p-group, metabelian, nilpotent (class 4), monomial
Aliases: (C22×Q8)⋊8C4, C23.6(C2×D4), C42⋊C2⋊8C4, (C2×D4).128D4, (C2×Q8).116D4, (C22×C4).93D4, C4.17(C23⋊C4), (C2×D4).17C23, C23.D4⋊6C2, C22.D4⋊3C4, C23⋊C4.10C22, C23.56(C22×C4), C23.21(C22⋊C4), C4.D4.11C22, C23.C23.9C2, C22.D4.2C22, C23.38C23.6C2, M4(2).8C22.10C2, (C2×C4).6(C2×D4), C22⋊C4.3(C2×C4), C2.35(C2×C23⋊C4), (C2×D4).126(C2×C4), (C22×C4).30(C2×C4), (C2×C4○D4).73C22, C22.59(C2×C22⋊C4), (C2×C4).145(C22⋊C4), SmallGroup(128,855)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C23.(C2×D4)
G = < a,b,c,d,e,f | a2=b2=c2=1, d2=e4=c, f2=ca=dad-1=ac, ab=ba, eae-1=abc, af=fa, ebe-1=fbf-1=bc=cb, ede-1=bd=db, cd=dc, ce=ec, cf=fc, fdf-1=bcd, fef-1=ace3 >
Subgroups: 268 in 117 conjugacy classes, 42 normal (26 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, D4, Q8, C23, C42, C22⋊C4, C22⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, C22×C4, C2×D4, C2×Q8, C2×Q8, C4○D4, C23⋊C4, C23⋊C4, C4.D4, C4.10D4, C42⋊C2, C42⋊C2, C22⋊Q8, C22.D4, C4.4D4, C4⋊Q8, C2×M4(2), C22×Q8, C2×C4○D4, C23.D4, C23.C23, M4(2).8C22, C23.38C23, C23.(C2×D4)
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C23⋊C4, C2×C22⋊C4, C2×C23⋊C4, C23.(C2×D4)
Character table of C23.(C2×D4)
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 8A | 8B | 8C | 8D | |
| size | 1 | 1 | 2 | 4 | 4 | 4 | 2 | 2 | 4 | 4 | 4 | 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 | -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 | 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 | 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 | linear of order 2 |
| ρ9 | 1 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | -i | -1 | 1 | -1 | -i | i | 1 | i | -i | i | i | -i | linear of order 4 |
| ρ10 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | i | 1 | -1 | -1 | -i | -i | 1 | i | -i | i | -i | i | linear of order 4 |
| ρ11 | 1 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | i | 1 | -1 | 1 | i | -i | -1 | -i | -i | i | i | -i | linear of order 4 |
| ρ12 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | -i | -1 | 1 | 1 | i | i | -1 | -i | -i | i | -i | i | linear of order 4 |
| ρ13 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | -i | 1 | -1 | -1 | i | i | 1 | -i | i | -i | i | -i | linear of order 4 |
| ρ14 | 1 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | i | -1 | 1 | -1 | i | -i | 1 | -i | i | -i | -i | i | linear of order 4 |
| ρ15 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | i | -1 | 1 | 1 | -i | -i | -1 | i | i | -i | i | -i | linear of order 4 |
| ρ16 | 1 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | -i | 1 | -1 | 1 | -i | i | -1 | i | i | -i | -i | i | linear of order 4 |
| ρ17 | 2 | 2 | 2 | 2 | -2 | -2 | 2 | 2 | -2 | -2 | 2 | 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 | 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 | 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 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ21 | 4 | 4 | -4 | 0 | 0 | 0 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C23⋊C4 |
| ρ22 | 4 | 4 | -4 | 0 | 0 | 0 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C23⋊C4 |
| ρ23 | 8 | -8 | 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
Generators in S32
(2 32)(3 7)(4 30)(6 28)(8 26)(9 13)(10 22)(12 20)(14 18)(16 24)(17 21)(25 29)
(1 27)(2 32)(3 29)(4 26)(5 31)(6 28)(7 25)(8 30)(9 21)(10 18)(11 23)(12 20)(13 17)(14 22)(15 19)(16 24)
(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 21 5 17)(2 10 6 14)(3 19 7 23)(4 16 8 12)(9 31 13 27)(11 29 15 25)(18 28 22 32)(20 26 24 30)
(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 23 5 19)(2 10 28 18)(3 13)(4 20 26 16)(6 14 32 22)(7 9)(8 24 30 12)(11 27 15 31)(17 25)(21 29)
G:=sub<Sym(32)| (2,32)(3,7)(4,30)(6,28)(8,26)(9,13)(10,22)(12,20)(14,18)(16,24)(17,21)(25,29), (1,27)(2,32)(3,29)(4,26)(5,31)(6,28)(7,25)(8,30)(9,21)(10,18)(11,23)(12,20)(13,17)(14,22)(15,19)(16,24), (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), (1,21,5,17)(2,10,6,14)(3,19,7,23)(4,16,8,12)(9,31,13,27)(11,29,15,25)(18,28,22,32)(20,26,24,30), (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), (1,23,5,19)(2,10,28,18)(3,13)(4,20,26,16)(6,14,32,22)(7,9)(8,24,30,12)(11,27,15,31)(17,25)(21,29)>;
G:=Group( (2,32)(3,7)(4,30)(6,28)(8,26)(9,13)(10,22)(12,20)(14,18)(16,24)(17,21)(25,29), (1,27)(2,32)(3,29)(4,26)(5,31)(6,28)(7,25)(8,30)(9,21)(10,18)(11,23)(12,20)(13,17)(14,22)(15,19)(16,24), (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), (1,21,5,17)(2,10,6,14)(3,19,7,23)(4,16,8,12)(9,31,13,27)(11,29,15,25)(18,28,22,32)(20,26,24,30), (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), (1,23,5,19)(2,10,28,18)(3,13)(4,20,26,16)(6,14,32,22)(7,9)(8,24,30,12)(11,27,15,31)(17,25)(21,29) );
G=PermutationGroup([[(2,32),(3,7),(4,30),(6,28),(8,26),(9,13),(10,22),(12,20),(14,18),(16,24),(17,21),(25,29)], [(1,27),(2,32),(3,29),(4,26),(5,31),(6,28),(7,25),(8,30),(9,21),(10,18),(11,23),(12,20),(13,17),(14,22),(15,19),(16,24)], [(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)], [(1,21,5,17),(2,10,6,14),(3,19,7,23),(4,16,8,12),(9,31,13,27),(11,29,15,25),(18,28,22,32),(20,26,24,30)], [(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)], [(1,23,5,19),(2,10,28,18),(3,13),(4,20,26,16),(6,14,32,22),(7,9),(8,24,30,12),(11,27,15,31),(17,25),(21,29)]])
Matrix representation of C23.(C2×D4) ►in GL8(𝔽17)
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 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 | 16 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 16 |
| 16 | 15 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 15 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 16 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 16 | 0 | 0 | 1 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 16 |
| 0 | 0 | 16 | 0 | 0 | 0 | 15 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 |
| 16 | 0 | 0 | 0 | 15 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 16 | 16 | 0 | 0 | 16 | 0 |
| 1 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 16 | 16 | 0 | 0 | 16 | 0 | 0 | 0 |
| 13 | 0 | 0 | 0 | 13 | 4 | 4 | 4 |
| 4 | 0 | 0 | 0 | 4 | 13 | 0 | 0 |
| 0 | 0 | 13 | 0 | 13 | 13 | 13 | 4 |
| 0 | 0 | 4 | 0 | 0 | 0 | 4 | 13 |
| 0 | 15 | 0 | 15 | 2 | 15 | 15 | 15 |
| 13 | 15 | 0 | 2 | 15 | 2 | 2 | 2 |
| 0 | 2 | 0 | 15 | 2 | 2 | 2 | 15 |
| 0 | 15 | 13 | 15 | 15 | 15 | 15 | 2 |
| 16 | 0 | 0 | 0 | 15 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 1 | 16 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 | 0 | 2 |
| 0 | 0 | 1 | 0 | 0 | 0 | 1 | 16 |
| 1 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 | 0 | 16 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(8,GF(17))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16],[16,0,0,0,0,0,0,0,15,1,0,0,1,16,0,0,0,0,16,0,0,0,0,0,0,0,15,1,0,0,1,16,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16],[0,0,16,0,0,0,1,16,0,0,0,0,0,0,0,16,16,0,0,0,1,16,0,0,0,0,0,0,0,16,0,0,0,0,15,1,0,0,1,16,0,0,0,1,0,0,0,0,15,1,0,0,1,16,0,0,0,1,0,0,0,0,0,0],[13,4,0,0,0,13,0,0,0,0,0,0,15,15,2,15,0,0,13,4,0,0,0,13,0,0,0,0,15,2,15,15,13,4,13,0,2,15,2,15,4,13,13,0,15,2,2,15,4,0,13,4,15,2,2,15,4,0,4,13,15,2,15,2],[16,1,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,16,1,0,0,1,0,0,0,0,0,0,0,1,0,15,1,0,0,1,16,0,0,0,16,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,2,16,0,0,16,1] >;
C23.(C2×D4) in GAP, Magma, Sage, TeX
C_2^3.(C_2\times D_4)
% in TeX
G:=Group("C2^3.(C2xD4)"); // GroupNames label
G:=SmallGroup(128,855);
// by ID
G=gap.SmallGroup(128,855);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,-2,112,141,456,723,352,1123,851,375,4037]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=1,d^2=e^4=c,f^2=c*a=d*a*d^-1=a*c,a*b=b*a,e*a*e^-1=a*b*c,a*f=f*a,e*b*e^-1=f*b*f^-1=b*c=c*b,e*d*e^-1=b*d=d*b,c*d=d*c,c*e=e*c,c*f=f*c,f*d*f^-1=b*c*d,f*e*f^-1=a*c*e^3>;
// generators/relations
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