p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.60D4, C4⋊Q8⋊11C4, (C4×Q8)⋊6C4, C42.77(C2×C4), C42⋊5C4.2C2, C23.505(C2×D4), (C22×C4).217D4, C22⋊C8.136C22, C42.6C4.18C2, (C2×C42).180C22, (C22×C4).637C23, C23.31D4.6C2, C22⋊Q8.144C22, C22.26(C8.C22), C2.C42.5C22, C2.20(C42⋊C22), C2.10(C23.38D4), C23.37C23.7C2, C2.20(C23.C23), C4⋊C4.15(C2×C4), (C2×Q8).12(C2×C4), (C2×C4).1161(C2×D4), (C2×C4).94(C22⋊C4), (C2×C4).127(C22×C4), C22.191(C2×C22⋊C4), SmallGroup(128,247)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.60D4
G = < a,b,c,d | a4=b4=c4=1, d2=a2b, ab=ba, cac-1=dad-1=ab2, cbc-1=a2b-1, bd=db, dcd-1=b-1c-1 >
Subgroups: 212 in 104 conjugacy classes, 44 normal (26 characteristic)
C1, C2, C2, C4, C22, C22, C22, C8, C2×C4, C2×C4, Q8, C23, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C2×Q8, C2×Q8, C2.C42, C2.C42, C8⋊C4, C22⋊C8, C4⋊C8, C2×C42, C42⋊C2, C4×Q8, C4×Q8, C22⋊Q8, C22⋊Q8, C42.C2, C4⋊Q8, C23.31D4, C42⋊5C4, C42.6C4, C23.37C23, C42.60D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C2×C22⋊C4, C8.C22, C23.C23, C23.38D4, C42⋊C22, C42.60D4
Character table of C42.60D4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | 8A | 8B | 8C | 8D | |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 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 | 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 | 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 | -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 | -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 | 1 | -1 | 1 | linear of order 2 |
| ρ9 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | i | i | -i | -i | -1 | 1 | i | i | -i | -i | linear of order 4 |
| ρ10 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | -1 | 1 | -1 | 1 | -i | i | i | -i | 1 | -1 | i | -i | -i | i | linear of order 4 |
| ρ11 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | i | i | -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 | i | -i | -1 | 1 | -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 | -i | i | -1 | 1 | i | -i | -i | i | linear of order 4 |
| ρ14 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | -i | -i | 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 | -i | i | 1 | -1 | -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 | i | i | -1 | 1 | -i | -i | i | i | linear of order 4 |
| ρ17 | 2 | 2 | 2 | 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 | 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 | 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 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| 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 | symplectic lifted from C8.C22, Schur index 2 |
| ρ22 | 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 C8.C22, Schur index 2 |
| ρ23 | 4 | -4 | 4 | -4 | 0 | 0 | 0 | -4i | 0 | 4i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C23.C23 |
| ρ24 | 4 | -4 | 4 | -4 | 0 | 0 | 0 | 4i | 0 | -4i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C23.C23 |
| ρ25 | 4 | 4 | -4 | -4 | 0 | 0 | 4i | 0 | -4i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C42⋊C22 |
| ρ26 | 4 | 4 | -4 | -4 | 0 | 0 | -4i | 0 | 4i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C42⋊C22 |
►On 32 points
(1 31 15 24)(2 28 16 21)(3 25 9 18)(4 30 10 23)(5 27 11 20)(6 32 12 17)(7 29 13 22)(8 26 14 19)
(1 9 5 13)(2 10 6 14)(3 11 7 15)(4 12 8 16)(17 26 21 30)(18 27 22 31)(19 28 23 32)(20 29 24 25)
(2 4 16 10)(3 13)(6 8 12 14)(7 9)(17 23 32 30)(18 25)(19 28 26 21)(20 24)(22 29)(27 31)
(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)
G:=sub<Sym(32)| (1,31,15,24)(2,28,16,21)(3,25,9,18)(4,30,10,23)(5,27,11,20)(6,32,12,17)(7,29,13,22)(8,26,14,19), (1,9,5,13)(2,10,6,14)(3,11,7,15)(4,12,8,16)(17,26,21,30)(18,27,22,31)(19,28,23,32)(20,29,24,25), (2,4,16,10)(3,13)(6,8,12,14)(7,9)(17,23,32,30)(18,25)(19,28,26,21)(20,24)(22,29)(27,31), (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)>;
G:=Group( (1,31,15,24)(2,28,16,21)(3,25,9,18)(4,30,10,23)(5,27,11,20)(6,32,12,17)(7,29,13,22)(8,26,14,19), (1,9,5,13)(2,10,6,14)(3,11,7,15)(4,12,8,16)(17,26,21,30)(18,27,22,31)(19,28,23,32)(20,29,24,25), (2,4,16,10)(3,13)(6,8,12,14)(7,9)(17,23,32,30)(18,25)(19,28,26,21)(20,24)(22,29)(27,31), (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) );
G=PermutationGroup([[(1,31,15,24),(2,28,16,21),(3,25,9,18),(4,30,10,23),(5,27,11,20),(6,32,12,17),(7,29,13,22),(8,26,14,19)], [(1,9,5,13),(2,10,6,14),(3,11,7,15),(4,12,8,16),(17,26,21,30),(18,27,22,31),(19,28,23,32),(20,29,24,25)], [(2,4,16,10),(3,13),(6,8,12,14),(7,9),(17,23,32,30),(18,25),(19,28,26,21),(20,24),(22,29),(27,31)], [(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)]])
Matrix representation of C42.60D4 ►in GL8(𝔽17)
| 0 | 0 | 1 | 0 | 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 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 15 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 9 | 16 | 15 |
| 0 | 0 | 0 | 0 | 2 | 6 | 1 | 1 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 4 |
| 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 | 16 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 6 | 2 | 13 | 0 |
| 0 | 0 | 0 | 0 | 8 | 14 | 4 | 4 |
| 3 | 14 | 4 | 4 | 0 | 0 | 0 | 0 |
| 3 | 3 | 13 | 4 | 0 | 0 | 0 | 0 |
| 13 | 13 | 14 | 3 | 0 | 0 | 0 | 0 |
| 4 | 13 | 14 | 14 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 11 | 9 | 0 |
| 0 | 0 | 0 | 0 | 16 | 10 | 8 | 8 |
| 0 | 0 | 0 | 0 | 15 | 10 | 6 | 11 |
| 0 | 0 | 0 | 0 | 6 | 6 | 0 | 13 |
G:=sub<GL(8,GF(17))| [0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,16,1,11,2,0,0,0,0,15,1,9,6,0,0,0,0,0,0,16,1,0,0,0,0,0,0,15,1],[0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,16,6,8,0,0,0,0,0,16,2,14,0,0,0,0,0,0,13,4,0,0,0,0,0,0,0,4],[3,3,13,4,0,0,0,0,14,3,13,13,0,0,0,0,4,13,14,14,0,0,0,0,4,4,3,14,0,0,0,0,0,0,0,0,5,16,15,6,0,0,0,0,11,10,10,6,0,0,0,0,9,8,6,0,0,0,0,0,0,8,11,13] >;
C42.60D4 in GAP, Magma, Sage, TeX
C_4^2._{60}D_4 % in TeX
G:=Group("C4^2.60D4"); // GroupNames label
G:=SmallGroup(128,247);
// by ID
G=gap.SmallGroup(128,247);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,232,1430,520,1123,1018,248,1971]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^4=1,d^2=a^2*b,a*b=b*a,c*a*c^-1=d*a*d^-1=a*b^2,c*b*c^-1=a^2*b^-1,b*d=d*b,d*c*d^-1=b^-1*c^-1>;
// generators/relations
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