p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.7D4, C4⋊Q8⋊4C4, C4.8(C4×D4), C4.4D4⋊4C4, C42.9(C2×C4), (C2×Q8).74D4, (C22×C4).69D4, C4.9C42⋊12C2, C23.135(C2×D4), C4.140(C4⋊D4), C22.35C22≀C2, (C22×C4).36C23, C42⋊C22.6C2, (C22×Q8).25C22, C42⋊C2.34C22, C23.32C23⋊3C2, C2.51(C23.23D4), (C2×M4(2)).195C22, C23.38C23.2C2, C22.10(C22.D4), (C2×D4).90(C2×C4), (C2×C4).247(C2×D4), (C2×Q8).78(C2×C4), (C2×C4).331(C4○D4), (C2×C4.10D4)⋊20C2, (C2×C4).19(C22⋊C4), (C2×C4).196(C22×C4), (C2×C4○D4).30C22, C22.50(C2×C22⋊C4), SmallGroup(128,644)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.7D4
G = < a,b,c,d | a4=b4=1, c4=b2, d2=b, ab=ba, cac-1=dad-1=a-1b-1, cbc-1=b-1, bd=db, dcd-1=bc3 >
Subgroups: 300 in 157 conjugacy classes, 54 normal (18 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C2×Q8, C4○D4, C4.10D4, C4≀C2, C42⋊C2, C42⋊C2, C42⋊C2, C4×Q8, C22⋊Q8, C22.D4, C4.4D4, C4⋊Q8, C2×M4(2), C22×Q8, C2×C4○D4, C4.9C42, C2×C4.10D4, C42⋊C22, C23.32C23, C23.38C23, C42.7D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C4○D4, C2×C22⋊C4, C4×D4, C22≀C2, C4⋊D4, C22.D4, C23.23D4, C42.7D4
Character table of C42.7D4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | 4Q | 4R | 4S | 8A | 8B | 8C | 8D | |
| size | 1 | 1 | 2 | 2 | 2 | 8 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 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 | 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 | -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 | 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 | -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 | -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 | 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 | -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 | 1 | -1 | 1 | linear of order 2 |
| ρ9 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | i | i | -1 | 1 | -1 | -i | i | -i | -i | i | -i | 1 | -1 | 1 | 1 | i | i | -i | -i | linear of order 4 |
| ρ10 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | -i | -i | -1 | 1 | -1 | i | -i | i | i | -i | i | 1 | 1 | -1 | -1 | i | i | -i | -i | linear of order 4 |
| ρ11 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | -i | i | 1 | -1 | 1 | i | -i | i | -i | i | -i | -1 | 1 | 1 | -1 | -i | i | i | -i | linear of order 4 |
| ρ12 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | i | -i | 1 | -1 | 1 | -i | i | -i | i | -i | i | -1 | -1 | -1 | 1 | -i | i | i | -i | linear of order 4 |
| ρ13 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | i | -i | 1 | -1 | 1 | -i | i | -i | i | -i | i | -1 | 1 | 1 | -1 | i | -i | -i | i | linear of order 4 |
| ρ14 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | -i | i | 1 | -1 | 1 | i | -i | i | -i | i | -i | -1 | -1 | -1 | 1 | i | -i | -i | i | linear of order 4 |
| ρ15 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | -i | -i | -1 | 1 | -1 | i | -i | i | i | -i | i | 1 | -1 | 1 | 1 | -i | -i | i | i | linear of order 4 |
| ρ16 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | i | i | -1 | 1 | -1 | -i | i | -i | -i | i | -i | 1 | 1 | -1 | -1 | -i | -i | i | i | linear of order 4 |
| ρ17 | 2 | 2 | -2 | -2 | 2 | 0 | 2 | -2 | -2 | 2 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ18 | 2 | 2 | 2 | 2 | 2 | 0 | -2 | -2 | -2 | -2 | 0 | 0 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ19 | 2 | 2 | 2 | 2 | 2 | 0 | -2 | -2 | -2 | -2 | 0 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ20 | 2 | 2 | -2 | 2 | -2 | 0 | -2 | -2 | 2 | 2 | 0 | 0 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ21 | 2 | 2 | -2 | 2 | -2 | 0 | -2 | -2 | 2 | 2 | 0 | 0 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ22 | 2 | 2 | -2 | -2 | 2 | 0 | -2 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ23 | 2 | 2 | -2 | -2 | 2 | 0 | 2 | -2 | -2 | 2 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ24 | 2 | 2 | -2 | -2 | 2 | 0 | -2 | 2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ25 | 2 | 2 | 2 | -2 | -2 | 0 | -2 | 2 | -2 | 2 | -2i | 0 | 0 | 0 | 0 | 2i | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ26 | 2 | 2 | 2 | -2 | -2 | 0 | 2 | -2 | 2 | -2 | 0 | -2i | 0 | 0 | 0 | 0 | 0 | 0 | -2i | 2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ27 | 2 | 2 | 2 | -2 | -2 | 0 | -2 | 2 | -2 | 2 | 2i | 0 | 0 | 0 | 0 | -2i | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ28 | 2 | 2 | 2 | -2 | -2 | 0 | 2 | -2 | 2 | -2 | 0 | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 2i | -2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ29 | 8 | -8 | 0 | 0 | 0 | 0 | 0 | 0 | 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
(1 28 31 2)(3 26 25 8)(4 7 30 29)(5 32 27 6)(9 19 22 10)(11 17 24 16)(12 15 21 20)(13 23 18 14)
(1 7 5 3)(2 4 6 8)(9 15 13 11)(10 12 14 16)(17 19 21 23)(18 24 22 20)(25 31 29 27)(26 28 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 12 7 14 5 16 3 10)(2 13 4 11 6 9 8 15)(17 29 19 27 21 25 23 31)(18 26 24 28 22 30 20 32)
G:=sub<Sym(32)| (1,28,31,2)(3,26,25,8)(4,7,30,29)(5,32,27,6)(9,19,22,10)(11,17,24,16)(12,15,21,20)(13,23,18,14), (1,7,5,3)(2,4,6,8)(9,15,13,11)(10,12,14,16)(17,19,21,23)(18,24,22,20)(25,31,29,27)(26,28,30,32), (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), (1,12,7,14,5,16,3,10)(2,13,4,11,6,9,8,15)(17,29,19,27,21,25,23,31)(18,26,24,28,22,30,20,32)>;
G:=Group( (1,28,31,2)(3,26,25,8)(4,7,30,29)(5,32,27,6)(9,19,22,10)(11,17,24,16)(12,15,21,20)(13,23,18,14), (1,7,5,3)(2,4,6,8)(9,15,13,11)(10,12,14,16)(17,19,21,23)(18,24,22,20)(25,31,29,27)(26,28,30,32), (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), (1,12,7,14,5,16,3,10)(2,13,4,11,6,9,8,15)(17,29,19,27,21,25,23,31)(18,26,24,28,22,30,20,32) );
G=PermutationGroup([[(1,28,31,2),(3,26,25,8),(4,7,30,29),(5,32,27,6),(9,19,22,10),(11,17,24,16),(12,15,21,20),(13,23,18,14)], [(1,7,5,3),(2,4,6,8),(9,15,13,11),(10,12,14,16),(17,19,21,23),(18,24,22,20),(25,31,29,27),(26,28,30,32)], [(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)], [(1,12,7,14,5,16,3,10),(2,13,4,11,6,9,8,15),(17,29,19,27,21,25,23,31),(18,26,24,28,22,30,20,32)]])
Matrix representation of C42.7D4 ►in GL8(𝔽17)
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 4 | 13 | 4 | 13 | 1 | 16 | 1 | 15 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 1 | 0 | 0 | 0 | 0 | 4 |
| 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 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 | 0 | 0 |
| 4 | 13 | 4 | 13 | 1 | 16 | 1 | 15 |
| 4 | 0 | 4 | 0 | 1 | 0 | 1 | 16 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 4 | 13 | 4 | 13 | 1 | 16 | 1 | 15 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 1 | 0 | 0 | 0 | 13 |
| 0 | 0 | 0 | 0 | 10 | 0 | 16 | 0 |
| 13 | 4 | 13 | 4 | 16 | 11 | 16 | 2 |
| 0 | 0 | 0 | 0 | 16 | 0 | 7 | 0 |
| 11 | 6 | 11 | 6 | 7 | 9 | 7 | 3 |
| 0 | 7 | 0 | 1 | 0 | 0 | 0 | 0 |
| 10 | 0 | 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 10 | 0 | 0 | 0 | 0 |
| 7 | 1 | 0 | 11 | 6 | 11 | 0 | 7 |
G:=sub<GL(8,GF(17))| [0,0,0,4,0,0,0,0,0,0,0,13,0,0,16,16,0,0,0,4,0,16,0,1,0,0,0,13,1,0,0,0,1,0,0,1,0,0,0,0,0,1,0,16,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,15,0,0,0,4],[0,16,0,0,0,0,4,4,1,0,0,0,0,0,13,0,0,0,0,16,0,0,4,4,0,0,1,0,0,0,13,0,0,0,0,0,0,16,1,1,0,0,0,0,1,0,16,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,15,16],[0,0,4,0,1,0,0,0,0,0,13,0,0,16,0,1,0,0,4,0,0,0,1,0,0,0,13,0,0,0,0,1,0,1,1,0,0,0,0,0,1,0,16,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,15,0,0,0,0,13],[0,13,0,11,0,10,0,7,0,4,0,6,7,0,1,1,0,13,0,11,0,16,0,0,0,4,0,6,1,0,10,11,10,16,16,7,0,0,0,6,0,11,0,9,0,0,0,11,16,16,7,7,0,0,0,0,0,2,0,3,0,0,0,7] >;
C42.7D4 in GAP, Magma, Sage, TeX
C_4^2._7D_4
% in TeX
G:=Group("C4^2.7D4"); // GroupNames label
G:=SmallGroup(128,644);
// by ID
G=gap.SmallGroup(128,644);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,288,422,352,2019,521,248,2804,1411,1027,124]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^4=b^2,d^2=b,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1*b^-1,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=b*c^3>;
// generators/relations
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