | -i | i | 1 | -1 | 1 | i | -i | i | -i | i | -i | -1 | -1 | -i | i | -1 | -i | 1 | i | linear of order 4 |
| ρ16 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | i | -i | 1 | -1 | 1 | -i | i | -i | i | -i | i | -1 | -1 | i | -i | -1 | i | 1 | -i | linear of order 4 |
| ρ17 | 2 | 2 | 2 | 2 | 2 | 0 | -2 | -2 | -2 | -2 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ18 | 2 | 2 | -2 | 2 | -2 | 0 | 2 | -2 | 2 | -2 | 0 | 0 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ19 | 2 | 2 | -2 | 2 | -2 | 0 | 2 | -2 | 2 | -2 | 0 | 0 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ20 | 2 | 2 | -2 | -2 | 2 | 0 | -2 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ21 | 2 | 2 | 2 | 2 | 2 | 0 | -2 | -2 | -2 | -2 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ22 | 2 | 2 | -2 | -2 | 2 | 0 | -2 | 2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ23 | 2 | 2 | -2 | -2 | 2 | 0 | 2 | -2 | -2 | 2 | 0 | 0 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ24 | 2 | 2 | -2 | -2 | 2 | 0 | 2 | -2 | -2 | 2 | 0 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ25 | 2 | 2 | 2 | -2 | -2 | 0 | -2 | -2 | 2 | 2 | 0 | -2i | 0 | 0 | 0 | 0 | 0 | 0 | -2i | 2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ26 | 2 | 2 | 2 | -2 | -2 | 0 | -2 | -2 | 2 | 2 | 0 | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 2i | -2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ27 | 2 | 2 | -2 | 2 | -2 | 0 | -2 | 2 | -2 | 2 | -2i | 0 | 0 | 0 | 0 | 2i | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ28 | 2 | 2 | -2 | 2 | -2 | 0 | -2 | 2 | -2 | 2 | 2i | 0 | 0 | 0 | 0 | -2i | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ29 | 8 | -8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
►On 32 points
(1 32 27 2)(3 26 29 4)(5 28 31 6)(7 30 25 8)(9 22 17 10)(11 24 19 12)(13 18 21 14)(15 20 23 16)
(1 3 5 7)(2 4 6 8)(9 11 13 15)(10 12 14 16)(17 19 21 23)(18 20 22 24)(25 27 29 31)(26 28 30 32)
(1 13 31 21)(2 24 32 16)(3 11 25 19)(4 22 26 14)(5 9 27 17)(6 20 28 12)(7 15 29 23)(8 18 30 10)
(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)
G:=sub<Sym(32)| (1,32,27,2)(3,26,29,4)(5,28,31,6)(7,30,25,8)(9,22,17,10)(11,24,19,12)(13,18,21,14)(15,20,23,16), (1,3,5,7)(2,4,6,8)(9,11,13,15)(10,12,14,16)(17,19,21,23)(18,20,22,24)(25,27,29,31)(26,28,30,32), (1,13,31,21)(2,24,32,16)(3,11,25,19)(4,22,26,14)(5,9,27,17)(6,20,28,12)(7,15,29,23)(8,18,30,10), (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)>;
G:=Group( (1,32,27,2)(3,26,29,4)(5,28,31,6)(7,30,25,8)(9,22,17,10)(11,24,19,12)(13,18,21,14)(15,20,23,16), (1,3,5,7)(2,4,6,8)(9,11,13,15)(10,12,14,16)(17,19,21,23)(18,20,22,24)(25,27,29,31)(26,28,30,32), (1,13,31,21)(2,24,32,16)(3,11,25,19)(4,22,26,14)(5,9,27,17)(6,20,28,12)(7,15,29,23)(8,18,30,10), (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) );
G=PermutationGroup([[(1,32,27,2),(3,26,29,4),(5,28,31,6),(7,30,25,8),(9,22,17,10),(11,24,19,12),(13,18,21,14),(15,20,23,16)], [(1,3,5,7),(2,4,6,8),(9,11,13,15),(10,12,14,16),(17,19,21,23),(18,20,22,24),(25,27,29,31),(26,28,30,32)], [(1,13,31,21),(2,24,32,16),(3,11,25,19),(4,22,26,14),(5,9,27,17),(6,20,28,12),(7,15,29,23),(8,18,30,10)], [(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)]])
Matrix representation of C42.6D4 ►in GL8(𝔽17)
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 6 | 11 | 16 | 1 | 1 | 15 | 0 | 0 |
| 5 | 1 | 9 | 8 | 1 | 0 | 15 | 0 |
| 3 | 3 | 14 | 3 | 1 | 0 | 0 | 15 |
| 0 | 0 | 0 | 16 | 0 | 0 | 0 | 0 |
| 15 | 2 | 5 | 11 | 0 | 6 | 1 | 16 |
| 12 | 13 | 11 | 14 | 3 | 16 | 8 | 9 |
| 15 | 11 | 6 | 2 | 3 | 14 | 3 | 14 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 | 0 | 0 |
| 6 | 11 | 16 | 1 | 1 | 15 | 0 | 0 |
| 6 | 0 | 16 | 0 | 1 | 16 | 0 | 0 |
| 1 | 15 | 14 | 12 | 0 | 16 | 0 | 1 |
| 4 | 16 | 11 | 3 | 1 | 16 | 16 | 0 |
| 8 | 5 | 11 | 6 | 10 | 0 | 7 | 0 |
| 6 | 15 | 3 | 14 | 7 | 10 | 0 | 10 |
| 8 | 5 | 11 | 6 | 0 | 0 | 7 | 0 |
| 15 | 6 | 10 | 7 | 0 | 7 | 0 | 10 |
| 0 | 12 | 0 | 12 | 0 | 0 | 0 | 0 |
| 12 | 2 | 9 | 8 | 9 | 16 | 9 | 9 |
| 0 | 2 | 8 | 9 | 3 | 16 | 15 | 11 |
| 12 | 12 | 5 | 12 | 0 | 0 | 0 | 13 |
| 6 | 11 | 16 | 1 | 1 | 15 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 | 0 | 0 |
| 14 | 14 | 3 | 14 | 16 | 0 | 0 | 2 |
| 5 | 1 | 9 | 8 | 1 | 0 | 15 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 14 | 9 | 0 | 0 | 7 | 11 | 16 | 16 |
| 5 | 1 | 5 | 4 | 10 | 12 | 9 | 9 |
| 13 | 14 | 16 | 10 | 3 | 14 | 14 | 14 |
G:=sub<GL(8,GF(17))| [0,6,5,3,0,15,12,15,0,11,1,3,0,2,13,11,0,16,9,14,0,5,11,6,0,1,8,3,16,11,14,2,1,1,1,1,0,0,3,3,0,15,0,0,0,6,16,14,0,0,15,0,0,1,8,3,0,0,0,15,0,16,9,14],[0,16,0,0,6,6,1,4,1,0,0,0,11,0,15,16,0,0,0,16,16,16,14,11,0,0,1,0,1,0,12,3,0,0,0,0,1,1,0,1,0,0,0,0,15,16,16,16,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0],[8,6,8,15,0,12,0,12,5,15,5,6,12,2,2,12,11,3,11,10,0,9,8,5,6,14,6,7,12,8,9,12,10,7,0,0,0,9,3,0,0,10,0,7,0,16,16,0,7,0,7,0,0,9,15,0,0,10,0,10,0,9,11,13],[6,0,14,5,1,14,5,13,11,0,14,1,0,9,1,14,16,0,3,9,0,0,5,16,1,0,14,8,0,0,4,10,1,16,16,1,0,7,10,3,15,0,0,0,0,11,12,14,0,0,0,15,0,16,9,14,0,0,2,0,0,16,9,14] >;
C42.6D4 in GAP, Magma, Sage, TeX
C_4^2._6D_4
% in TeX
G:=Group("C4^2.6D4"); // GroupNames label
G:=SmallGroup(128,637);
// by ID
G=gap.SmallGroup(128,637);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,288,422,352,2019,521,248,1411,718,172,1027,124]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^4=1,d^2=b,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1*b^-1,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=b^-1*c^-1>;
// generators/relations
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