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 [×3], C2 [×6], C4 [×9], C22 [×3], C22 [×16], C2×C4 [×2], C2×C4 [×17], D4 [×4], C23, C23 [×4], C23 [×8], C42, C22⋊C4 [×10], C4⋊C4, C22×C4, C22×C4 [×4], C2×D4 [×2], C2×D4 [×2], C2×D4 [×2], C24 [×2], C23⋊C4 [×4], C23⋊C4 [×2], C2×C42, C2×C22⋊C4 [×4], C2×C4⋊C4, C22×D4, C24.3C22, C2×C23⋊C4 [×2], C24.6D4
Quotients: C1, C2 [×3], C4 [×6], C22, C2×C4 [×3], D4 [×3], Q8, C42, C22⋊C4 [×3], C4⋊C4 [×3], C2.C42, C23⋊C4 [×2], 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 11)(2 16)(3 12)(4 15)(5 9)(6 14)(7 13)(8 10)(17 21)(18 24)(19 20)(22 23)(25 32)(26 27)(28 29)(30 31)
(1 3)(2 7)(4 6)(5 8)(9 10)(11 12)(13 16)(14 15)(17 30)(18 27)(19 32)(20 25)(21 31)(22 28)(23 29)(24 26)
(1 4)(2 8)(3 6)(5 7)(9 13)(10 16)(11 15)(12 14)(17 22)(18 20)(19 24)(21 23)(25 27)(26 32)(28 30)(29 31)
(1 5)(2 6)(3 8)(4 7)(9 11)(10 12)(13 15)(14 16)(17 24)(18 21)(19 22)(20 23)(25 29)(26 30)(27 31)(28 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 30 11 31)(2 19 16 20)(3 24 12 18)(4 32 15 25)(5 26 9 27)(6 22 14 23)(7 28 13 29)(8 17 10 21)
G:=sub<Sym(32)| (1,11)(2,16)(3,12)(4,15)(5,9)(6,14)(7,13)(8,10)(17,21)(18,24)(19,20)(22,23)(25,32)(26,27)(28,29)(30,31), (1,3)(2,7)(4,6)(5,8)(9,10)(11,12)(13,16)(14,15)(17,30)(18,27)(19,32)(20,25)(21,31)(22,28)(23,29)(24,26), (1,4)(2,8)(3,6)(5,7)(9,13)(10,16)(11,15)(12,14)(17,22)(18,20)(19,24)(21,23)(25,27)(26,32)(28,30)(29,31), (1,5)(2,6)(3,8)(4,7)(9,11)(10,12)(13,15)(14,16)(17,24)(18,21)(19,22)(20,23)(25,29)(26,30)(27,31)(28,32), (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), (1,30,11,31)(2,19,16,20)(3,24,12,18)(4,32,15,25)(5,26,9,27)(6,22,14,23)(7,28,13,29)(8,17,10,21)>;
G:=Group( (1,11)(2,16)(3,12)(4,15)(5,9)(6,14)(7,13)(8,10)(17,21)(18,24)(19,20)(22,23)(25,32)(26,27)(28,29)(30,31), (1,3)(2,7)(4,6)(5,8)(9,10)(11,12)(13,16)(14,15)(17,30)(18,27)(19,32)(20,25)(21,31)(22,28)(23,29)(24,26), (1,4)(2,8)(3,6)(5,7)(9,13)(10,16)(11,15)(12,14)(17,22)(18,20)(19,24)(21,23)(25,27)(26,32)(28,30)(29,31), (1,5)(2,6)(3,8)(4,7)(9,11)(10,12)(13,15)(14,16)(17,24)(18,21)(19,22)(20,23)(25,29)(26,30)(27,31)(28,32), (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), (1,30,11,31)(2,19,16,20)(3,24,12,18)(4,32,15,25)(5,26,9,27)(6,22,14,23)(7,28,13,29)(8,17,10,21) );
G=PermutationGroup([(1,11),(2,16),(3,12),(4,15),(5,9),(6,14),(7,13),(8,10),(17,21),(18,24),(19,20),(22,23),(25,32),(26,27),(28,29),(30,31)], [(1,3),(2,7),(4,6),(5,8),(9,10),(11,12),(13,16),(14,15),(17,30),(18,27),(19,32),(20,25),(21,31),(22,28),(23,29),(24,26)], [(1,4),(2,8),(3,6),(5,7),(9,13),(10,16),(11,15),(12,14),(17,22),(18,20),(19,24),(21,23),(25,27),(26,32),(28,30),(29,31)], [(1,5),(2,6),(3,8),(4,7),(9,11),(10,12),(13,15),(14,16),(17,24),(18,21),(19,22),(20,23),(25,29),(26,30),(27,31),(28,32)], [(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)], [(1,30,11,31),(2,19,16,20),(3,24,12,18),(4,32,15,25),(5,26,9,27),(6,22,14,23),(7,28,13,29),(8,17,10,21)])
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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