p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.87D4, C42.178C23, C4.D8⋊26C2, C4⋊C8.212C22, C4.87(C8⋊C22), C42.119(C2×C4), C4.6Q16⋊26C2, C4.4D4.12C4, (C22×C4).247D4, C4⋊Q8.250C22, C4⋊M4(2)⋊22C2, C4.17(C4.D4), C4.89(C8.C22), C4⋊1D4.132C22, C23.68(C22⋊C4), (C2×C42).222C22, C2.18(C23.36D4), C22.26C24.18C2, (C2×C4○D4).8C4, (C2×D4).36(C2×C4), (C2×Q8).36(C2×C4), (C2×C4).1249(C2×D4), C2.23(C2×C4.D4), (C2×C4).172(C22×C4), (C22×C4).244(C2×C4), (C2×C4).109(C22⋊C4), C22.236(C2×C22⋊C4), SmallGroup(128,292)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.87D4
G = < a,b,c,d | a4=b4=1, c4=a2b2, d2=b, ab=ba, cac-1=dad-1=a-1, cbc-1=b-1, bd=db, dcd-1=a2b-1c3 >
Subgroups: 292 in 127 conjugacy classes, 48 normal (14 characteristic)
C1, C2, 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, C4○D4, C4⋊C8, C4⋊C8, C2×C42, C4×D4, C4⋊D4, C4.4D4, C4⋊1D4, C4⋊Q8, C2×M4(2), C2×C4○D4, C4.D8, C4.6Q16, C4⋊M4(2), C22.26C24, C42.87D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C4.D4, C2×C22⋊C4, C8⋊C22, C8.C22, C2×C4.D4, C23.36D4, C42.87D4
Character table of C42.87D4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | |
| size | 1 | 1 | 1 | 1 | 4 | 8 | 8 | 2 | 2 | 2 | 2 | 2 | 2 | 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 | 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 | 1 | -1 | i | i | -i | -i | -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 | -1 | 1 | -i | -i | i | -i | -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 | -1 | 1 | -i | i | -i | -i | 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 | 1 | -1 | i | -i | i | -i | 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 | -1 | 1 | i | i | -i | i | 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 | 1 | -1 | -i | -i | i | i | 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 | 1 | -1 | -i | i | -i | i | -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 | -1 | 1 | i | -i | i | i | -i | i | -i | -i | linear of order 4 |
| ρ17 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 2 | -2 | -2 | -2 | -2 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ18 | 2 | 2 | 2 | 2 | -2 | 0 | 0 | 2 | 2 | -2 | 2 | -2 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ19 | 2 | 2 | 2 | 2 | -2 | 0 | 0 | -2 | 2 | 2 | 2 | 2 | -2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ20 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | -2 | -2 | 2 | -2 | 2 | -2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ21 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | -4 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C4.D4 |
| ρ22 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 4 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C4.D4 |
| ρ23 | 4 | -4 | 4 | -4 | 0 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C8⋊C22 |
| ρ24 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 4 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C8⋊C22 |
| ρ25 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | -4 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from C8.C22, Schur index 2 |
| ρ26 | 4 | -4 | 4 | -4 | 0 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from C8.C22, Schur index 2 |
►On 64 points
Generators in S64
(1 11 61 25)(2 26 62 12)(3 13 63 27)(4 28 64 14)(5 15 57 29)(6 30 58 16)(7 9 59 31)(8 32 60 10)(17 46 40 55)(18 56 33 47)(19 48 34 49)(20 50 35 41)(21 42 36 51)(22 52 37 43)(23 44 38 53)(24 54 39 45)
(1 21 57 40)(2 33 58 22)(3 23 59 34)(4 35 60 24)(5 17 61 36)(6 37 62 18)(7 19 63 38)(8 39 64 20)(9 48 27 53)(10 54 28 41)(11 42 29 55)(12 56 30 43)(13 44 31 49)(14 50 32 45)(15 46 25 51)(16 52 26 47)
(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)(33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56)(57 58 59 60 61 62 63 64)
(1 64 21 20 57 8 40 39)(2 38 33 7 58 19 22 63)(3 62 23 18 59 6 34 37)(4 36 35 5 60 17 24 61)(9 30 48 43 27 12 53 56)(10 55 54 11 28 42 41 29)(13 26 44 47 31 16 49 52)(14 51 50 15 32 46 45 25)
G:=sub<Sym(64)| (1,11,61,25)(2,26,62,12)(3,13,63,27)(4,28,64,14)(5,15,57,29)(6,30,58,16)(7,9,59,31)(8,32,60,10)(17,46,40,55)(18,56,33,47)(19,48,34,49)(20,50,35,41)(21,42,36,51)(22,52,37,43)(23,44,38,53)(24,54,39,45), (1,21,57,40)(2,33,58,22)(3,23,59,34)(4,35,60,24)(5,17,61,36)(6,37,62,18)(7,19,63,38)(8,39,64,20)(9,48,27,53)(10,54,28,41)(11,42,29,55)(12,56,30,43)(13,44,31,49)(14,50,32,45)(15,46,25,51)(16,52,26,47), (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)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64), (1,64,21,20,57,8,40,39)(2,38,33,7,58,19,22,63)(3,62,23,18,59,6,34,37)(4,36,35,5,60,17,24,61)(9,30,48,43,27,12,53,56)(10,55,54,11,28,42,41,29)(13,26,44,47,31,16,49,52)(14,51,50,15,32,46,45,25)>;
G:=Group( (1,11,61,25)(2,26,62,12)(3,13,63,27)(4,28,64,14)(5,15,57,29)(6,30,58,16)(7,9,59,31)(8,32,60,10)(17,46,40,55)(18,56,33,47)(19,48,34,49)(20,50,35,41)(21,42,36,51)(22,52,37,43)(23,44,38,53)(24,54,39,45), (1,21,57,40)(2,33,58,22)(3,23,59,34)(4,35,60,24)(5,17,61,36)(6,37,62,18)(7,19,63,38)(8,39,64,20)(9,48,27,53)(10,54,28,41)(11,42,29,55)(12,56,30,43)(13,44,31,49)(14,50,32,45)(15,46,25,51)(16,52,26,47), (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)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64), (1,64,21,20,57,8,40,39)(2,38,33,7,58,19,22,63)(3,62,23,18,59,6,34,37)(4,36,35,5,60,17,24,61)(9,30,48,43,27,12,53,56)(10,55,54,11,28,42,41,29)(13,26,44,47,31,16,49,52)(14,51,50,15,32,46,45,25) );
G=PermutationGroup([[(1,11,61,25),(2,26,62,12),(3,13,63,27),(4,28,64,14),(5,15,57,29),(6,30,58,16),(7,9,59,31),(8,32,60,10),(17,46,40,55),(18,56,33,47),(19,48,34,49),(20,50,35,41),(21,42,36,51),(22,52,37,43),(23,44,38,53),(24,54,39,45)], [(1,21,57,40),(2,33,58,22),(3,23,59,34),(4,35,60,24),(5,17,61,36),(6,37,62,18),(7,19,63,38),(8,39,64,20),(9,48,27,53),(10,54,28,41),(11,42,29,55),(12,56,30,43),(13,44,31,49),(14,50,32,45),(15,46,25,51),(16,52,26,47)], [(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),(33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56),(57,58,59,60,61,62,63,64)], [(1,64,21,20,57,8,40,39),(2,38,33,7,58,19,22,63),(3,62,23,18,59,6,34,37),(4,36,35,5,60,17,24,61),(9,30,48,43,27,12,53,56),(10,55,54,11,28,42,41,29),(13,26,44,47,31,16,49,52),(14,51,50,15,32,46,45,25)]])
Matrix representation of C42.87D4 ►in GL8(𝔽17)
| 1 | 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 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 15 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 15 |
| 0 | 0 | 0 | 0 | 1 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 16 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 16 | 1 | 15 | 0 | 0 | 0 | 0 |
| 1 | 0 | 1 | 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 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 13 | 0 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 13 | 0 | 0 | 0 | 0 |
| 4 | 13 | 4 | 13 | 0 | 0 | 0 | 0 |
| 4 | 13 | 0 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 6 | 11 | 7 | 10 |
| 0 | 0 | 0 | 0 | 6 | 6 | 7 | 7 |
| 0 | 0 | 0 | 0 | 1 | 16 | 11 | 6 |
| 0 | 0 | 0 | 0 | 1 | 1 | 11 | 11 |
| 0 | 0 | 4 | 13 | 0 | 0 | 0 | 0 |
| 0 | 13 | 0 | 13 | 0 | 0 | 0 | 0 |
| 13 | 0 | 0 | 4 | 0 | 0 | 0 | 0 |
| 13 | 4 | 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 6 | 11 | 7 | 10 |
| 0 | 0 | 0 | 0 | 11 | 11 | 10 | 10 |
| 0 | 0 | 0 | 0 | 1 | 16 | 11 | 6 |
| 0 | 0 | 0 | 0 | 16 | 16 | 6 | 6 |
G:=sub<GL(8,GF(17))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,1,0,0,0,0,0,0,1,0,1,0,0,0,0,15,0,16,0,0,0,0,0,0,15,0,16],[0,16,1,1,0,0,0,0,1,0,16,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,15,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,0,4,4,0,0,0,0,13,0,13,13,0,0,0,0,0,4,4,0,0,0,0,0,13,13,13,13,0,0,0,0,0,0,0,0,6,6,1,1,0,0,0,0,11,6,16,1,0,0,0,0,7,7,11,11,0,0,0,0,10,7,6,11],[0,0,13,13,0,0,0,0,0,13,0,4,0,0,0,0,4,0,0,0,0,0,0,0,13,13,4,4,0,0,0,0,0,0,0,0,6,11,1,16,0,0,0,0,11,11,16,16,0,0,0,0,7,10,11,6,0,0,0,0,10,10,6,6] >;
C42.87D4 in GAP, Magma, Sage, TeX
C_4^2._{87}D_4 % in TeX
G:=Group("C4^2.87D4"); // GroupNames label
G:=SmallGroup(128,292);
// by ID
G=gap.SmallGroup(128,292);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,1430,352,1123,1018,248,1971,242]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^4=a^2*b^2,d^2=b,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=a^2*b^-1*c^3>;
// generators/relations
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