p-group, metabelian, nilpotent (class 4), monomial
Aliases: C24.5D4, C23.1C42, C2.3C2≀C4, C23⋊C4⋊1C4, (C23×C4)⋊1C4, (C2×D4).1Q8, (C2×D4).39D4, C4.D4⋊5C4, C23.7(C4⋊C4), C24.18(C2×C4), (C22×C4).35D4, C23.1(C22⋊C4), (C22×D4).1C22, C22.42(C23⋊C4), C23.23D4.1C2, C2.3(C23.D4), C2.12(C23.9D4), C22.1(C2.C42), (C2×C4).1(C4⋊C4), (C2×C22⋊C4)⋊2C4, (C2×D4).43(C2×C4), (C2×C23⋊C4).1C2, (C2×C4).1(C22⋊C4), (C2×C4.D4).5C2, SmallGroup(128,122)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C24.5D4
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e4=d, f2=a, ab=ba, ac=ca, ad=da, eae-1=acd, af=fa, bc=cb, bd=db, be=eb, bf=fb, ece-1=fcf-1=cd=dc, de=ed, df=fd, fef-1=abde3 >
Subgroups: 336 in 117 conjugacy classes, 32 normal (28 characteristic)
C1, C2 [×3], C2 [×6], C4 [×7], C22 [×3], C22 [×17], C8 [×2], C2×C4 [×2], C2×C4 [×17], D4 [×4], C23 [×5], C23 [×8], C22⋊C4 [×6], C2×C8, M4(2) [×3], C22×C4, C22×C4 [×6], C2×D4 [×4], C2×D4 [×2], C24 [×2], C2.C42, C23⋊C4 [×2], C23⋊C4, C4.D4 [×2], C4.D4, C2×C22⋊C4, C2×C22⋊C4 [×2], C2×M4(2), C23×C4, C22×D4, C23.23D4, C2×C23⋊C4, C2×C4.D4, C24.5D4
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, C2≀C4, C23.D4, C24.5D4
Character table of C24.5D4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 8A | 8B | 8C | 8D | |
| 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 | i | 1 | -1 | -i | i | -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 | 1 | i | -i | i | i | i | -i | i | -i | -i | 1 | -1 | 1 | -1 | linear of order 4 |
| ρ7 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | -i | -1 | 1 | i | -i | i | -i | -i | -i | i | i | i | -1 | 1 | -1 | 1 | linear of order 4 |
| ρ8 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | 1 | -1 | -i | 1 | -1 | i | -i | i | -1 | 1 | i | -i | -1 | 1 | i | -i | -i | i | linear of order 4 |
| ρ9 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | i | -1 | 1 | -i | 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 | -1 | 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 | i | 1 | -1 | -i | i | -i | 1 | -1 | -i | i | 1 | -1 | i | -i | -i | i | linear of order 4 |
| ρ12 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -i | i | 1 | 1 | i | -i | i | i | -i | -i | linear of order 4 |
| ρ13 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | i | -i | 1 | 1 | -i | i | -i | -i | i | i | linear of order 4 |
| ρ14 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | 1 | -1 | -i | 1 | -1 | i | -i | i | 1 | -1 | i | -i | 1 | -1 | -i | i | i | -i | linear of order 4 |
| ρ15 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | i | -i | -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 | -1 | -1 | i | -i | -i | -i | i | i | linear of order 4 |
| ρ17 | 2 | 2 | 2 | 2 | 2 | 2 | -2 | -2 | -2 | -2 | 0 | 2 | 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 | 0 | -2 | -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 | 0 | -2 | 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 | 0 | 2 | -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 | 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 |
| ρ22 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2≀C4 |
| ρ23 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2≀C4 |
| ρ24 | 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 |
| ρ25 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 2i | 0 | 0 | -2i | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C23.D4 |
| ρ26 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | -2i | 0 | 0 | 2i | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C23.D4 |
►On 32 points
(2 30)(3 7)(4 28)(6 26)(8 32)(9 23)(10 14)(11 21)(13 19)(15 17)(20 24)(27 31)
(1 16)(2 9)(3 10)(4 11)(5 12)(6 13)(7 14)(8 15)(17 32)(18 25)(19 26)(20 27)(21 28)(22 29)(23 30)(24 31)
(1 25)(2 30)(3 27)(4 32)(5 29)(6 26)(7 31)(8 28)(9 23)(10 20)(11 17)(12 22)(13 19)(14 24)(15 21)(16 18)
(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 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 16)(2 32 30 8)(3 20 7 24)(4 6 28 26)(5 12)(9 17 23 15)(10 27 14 31)(11 13 21 19)(18 29)(22 25)
G:=sub<Sym(32)| (2,30)(3,7)(4,28)(6,26)(8,32)(9,23)(10,14)(11,21)(13,19)(15,17)(20,24)(27,31), (1,16)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14)(8,15)(17,32)(18,25)(19,26)(20,27)(21,28)(22,29)(23,30)(24,31), (1,25)(2,30)(3,27)(4,32)(5,29)(6,26)(7,31)(8,28)(9,23)(10,20)(11,17)(12,22)(13,19)(14,24)(15,21)(16,18), (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), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32), (1,16)(2,32,30,8)(3,20,7,24)(4,6,28,26)(5,12)(9,17,23,15)(10,27,14,31)(11,13,21,19)(18,29)(22,25)>;
G:=Group( (2,30)(3,7)(4,28)(6,26)(8,32)(9,23)(10,14)(11,21)(13,19)(15,17)(20,24)(27,31), (1,16)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14)(8,15)(17,32)(18,25)(19,26)(20,27)(21,28)(22,29)(23,30)(24,31), (1,25)(2,30)(3,27)(4,32)(5,29)(6,26)(7,31)(8,28)(9,23)(10,20)(11,17)(12,22)(13,19)(14,24)(15,21)(16,18), (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), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32), (1,16)(2,32,30,8)(3,20,7,24)(4,6,28,26)(5,12)(9,17,23,15)(10,27,14,31)(11,13,21,19)(18,29)(22,25) );
G=PermutationGroup([(2,30),(3,7),(4,28),(6,26),(8,32),(9,23),(10,14),(11,21),(13,19),(15,17),(20,24),(27,31)], [(1,16),(2,9),(3,10),(4,11),(5,12),(6,13),(7,14),(8,15),(17,32),(18,25),(19,26),(20,27),(21,28),(22,29),(23,30),(24,31)], [(1,25),(2,30),(3,27),(4,32),(5,29),(6,26),(7,31),(8,28),(9,23),(10,20),(11,17),(12,22),(13,19),(14,24),(15,21),(16,18)], [(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)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32)], [(1,16),(2,32,30,8),(3,20,7,24),(4,6,28,26),(5,12),(9,17,23,15),(10,27,14,31),(11,13,21,19),(18,29),(22,25)])
Matrix representation of C24.5D4 ►in GL6(𝔽17)
| 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 | 16 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 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 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 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 | 16 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 1 | 15 | 0 | 0 | 0 | 0 |
| 1 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 9 | 9 | 9 |
| 0 | 0 | 8 | 9 | 8 | 8 |
| 0 | 0 | 8 | 8 | 8 | 9 |
| 0 | 0 | 9 | 9 | 8 | 9 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 4 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[1,1,0,0,0,0,15,16,0,0,0,0,0,0,8,8,8,9,0,0,9,9,8,9,0,0,9,8,8,8,0,0,9,8,9,9],[4,4,0,0,0,0,0,13,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,0,1,0,0,0,0,16,0] >;
C24.5D4 in GAP, Magma, Sage, TeX
C_2^4._5D_4
% in TeX
G:=Group("C2^4.5D4"); // GroupNames label
G:=SmallGroup(128,122);
// by ID
G=gap.SmallGroup(128,122);
# by ID
G:=PCGroup([7,-2,2,-2,2,2,-2,-2,56,85,120,758,570,521,4037]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^4=d,f^2=a,a*b=b*a,a*c=c*a,a*d=d*a,e*a*e^-1=a*c*d,a*f=f*a,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*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*d*e^3>;
// generators/relations
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