p-group, metabelian, nilpotent (class 4), monomial
Aliases: C24.6D4, C23⋊C4⋊3C4, (C2×C42)⋊3C4, (C2×D4).3Q8, (C2×D4).41D4, (C2×C4).1C42, C23.2(C4⋊C4), C2.3(C42⋊C4), C2.3(C42⋊3C4), C23.3(C22⋊C4), (C22×D4).2C22, C22.14(C23⋊C4), C2.15(C23.9D4), C24.3C22.3C2, C22.4(C2.C42), (C2×C4⋊C4)⋊4C4, (C2×C4).9(C4⋊C4), (C2×D4).46(C2×C4), (C2×C23⋊C4).2C2, (C22×C4).66(C2×C4), SmallGroup(128,125)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C24.6D4
G = < a,b,c,d,e,f | a2=b2=c2=d2=e4=1, f2=a, ab=ba, eae-1=ac=ca, ad=da, af=fa, bc=cb, ebe-1=fbf-1=bd=db, ece-1=fcf-1=cd=dc, de=ed, df=fd, fef-1=abe-1 >
Subgroups: 352 in 116 conjugacy classes, 32 normal (18 characteristic)
C1, C2, C2, C4, C22, C22, C2×C4, C2×C4, D4, C23, C23, C23, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C24, C23⋊C4, C23⋊C4, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C22×D4, C24.3C22, C2×C23⋊C4, C24.6D4
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C42, C22⋊C4, C4⋊C4, C2.C42, C23⋊C4, C23.9D4, C42⋊C4, C42⋊3C4, C24.6D4
Character table of C24.6D4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 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 | 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 | 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 | 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 | linear of order 2 |
| ρ5 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | -1 | 1 | -1 | -1 | -1 | i | i | -i | -i | 1 | 1 | -i | -i | i | i | linear of order 4 |
| ρ6 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | -i | -i | i | i | -1 | -1 | -i | -i | i | i | linear of order 4 |
| ρ7 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | i | 1 | -i | -i | i | -1 | 1 | 1 | -1 | i | -i | -i | i | i | -i | linear of order 4 |
| ρ8 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | i | 1 | -i | -i | i | -i | i | -i | i | -i | i | 1 | -1 | 1 | -1 | linear of order 4 |
| ρ9 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | i | 1 | -i | -i | i | 1 | -1 | -1 | 1 | i | -i | i | -i | -i | i | linear of order 4 |
| ρ10 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | -i | 1 | i | 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 | -1 | -i | 1 | i | 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 | -1 | -i | 1 | i | i | -i | 1 | -1 | -1 | 1 | -i | i | -i | i | i | -i | linear of order 4 |
| ρ13 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | i | 1 | -i | -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 | 1 | 1 | 1 | 1 | 1 | 1 | i | i | -i | -i | -1 | -1 | i | i | -i | -i | linear of order 4 |
| ρ15 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | -i | 1 | i | i | -i | -1 | 1 | 1 | -1 | -i | i | i | -i | -i | i | linear of order 4 |
| ρ16 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | -1 | 1 | -1 | -1 | -1 | -i | -i | i | i | 1 | 1 | i | i | -i | -i | linear of order 4 |
| ρ17 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | -2 | -2 | -2 | 0 | -2 | 0 | 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 | 2 | 0 | -2 | 0 | 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 | -2 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ20 | 2 | -2 | 2 | -2 | 2 | -2 | -2 | 2 | -2 | 2 | 2 | 0 | -2 | 0 | 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 | 2 | 0 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C42⋊C4 |
| ρ22 | 4 | -4 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C23⋊C4 |
| ρ23 | 4 | 4 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C23⋊C4 |
| ρ24 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C42⋊C4 |
| ρ25 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2i | 0 | 2i | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C42⋊3C4 |
| ρ26 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2i | 0 | -2i | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C42⋊3C4 |
►On 32 points
Generators in S32
(1 9)(2 15)(3 10)(4 14)(5 16)(6 12)(7 11)(8 13)(17 28)(18 19)(20 25)(21 29)(22 23)(24 30)(26 27)(31 32)
(1 6)(2 4)(3 7)(5 8)(9 12)(10 11)(13 16)(14 15)(17 23)(18 29)(19 21)(20 31)(22 28)(24 26)(25 32)(27 30)
(1 4)(2 6)(3 8)(5 7)(9 14)(10 13)(11 16)(12 15)(17 19)(18 28)(20 26)(21 23)(22 29)(24 31)(25 27)(30 32)
(1 7)(2 8)(3 6)(4 5)(9 11)(10 12)(13 15)(14 16)(17 25)(18 26)(19 27)(20 28)(21 30)(22 31)(23 32)(24 29)
(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 29 9 21)(2 28 15 17)(3 18 10 19)(4 31 14 32)(5 22 16 23)(6 26 12 27)(7 24 11 30)(8 20 13 25)
G:=sub<Sym(32)| (1,9)(2,15)(3,10)(4,14)(5,16)(6,12)(7,11)(8,13)(17,28)(18,19)(20,25)(21,29)(22,23)(24,30)(26,27)(31,32), (1,6)(2,4)(3,7)(5,8)(9,12)(10,11)(13,16)(14,15)(17,23)(18,29)(19,21)(20,31)(22,28)(24,26)(25,32)(27,30), (1,4)(2,6)(3,8)(5,7)(9,14)(10,13)(11,16)(12,15)(17,19)(18,28)(20,26)(21,23)(22,29)(24,31)(25,27)(30,32), (1,7)(2,8)(3,6)(4,5)(9,11)(10,12)(13,15)(14,16)(17,25)(18,26)(19,27)(20,28)(21,30)(22,31)(23,32)(24,29), (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), (1,29,9,21)(2,28,15,17)(3,18,10,19)(4,31,14,32)(5,22,16,23)(6,26,12,27)(7,24,11,30)(8,20,13,25)>;
G:=Group( (1,9)(2,15)(3,10)(4,14)(5,16)(6,12)(7,11)(8,13)(17,28)(18,19)(20,25)(21,29)(22,23)(24,30)(26,27)(31,32), (1,6)(2,4)(3,7)(5,8)(9,12)(10,11)(13,16)(14,15)(17,23)(18,29)(19,21)(20,31)(22,28)(24,26)(25,32)(27,30), (1,4)(2,6)(3,8)(5,7)(9,14)(10,13)(11,16)(12,15)(17,19)(18,28)(20,26)(21,23)(22,29)(24,31)(25,27)(30,32), (1,7)(2,8)(3,6)(4,5)(9,11)(10,12)(13,15)(14,16)(17,25)(18,26)(19,27)(20,28)(21,30)(22,31)(23,32)(24,29), (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), (1,29,9,21)(2,28,15,17)(3,18,10,19)(4,31,14,32)(5,22,16,23)(6,26,12,27)(7,24,11,30)(8,20,13,25) );
G=PermutationGroup([[(1,9),(2,15),(3,10),(4,14),(5,16),(6,12),(7,11),(8,13),(17,28),(18,19),(20,25),(21,29),(22,23),(24,30),(26,27),(31,32)], [(1,6),(2,4),(3,7),(5,8),(9,12),(10,11),(13,16),(14,15),(17,23),(18,29),(19,21),(20,31),(22,28),(24,26),(25,32),(27,30)], [(1,4),(2,6),(3,8),(5,7),(9,14),(10,13),(11,16),(12,15),(17,19),(18,28),(20,26),(21,23),(22,29),(24,31),(25,27),(30,32)], [(1,7),(2,8),(3,6),(4,5),(9,11),(10,12),(13,15),(14,16),(17,25),(18,26),(19,27),(20,28),(21,30),(22,31),(23,32),(24,29)], [(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)], [(1,29,9,21),(2,28,15,17),(3,18,10,19),(4,31,14,32),(5,22,16,23),(6,26,12,27),(7,24,11,30),(8,20,13,25)]])
Matrix representation of C24.6D4 ►in GL8(ℤ)
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 |
| 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 2 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 |
| -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | -1 | -2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | -1 | -2 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | -1 | -2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 |
| 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | -1 | -1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 |
| 0 | 0 | 0 | 0 | -1 | -1 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 1 | 0 | 0 |
G:=sub<GL(8,Integers())| [0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,2,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,2,-1],[-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,-2,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,-2,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1],[1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,-2,1,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,-1],[0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,1,0,-1,1,0,0,0,0,-1,1,0,0,0,0,0,0,-1,1,1,0] >;
C24.6D4 in GAP, Magma, Sage, TeX
C_2^4._6D_4
% in TeX
G:=Group("C2^4.6D4"); // GroupNames label
G:=SmallGroup(128,125);
// by ID
G=gap.SmallGroup(128,125);
# by ID
G:=PCGroup([7,-2,2,-2,2,2,-2,-2,56,85,120,758,352,1466,521,2804,4037]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=e^4=1,f^2=a,a*b=b*a,e*a*e^-1=a*c=c*a,a*d=d*a,a*f=f*a,b*c=c*b,e*b*e^-1=f*b*f^-1=b*d=d*b,e*c*e^-1=f*c*f^-1=c*d=d*c,d*e=e*d,d*f=f*d,f*e*f^-1=a*b*e^-1>;
// generators/relations
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