p-group, metabelian, nilpotent (class 4), monomial
Aliases: C23.2M4(2), C22⋊C4⋊C8, C2.2C2≀C4, C23⋊C8.2C2, (C23×C4).4C4, C23.1(C2×C8), (C22×C4).7D4, C24.17(C2×C4), C23.8Q8.1C2, C22.41(C23⋊C4), C22.19(C22⋊C8), C2.2(C23.D4), C23.153(C22⋊C4), C22.8(C4.10D4), C2.7(C22.M4(2)), (C2×C22⋊C4).6C4, (C2×C22⋊C4).3C22, SmallGroup(128,58)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C23.2M4(2)
G = < a,b,c,d,e | a2=b2=c2=d8=1, e2=b, dad-1=ab=ba, eae-1=ac=ca, dbd-1=bc=cb, be=eb, cd=dc, ce=ec, ede-1=abcd5 >
Subgroups: 216 in 79 conjugacy classes, 22 normal (20 characteristic)
C1, C2, C2, C4, C22, C22, C8, C2×C4, C23, C23, C22⋊C4, C22⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C24, C2.C42, C22⋊C8, C2×C22⋊C4, C2×C4⋊C4, C23×C4, C23⋊C8, C23.8Q8, C23.2M4(2)
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, C22⋊C4, C2×C8, M4(2), C22⋊C8, C23⋊C4, C4.10D4, C22.M4(2), C2≀C4, C23.D4, C23.2M4(2)
Character table of C23.2M4(2)
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | |
| 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 | 1 | 1 | i | -i | i | i | 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 | 1 | 1 | -i | i | -i | -i | -i | i | i | i | linear of order 4 |
| ρ7 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | -i | -i | -i | i | i | i | i | -i | linear of order 4 |
| ρ8 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | i | i | i | -i | -i | -i | -i | i | linear of order 4 |
| ρ9 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | -i | -i | i | -i | -i | i | i | i | 1 | -1 | ζ85 | ζ83 | ζ8 | ζ85 | ζ8 | ζ87 | ζ83 | ζ87 | linear of order 8 |
| ρ10 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | -i | -i | i | -i | -i | i | i | i | 1 | -1 | ζ8 | ζ87 | ζ85 | ζ8 | ζ85 | ζ83 | ζ87 | ζ83 | linear of order 8 |
| ρ11 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | -i | i | -i | -i | i | i | i | -i | -1 | 1 | ζ87 | ζ85 | ζ83 | ζ83 | ζ87 | ζ85 | ζ8 | ζ8 | linear of order 8 |
| ρ12 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | -i | i | -i | -i | i | i | i | -i | -1 | 1 | ζ83 | ζ8 | ζ87 | ζ87 | ζ83 | ζ8 | ζ85 | ζ85 | linear of order 8 |
| ρ13 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | i | i | -i | i | i | -i | -i | -i | 1 | -1 | ζ83 | ζ85 | ζ87 | ζ83 | ζ87 | ζ8 | ζ85 | ζ8 | linear of order 8 |
| ρ14 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | i | -i | i | i | -i | -i | -i | i | -1 | 1 | ζ8 | ζ83 | ζ85 | ζ85 | ζ8 | ζ83 | ζ87 | ζ87 | linear of order 8 |
| ρ15 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | i | i | -i | i | i | -i | -i | -i | 1 | -1 | ζ87 | ζ8 | ζ83 | ζ87 | ζ83 | ζ85 | ζ8 | ζ85 | linear of order 8 |
| ρ16 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | i | -i | i | i | -i | -i | -i | i | -1 | 1 | ζ85 | ζ87 | ζ8 | ζ8 | ζ85 | ζ87 | ζ83 | ζ83 | linear of order 8 |
| ρ17 | 2 | 2 | 2 | 2 | 2 | 2 | -2 | -2 | 0 | 2 | -2 | 0 | -2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ18 | 2 | 2 | 2 | 2 | 2 | 2 | -2 | -2 | 0 | -2 | 2 | 0 | 2 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ19 | 2 | -2 | 2 | -2 | -2 | 2 | -2 | 2 | 0 | -2i | -2i | 0 | 2i | 0 | 0 | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from M4(2) |
| ρ20 | 2 | -2 | 2 | -2 | -2 | 2 | -2 | 2 | 0 | 2i | 2i | 0 | -2i | 0 | 0 | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from M4(2) |
| ρ21 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 2 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2≀C4 |
| ρ22 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | -2 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2≀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 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from C4.10D4, Schur index 2 |
| ρ25 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 2i | 0 | 0 | -2i | 0 | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C23.D4 |
| ρ26 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | -2i | 0 | 0 | 2i | 0 | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C23.D4 |
►On 32 points
Generators in S32
(1 30)(2 31)(3 19)(4 20)(5 26)(6 27)(7 23)(8 24)(9 28)(10 29)(11 17)(12 18)(13 32)(14 25)(15 21)(16 22)
(2 12)(4 14)(6 16)(8 10)(18 31)(20 25)(22 27)(24 29)
(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 9)(8 10)(17 30)(18 31)(19 32)(20 25)(21 26)(22 27)(23 28)(24 29)
(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 15)(2 18 12 31)(3 7)(4 20 14 25)(5 11)(6 22 16 27)(8 24 10 29)(9 13)(17 21)(19 28)(23 32)(26 30)
G:=sub<Sym(32)| (1,30)(2,31)(3,19)(4,20)(5,26)(6,27)(7,23)(8,24)(9,28)(10,29)(11,17)(12,18)(13,32)(14,25)(15,21)(16,22), (2,12)(4,14)(6,16)(8,10)(18,31)(20,25)(22,27)(24,29), (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,9)(8,10)(17,30)(18,31)(19,32)(20,25)(21,26)(22,27)(23,28)(24,29), (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), (1,15)(2,18,12,31)(3,7)(4,20,14,25)(5,11)(6,22,16,27)(8,24,10,29)(9,13)(17,21)(19,28)(23,32)(26,30)>;
G:=Group( (1,30)(2,31)(3,19)(4,20)(5,26)(6,27)(7,23)(8,24)(9,28)(10,29)(11,17)(12,18)(13,32)(14,25)(15,21)(16,22), (2,12)(4,14)(6,16)(8,10)(18,31)(20,25)(22,27)(24,29), (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,9)(8,10)(17,30)(18,31)(19,32)(20,25)(21,26)(22,27)(23,28)(24,29), (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), (1,15)(2,18,12,31)(3,7)(4,20,14,25)(5,11)(6,22,16,27)(8,24,10,29)(9,13)(17,21)(19,28)(23,32)(26,30) );
G=PermutationGroup([[(1,30),(2,31),(3,19),(4,20),(5,26),(6,27),(7,23),(8,24),(9,28),(10,29),(11,17),(12,18),(13,32),(14,25),(15,21),(16,22)], [(2,12),(4,14),(6,16),(8,10),(18,31),(20,25),(22,27),(24,29)], [(1,11),(2,12),(3,13),(4,14),(5,15),(6,16),(7,9),(8,10),(17,30),(18,31),(19,32),(20,25),(21,26),(22,27),(23,28),(24,29)], [(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)], [(1,15),(2,18,12,31),(3,7),(4,20,14,25),(5,11),(6,22,16,27),(8,24,10,29),(9,13),(17,21),(19,28),(23,32),(26,30)]])
Matrix representation of C23.2M4(2) ►in GL6(𝔽17)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 15 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 15 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 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 | 16 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 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 |
| 2 | 16 | 0 | 0 | 0 | 0 |
| 8 | 15 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 16 | 16 |
| 0 | 0 | 1 | 2 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 13 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 16 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 2 |
| 0 | 0 | 0 | 0 | 16 | 16 |
G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,15,1,0,0,0,0,0,0,16,0,0,0,0,0,15,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[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],[2,8,0,0,0,0,16,15,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,16,0,0,1,16,0,0,0,0,0,16,0,0],[16,13,0,0,0,0,0,1,0,0,0,0,0,0,1,16,0,0,0,0,0,16,0,0,0,0,0,0,1,16,0,0,0,0,2,16] >;
C23.2M4(2) in GAP, Magma, Sage, TeX
C_2^3._2M_4(2)
% in TeX
G:=Group("C2^3.2M4(2)"); // GroupNames label
G:=SmallGroup(128,58);
// by ID
G=gap.SmallGroup(128,58);
# by ID
G:=PCGroup([7,-2,2,-2,2,-2,2,-2,56,85,120,422,723,346,521,136,2804]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^8=1,e^2=b,d*a*d^-1=a*b=b*a,e*a*e^-1=a*c=c*a,d*b*d^-1=b*c=c*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=a*b*c*d^5>;
// generators/relations
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