p-group, metabelian, nilpotent (class 2), monomial, rational
Aliases: C23.41C23, C42.42C22, C22.38C24, C2.82- 1+4, C2.112+ 1+4, (C2×C4)⋊2Q8, C4⋊Q8⋊12C2, C4.10(C2×Q8), C42.C2⋊7C2, C22⋊Q8.9C2, C22.5(C2×Q8), C2.8(C22×Q8), C4⋊C4.32C22, (C2×C4).25C23, (C2×Q8).31C22, C42⋊C2.13C2, C22⋊C4.19C22, (C22×C4).67C22, (C2×C4⋊C4).20C2, SmallGroup(64,225)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C23.41C23
G = < a,b,c,d,e,f | a2=b2=c2=1, d2=f2=c, e2=b, dad-1=ab=ba, ac=ca, ae=ea, af=fa, bc=cb, ede-1=bd=db, fef-1=be=eb, bf=fb, fdf-1=cd=dc, ce=ec, cf=fc >
Subgroups: 137 in 103 conjugacy classes, 81 normal (9 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, Q8, C23, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×Q8, C2×C4⋊C4, C42⋊C2, C22⋊Q8, C42.C2, C4⋊Q8, C23.41C23
Quotients: C1, C2, C22, Q8, C23, C2×Q8, C24, C22×Q8, 2+ 1+4, 2- 1+4, C23.41C23
Character table of C23.41C23
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | |
| ρ1 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | linear of order 2 |
| ρ9 | 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 |
| ρ10 | 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 |
| ρ11 | 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 |
| ρ12 | 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 |
| ρ13 | 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 |
| ρ14 | 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 |
| ρ15 | 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 |
| ρ16 | 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 |
| ρ17 | 2 | 2 | -2 | -2 | -2 | 2 | -2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ18 | 2 | 2 | -2 | -2 | 2 | -2 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ19 | 2 | 2 | -2 | -2 | -2 | 2 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ20 | 2 | 2 | -2 | -2 | 2 | -2 | 2 | -2 | 2 | -2 | 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 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from 2+ 1+4 |
| ρ22 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from 2- 1+4, Schur index 2 |
►On 32 points
(2 28)(4 26)(5 20)(7 18)(10 14)(12 16)(22 30)(24 32)
(1 27)(2 28)(3 25)(4 26)(5 20)(6 17)(7 18)(8 19)(9 13)(10 14)(11 15)(12 16)(21 29)(22 30)(23 31)(24 32)
(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(13 15)(14 16)(17 19)(18 20)(21 23)(22 24)(25 27)(26 28)(29 31)(30 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 31 27 23)(2 24 28 32)(3 29 25 21)(4 22 26 30)(5 14 20 10)(6 11 17 15)(7 16 18 12)(8 9 19 13)
(1 9 3 11)(2 12 4 10)(5 32 7 30)(6 31 8 29)(13 25 15 27)(14 28 16 26)(17 23 19 21)(18 22 20 24)
G:=sub<Sym(32)| (2,28)(4,26)(5,20)(7,18)(10,14)(12,16)(22,30)(24,32), (1,27)(2,28)(3,25)(4,26)(5,20)(6,17)(7,18)(8,19)(9,13)(10,14)(11,15)(12,16)(21,29)(22,30)(23,31)(24,32), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24)(25,27)(26,28)(29,31)(30,32), (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), (1,31,27,23)(2,24,28,32)(3,29,25,21)(4,22,26,30)(5,14,20,10)(6,11,17,15)(7,16,18,12)(8,9,19,13), (1,9,3,11)(2,12,4,10)(5,32,7,30)(6,31,8,29)(13,25,15,27)(14,28,16,26)(17,23,19,21)(18,22,20,24)>;
G:=Group( (2,28)(4,26)(5,20)(7,18)(10,14)(12,16)(22,30)(24,32), (1,27)(2,28)(3,25)(4,26)(5,20)(6,17)(7,18)(8,19)(9,13)(10,14)(11,15)(12,16)(21,29)(22,30)(23,31)(24,32), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24)(25,27)(26,28)(29,31)(30,32), (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), (1,31,27,23)(2,24,28,32)(3,29,25,21)(4,22,26,30)(5,14,20,10)(6,11,17,15)(7,16,18,12)(8,9,19,13), (1,9,3,11)(2,12,4,10)(5,32,7,30)(6,31,8,29)(13,25,15,27)(14,28,16,26)(17,23,19,21)(18,22,20,24) );
G=PermutationGroup([[(2,28),(4,26),(5,20),(7,18),(10,14),(12,16),(22,30),(24,32)], [(1,27),(2,28),(3,25),(4,26),(5,20),(6,17),(7,18),(8,19),(9,13),(10,14),(11,15),(12,16),(21,29),(22,30),(23,31),(24,32)], [(1,3),(2,4),(5,7),(6,8),(9,11),(10,12),(13,15),(14,16),(17,19),(18,20),(21,23),(22,24),(25,27),(26,28),(29,31),(30,32)], [(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)], [(1,31,27,23),(2,24,28,32),(3,29,25,21),(4,22,26,30),(5,14,20,10),(6,11,17,15),(7,16,18,12),(8,9,19,13)], [(1,9,3,11),(2,12,4,10),(5,32,7,30),(6,31,8,29),(13,25,15,27),(14,28,16,26),(17,23,19,21),(18,22,20,24)]])
C23.41C23 is a maximal subgroup of
(C2×C4)⋊SD16 (C2×C4)⋊Q16 (C2×C4).SD16 C4⋊C4.20D4 (C2×D4)⋊2Q8 (C2×Q8)⋊2Q8 M4(2)⋊Q8 C42⋊3Q8 C42.20C23 C42.21C23 C42.22C23 C42.23C23 C42.366C23 C42.367C23 M4(2)⋊3Q8 M4(2)⋊4Q8 C22.47C25 C22.49C25 C22.81C25 C22.82C25 C22.83C25 C22.84C25 C22.90C25 C22.93C25 C22.124C25 C22.127C25 C22.153C25
C2p.2- 1+4: C42.423C23 C42.424C23 C42.425C23 C42.426C23 C22.50C25 C22.91C25 C22.92C25 C22.133C25 ...
C23.41C23 is a maximal quotient of
C24.545C23 C23.199C24 C23.211C24 C42.33Q8 C42⋊4Q8 C24.567C23 C42.36Q8 C42.37Q8 C42⋊8Q8 C42.38Q8 C42⋊9Q8 C23.527C24 C23.546C24 C42.39Q8 C23.559C24 C24.379C23 C42⋊11Q8 C23.567C24 C24.421C23 C23.634C24 C24.428C23 C23.655C24 C23.663C24 C23.668C24 C23.674C24 C23.688C24 C24.454C23 C23.691C24 C23.692C24 C23.702C24 C23.705C24 C23.706C24 C23.707C24 C23.741C24 C42⋊12Q8 C42⋊13Q8 C42.40Q8
C42.D2p: C42.187D4 C42.188D4 C42⋊10Q8 C42.90D6 C42.148D6 C42.174D6 C42.90D10 C42.148D10 ...
C4⋊C4.D2p: C24.267C23 C24.568C23 C24.268C23 C24.569C23 C42⋊6Q8 C42⋊7Q8 C42.35Q8 C24.385C23 ...
Matrix representation of C23.41C23 ►in GL6(𝔽5)
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 3 | 0 | 0 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 2 | 0 | 0 |
| 0 | 0 | 2 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 3 |
| 0 | 0 | 0 | 0 | 3 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(6,GF(5))| [4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[3,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,1,0,0,0,0,0,0,1,0,0],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,2,0,0,0,0,2,0,0,0,0,0,0,0,0,3,0,0,0,0,3,0],[0,4,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,4,0] >;
C23.41C23 in GAP, Magma, Sage, TeX
C_2^3._{41}C_2^3 % in TeX
G:=Group("C2^3.41C2^3"); // GroupNames label
G:=SmallGroup(64,225);
// by ID
G=gap.SmallGroup(64,225);
# by ID
G:=PCGroup([6,-2,2,2,2,-2,2,192,217,103,650,188,158,579]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=1,d^2=f^2=c,e^2=b,d*a*d^-1=a*b=b*a,a*c=c*a,a*e=e*a,a*f=f*a,b*c=c*b,e*d*e^-1=b*d=d*b,f*e*f^-1=b*e=e*b,b*f=f*b,f*d*f^-1=c*d=d*c,c*e=e*c,c*f=f*c>;
// generators/relations
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