p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.58D4, (C4×D4)⋊5C4, (C4×Q8)⋊5C4, C4.4D4⋊9C4, C42.C2⋊5C4, C42⋊8C4⋊1C2, C42.75(C2×C4), C23.502(C2×D4), (C22×C4).214D4, C42.6C4⋊31C2, C22.SD16.6C2, C23.31D4⋊21C2, C4⋊D4.137C22, C22⋊C8.133C22, C22.16(C8⋊C22), (C22×C4).634C23, (C2×C42).178C22, C22⋊Q8.142C22, C22.12(C8.C22), C2.C42.2C22, C2.17(C42⋊C22), C2.10(C23.36D4), C23.36C23.7C2, C2.18(C23.C23), C4⋊C4.12(C2×C4), (C2×D4).10(C2×C4), (C2×Q8).10(C2×C4), (C2×C4).1158(C2×D4), (C2×C4).92(C22⋊C4), (C2×C4).124(C22×C4), C22.188(C2×C22⋊C4), SmallGroup(128,244)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.58D4
G = < a,b,c,d | a4=b4=c4=1, d2=b-1, ab=ba, cac-1=dad-1=ab2, cbc-1=a2b, bd=db, dcd-1=b-1c-1 >
Subgroups: 244 in 111 conjugacy classes, 44 normal (36 characteristic)
C1, C2, C2, C4, C22, C22, C8, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2.C42, C2.C42, C8⋊C4, C22⋊C8, C4⋊C8, C2×C42, C2×C4⋊C4, C42⋊C2, C4×D4, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C42.C2, C42⋊2C2, C22.SD16, C23.31D4, C42⋊8C4, C42.6C4, C23.36C23, C42.58D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C2×C22⋊C4, C8⋊C22, C8.C22, C23.C23, C23.36D4, C42⋊C22, C42.58D4
Character table of C42.58D4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 8A | 8B | 8C | 8D | |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 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 | i | -i | i | -i | -1 | -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 | -i | i | i | -i | -1 | 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 | i | -i | -i | i | 1 | -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 | -i | i | -i | i | 1 | 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 | -i | i | i | -i | 1 | -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 | i | -i | i | -i | 1 | 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 | -i | i | -i | i | -1 | -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 | i | -i | -i | i | -1 | 1 | 1 | -i | i | i | -i | linear of order 4 |
| ρ17 | 2 | 2 | 2 | 2 | -2 | -2 | 0 | 2 | 2 | 2 | 2 | -2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ18 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | -2 | 2 | -2 | 2 | 2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ19 | 2 | 2 | 2 | 2 | -2 | -2 | 0 | 2 | -2 | 2 | -2 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ20 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | -2 | -2 | -2 | -2 | -2 | 2 | 2 | -2 | 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 | orthogonal lifted from C8⋊C22 |
| 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 | 0 | 4i | 0 | -4i | 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 C42⋊C22 |
| ρ25 | 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 C42⋊C22 |
| ρ26 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | -4i | 0 | 4i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C23.C23 |
►On 32 points
(1 29 11 22)(2 26 12 19)(3 31 13 24)(4 28 14 21)(5 25 15 18)(6 30 16 23)(7 27 9 20)(8 32 10 17)
(1 7 5 3)(2 8 6 4)(9 15 13 11)(10 16 14 12)(17 23 21 19)(18 24 22 20)(25 31 29 27)(26 32 30 28)
(2 14 16 8)(3 13)(4 6 10 12)(7 9)(17 23 28 26)(18 22)(19 32 30 21)(20 31)(24 27)(25 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)
G:=sub<Sym(32)| (1,29,11,22)(2,26,12,19)(3,31,13,24)(4,28,14,21)(5,25,15,18)(6,30,16,23)(7,27,9,20)(8,32,10,17), (1,7,5,3)(2,8,6,4)(9,15,13,11)(10,16,14,12)(17,23,21,19)(18,24,22,20)(25,31,29,27)(26,32,30,28), (2,14,16,8)(3,13)(4,6,10,12)(7,9)(17,23,28,26)(18,22)(19,32,30,21)(20,31)(24,27)(25,29), (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)>;
G:=Group( (1,29,11,22)(2,26,12,19)(3,31,13,24)(4,28,14,21)(5,25,15,18)(6,30,16,23)(7,27,9,20)(8,32,10,17), (1,7,5,3)(2,8,6,4)(9,15,13,11)(10,16,14,12)(17,23,21,19)(18,24,22,20)(25,31,29,27)(26,32,30,28), (2,14,16,8)(3,13)(4,6,10,12)(7,9)(17,23,28,26)(18,22)(19,32,30,21)(20,31)(24,27)(25,29), (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) );
G=PermutationGroup([[(1,29,11,22),(2,26,12,19),(3,31,13,24),(4,28,14,21),(5,25,15,18),(6,30,16,23),(7,27,9,20),(8,32,10,17)], [(1,7,5,3),(2,8,6,4),(9,15,13,11),(10,16,14,12),(17,23,21,19),(18,24,22,20),(25,31,29,27),(26,32,30,28)], [(2,14,16,8),(3,13),(4,6,10,12),(7,9),(17,23,28,26),(18,22),(19,32,30,21),(20,31),(24,27),(25,29)], [(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)]])
Matrix representation of C42.58D4 ►in GL8(𝔽17)
| 1 | 15 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 15 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 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 |
| 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 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 15 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 2 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 16 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 13 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 13 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 15 | 16 |
| 0 | 0 | 13 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 13 | 4 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 16 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 15 | 16 |
| 0 | 0 | 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 | 0 | 0 |
G:=sub<GL(8,GF(17))| [1,0,0,0,0,0,0,0,15,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,15,16,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,15,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,2,0,0,0,0,0,0,0,1],[1,1,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,13,13,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,16,0,0,0,0,0,0,0,0,1,15,0,0,0,0,0,0,1,16],[0,0,1,1,0,0,0,0,0,0,0,16,0,0,0,0,13,13,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,16,0,0,0,0,1,15,0,0,0,0,0,0,1,16,0,0] >;
C42.58D4 in GAP, Magma, Sage, TeX
C_4^2._{58}D_4 % in TeX
G:=Group("C4^2.58D4"); // GroupNames label
G:=SmallGroup(128,244);
// by ID
G=gap.SmallGroup(128,244);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,1430,184,1123,1018,248,1971]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^4=1,d^2=b^-1,a*b=b*a,c*a*c^-1=d*a*d^-1=a*b^2,c*b*c^-1=a^2*b,b*d=d*b,d*c*d^-1=b^-1*c^-1>;
// generators/relations
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