p-group, metabelian, nilpotent (class 4), monomial
Aliases: C24.4D4, C23.2Q16, C23.6SD16, C23⋊C8.5C2, C22.18C4≀C2, (C22×C4).11D4, C2.C42⋊6C4, C2.5(C42⋊C4), C23.9D4.5C2, C23.4Q8.1C2, C22.59(C23⋊C4), C2.6(C23.D4), C23.159(C22⋊C4), C22.17(Q8⋊C4), C2.5(C23.31D4), (C2×C4⋊C4)⋊3C4, (C22×C4).6(C2×C4), (C2×C22⋊C4).85C22, SmallGroup(128,84)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C24.4D4
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e4=c, f2=a, ab=ba, ac=ca, ad=da, eae-1=abc, af=fa, bc=cb, ebe-1=fbf-1=bd=db, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=abcde3 >
Subgroups: 236 in 77 conjugacy classes, 20 normal (all characteristic)
C1, C2 [×3], C2 [×4], C4 [×7], C22 [×3], C22 [×9], C8, C2×C4 [×15], C23 [×3], C23 [×4], C22⋊C4 [×6], C4⋊C4 [×3], C2×C8, C22×C4 [×2], C22×C4 [×3], C24, C2.C42, C22⋊C8, C2×C22⋊C4, C2×C22⋊C4 [×2], C2×C4⋊C4, C2×C4⋊C4, C23⋊C8, C23.9D4, C23.4Q8, C24.4D4
Quotients: C1, C2 [×3], C4 [×2], C22, C2×C4, D4 [×2], C22⋊C4, SD16, Q16, C23⋊C4, Q8⋊C4, C4≀C2, C23.31D4, C23.D4, C42⋊C4, C24.4D4
Character table of C24.4D4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 8A | 8B | 8C | 8D | |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 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 | 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 | 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 | 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 | linear of order 2 |
| ρ5 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -i | -1 | i | 1 | 1 | i | -i | 1 | -1 | -i | i | i | -i | linear of order 4 |
| ρ6 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | i | 1 | -i | -1 | 1 | -i | i | -1 | 1 | -i | i | i | -i | linear of order 4 |
| ρ7 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -i | 1 | i | -1 | 1 | i | -i | -1 | 1 | i | -i | -i | i | linear of order 4 |
| ρ8 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | i | -1 | -i | 1 | 1 | -i | i | 1 | -1 | i | -i | -i | i | linear of order 4 |
| ρ9 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | 2 | 2 | 2 | 2 | 2 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ11 | 2 | -2 | 2 | -2 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -√2 | -√2 | √2 | √2 | symplectic lifted from Q16, Schur index 2 |
| ρ12 | 2 | -2 | 2 | -2 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | √2 | √2 | -√2 | -√2 | symplectic lifted from Q16, Schur index 2 |
| ρ13 | 2 | -2 | 2 | -2 | -2 | 2 | 0 | 0 | 2i | -2i | 1-i | 0 | -1-i | 0 | 0 | 1+i | -1+i | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4≀C2 |
| ρ14 | 2 | -2 | 2 | -2 | -2 | 2 | 0 | 0 | -2i | 2i | -1-i | 0 | 1-i | 0 | 0 | -1+i | 1+i | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4≀C2 |
| ρ15 | 2 | -2 | 2 | -2 | -2 | 2 | 0 | 0 | 2i | -2i | -1+i | 0 | 1+i | 0 | 0 | -1-i | 1-i | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4≀C2 |
| ρ16 | 2 | -2 | 2 | -2 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | √-2 | -√-2 | √-2 | -√-2 | complex lifted from SD16 |
| ρ17 | 2 | -2 | 2 | -2 | -2 | 2 | 0 | 0 | -2i | 2i | 1+i | 0 | -1+i | 0 | 0 | 1-i | -1-i | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4≀C2 |
| ρ18 | 2 | -2 | 2 | -2 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -√-2 | √-2 | -√-2 | √-2 | complex lifted from SD16 |
| ρ19 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C42⋊C4 |
| ρ20 | 4 | 4 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C23⋊C4 |
| ρ21 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C42⋊C4 |
| ρ22 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 2i | 0 | 0 | 0 | 0 | complex lifted from C23.D4 |
| ρ23 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2i | 0 | 0 | 0 | 0 | 0 | 0 | -2i | 0 | 0 | 0 | 0 | complex lifted from C23.D4 |
►On 32 points
Generators in S32
(2 31)(3 17)(4 9)(6 27)(7 21)(8 13)(11 20)(12 28)(15 24)(16 32)(18 25)(22 29)
(1 26)(2 11)(3 28)(4 13)(5 30)(6 15)(7 32)(8 9)(10 23)(12 17)(14 19)(16 21)(18 29)(20 31)(22 25)(24 27)
(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)(25 29)(26 30)(27 31)(28 32)
(1 23)(2 24)(3 17)(4 18)(5 19)(6 20)(7 21)(8 22)(9 25)(10 26)(11 27)(12 28)(13 29)(14 30)(15 31)(16 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 4 31 9)(3 12 17 28)(6 8 27 13)(7 16 21 32)(10 26)(11 29 20 22)(14 30)(15 25 24 18)
G:=sub<Sym(32)| (2,31)(3,17)(4,9)(6,27)(7,21)(8,13)(11,20)(12,28)(15,24)(16,32)(18,25)(22,29), (1,26)(2,11)(3,28)(4,13)(5,30)(6,15)(7,32)(8,9)(10,23)(12,17)(14,19)(16,21)(18,29)(20,31)(22,25)(24,27), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32), (1,23)(2,24)(3,17)(4,18)(5,19)(6,20)(7,21)(8,22)(9,25)(10,26)(11,27)(12,28)(13,29)(14,30)(15,31)(16,32), (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), (2,4,31,9)(3,12,17,28)(6,8,27,13)(7,16,21,32)(10,26)(11,29,20,22)(14,30)(15,25,24,18)>;
G:=Group( (2,31)(3,17)(4,9)(6,27)(7,21)(8,13)(11,20)(12,28)(15,24)(16,32)(18,25)(22,29), (1,26)(2,11)(3,28)(4,13)(5,30)(6,15)(7,32)(8,9)(10,23)(12,17)(14,19)(16,21)(18,29)(20,31)(22,25)(24,27), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32), (1,23)(2,24)(3,17)(4,18)(5,19)(6,20)(7,21)(8,22)(9,25)(10,26)(11,27)(12,28)(13,29)(14,30)(15,31)(16,32), (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), (2,4,31,9)(3,12,17,28)(6,8,27,13)(7,16,21,32)(10,26)(11,29,20,22)(14,30)(15,25,24,18) );
G=PermutationGroup([(2,31),(3,17),(4,9),(6,27),(7,21),(8,13),(11,20),(12,28),(15,24),(16,32),(18,25),(22,29)], [(1,26),(2,11),(3,28),(4,13),(5,30),(6,15),(7,32),(8,9),(10,23),(12,17),(14,19),(16,21),(18,29),(20,31),(22,25),(24,27)], [(1,5),(2,6),(3,7),(4,8),(9,13),(10,14),(11,15),(12,16),(17,21),(18,22),(19,23),(20,24),(25,29),(26,30),(27,31),(28,32)], [(1,23),(2,24),(3,17),(4,18),(5,19),(6,20),(7,21),(8,22),(9,25),(10,26),(11,27),(12,28),(13,29),(14,30),(15,31),(16,32)], [(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)], [(2,4,31,9),(3,12,17,28),(6,8,27,13),(7,16,21,32),(10,26),(11,29,20,22),(14,30),(15,25,24,18)])
Matrix representation of C24.4D4 ►in GL6(𝔽17)
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 2 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 4 | 13 | 0 | 1 |
| 0 | 0 | 4 | 13 | 1 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 15 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 1 |
| 0 | 0 | 0 | 4 | 1 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 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 | 16 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 11 | 11 | 0 | 0 | 0 | 0 |
| 3 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 1 | 1 | 1 |
| 0 | 0 | 4 | 0 | 1 | 1 |
| 0 | 0 | 1 | 6 | 7 | 6 |
| 0 | 0 | 0 | 7 | 7 | 6 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 13 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 13 | 0 | 0 | 15 |
| 0 | 0 | 13 | 0 | 16 | 16 |
| 0 | 0 | 16 | 0 | 0 | 4 |
| 0 | 0 | 0 | 16 | 0 | 4 |
G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,4,4,0,0,2,1,13,13,0,0,0,0,0,1,0,0,0,0,1,0],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,15,16,4,4,0,0,0,0,0,1,0,0,0,0,1,0],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[11,3,0,0,0,0,11,0,0,0,0,0,0,0,4,4,1,0,0,0,1,0,6,7,0,0,1,1,7,7,0,0,1,1,6,6],[4,13,0,0,0,0,0,13,0,0,0,0,0,0,13,13,16,0,0,0,0,0,0,16,0,0,0,16,0,0,0,0,15,16,4,4] >;
C24.4D4 in GAP, Magma, Sage, TeX
C_2^4._4D_4
% in TeX
G:=Group("C2^4.4D4"); // GroupNames label
G:=SmallGroup(128,84);
// by ID
G=gap.SmallGroup(128,84);
# by ID
G:=PCGroup([7,-2,2,-2,2,-2,2,-2,56,85,568,422,387,520,1690,521,2804]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^4=c,f^2=a,a*b=b*a,a*c=c*a,a*d=d*a,e*a*e^-1=a*b*c,a*f=f*a,b*c=c*b,e*b*e^-1=f*b*f^-1=b*d=d*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=a*b*c*d*e^3>;
// generators/relations
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