p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.86D4, C42.177C23, (C4×Q8).9C4, C22⋊Q8.16C4, C4.10D8⋊41C2, C4⋊C8.211C22, C4.86(C8⋊C22), C42.118(C2×C4), C4.6Q16⋊25C2, (C22×C4).246D4, C4⋊Q8.249C22, C4.109(C8.C22), C4⋊M4(2).18C2, C23.67(C22⋊C4), C42.6C4.27C2, (C2×C42).221C22, C2.16(C23.38D4), C2.17(C23.36D4), C23.37C23.17C2, C2.21(M4(2).8C22), C4⋊C4.45(C2×C4), (C2×Q8).35(C2×C4), (C2×C4).1248(C2×D4), (C2×C4).171(C22×C4), (C22×C4).243(C2×C4), (C2×C4).324(C22⋊C4), C22.235(C2×C22⋊C4), SmallGroup(128,291)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.86D4
G = < a,b,c,d | a4=b4=1, c4=a2, d2=b, ab=ba, cac-1=dad-1=a-1b2, cbc-1=b-1, bd=db, dcd-1=b-1c3 >
Subgroups: 188 in 101 conjugacy classes, 46 normal (28 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C8, C2×C4, C2×C4, Q8, C23, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, C2×Q8, C2×Q8, C8⋊C4, C22⋊C8, C4⋊C8, C4⋊C8, C2×C42, C42⋊C2, C4×Q8, C4×Q8, C22⋊Q8, C22⋊Q8, C42.C2, C4⋊Q8, C2×M4(2), C4.10D8, C4.6Q16, C4⋊M4(2), C42.6C4, C23.37C23, C42.86D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C2×C22⋊C4, C8⋊C22, C8.C22, M4(2).8C22, C23.36D4, C23.38D4, C42.86D4
Character table of C42.86D4
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | |
| size | 1 | 1 | 1 | 1 | 4 | 2 | 2 | 2 | 2 | 2 | 2 | 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 | -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 | -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 | 0 | 0 | 4 | 0 | 0 | 0 | -4 | 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 | 0 | 4 | 0 | -4 | 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 | -4 | 0 | 4 | 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 |
| ρ24 | 4 | 4 | -4 | -4 | 0 | 0 | -4 | 0 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from C8.C22, Schur index 2 |
| ρ25 | 4 | -4 | 4 | -4 | 0 | 0 | 0 | 0 | -4i | 4i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from M4(2).8C22 |
| ρ26 | 4 | -4 | 4 | -4 | 0 | 0 | 0 | 0 | 4i | -4i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from M4(2).8C22 |
►On 64 points
Generators in S64
(1 7 5 3)(2 61 6 57)(4 63 8 59)(9 21 13 17)(10 16 14 12)(11 23 15 19)(18 24 22 20)(25 27 29 31)(26 55 30 51)(28 49 32 53)(33 48 37 44)(34 36 38 40)(35 42 39 46)(41 43 45 47)(50 52 54 56)(58 64 62 60)
(1 14 58 24)(2 17 59 15)(3 16 60 18)(4 19 61 9)(5 10 62 20)(6 21 63 11)(7 12 64 22)(8 23 57 13)(25 40 56 41)(26 42 49 33)(27 34 50 43)(28 44 51 35)(29 36 52 45)(30 46 53 37)(31 38 54 47)(32 48 55 39)
(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 48 14 55 58 39 24 32)(2 50 17 43 59 27 15 34)(3 46 16 53 60 37 18 30)(4 56 19 41 61 25 9 40)(5 44 10 51 62 35 20 28)(6 54 21 47 63 31 11 38)(7 42 12 49 64 33 22 26)(8 52 23 45 57 29 13 36)
G:=sub<Sym(64)| (1,7,5,3)(2,61,6,57)(4,63,8,59)(9,21,13,17)(10,16,14,12)(11,23,15,19)(18,24,22,20)(25,27,29,31)(26,55,30,51)(28,49,32,53)(33,48,37,44)(34,36,38,40)(35,42,39,46)(41,43,45,47)(50,52,54,56)(58,64,62,60), (1,14,58,24)(2,17,59,15)(3,16,60,18)(4,19,61,9)(5,10,62,20)(6,21,63,11)(7,12,64,22)(8,23,57,13)(25,40,56,41)(26,42,49,33)(27,34,50,43)(28,44,51,35)(29,36,52,45)(30,46,53,37)(31,38,54,47)(32,48,55,39), (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), (1,48,14,55,58,39,24,32)(2,50,17,43,59,27,15,34)(3,46,16,53,60,37,18,30)(4,56,19,41,61,25,9,40)(5,44,10,51,62,35,20,28)(6,54,21,47,63,31,11,38)(7,42,12,49,64,33,22,26)(8,52,23,45,57,29,13,36)>;
G:=Group( (1,7,5,3)(2,61,6,57)(4,63,8,59)(9,21,13,17)(10,16,14,12)(11,23,15,19)(18,24,22,20)(25,27,29,31)(26,55,30,51)(28,49,32,53)(33,48,37,44)(34,36,38,40)(35,42,39,46)(41,43,45,47)(50,52,54,56)(58,64,62,60), (1,14,58,24)(2,17,59,15)(3,16,60,18)(4,19,61,9)(5,10,62,20)(6,21,63,11)(7,12,64,22)(8,23,57,13)(25,40,56,41)(26,42,49,33)(27,34,50,43)(28,44,51,35)(29,36,52,45)(30,46,53,37)(31,38,54,47)(32,48,55,39), (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), (1,48,14,55,58,39,24,32)(2,50,17,43,59,27,15,34)(3,46,16,53,60,37,18,30)(4,56,19,41,61,25,9,40)(5,44,10,51,62,35,20,28)(6,54,21,47,63,31,11,38)(7,42,12,49,64,33,22,26)(8,52,23,45,57,29,13,36) );
G=PermutationGroup([[(1,7,5,3),(2,61,6,57),(4,63,8,59),(9,21,13,17),(10,16,14,12),(11,23,15,19),(18,24,22,20),(25,27,29,31),(26,55,30,51),(28,49,32,53),(33,48,37,44),(34,36,38,40),(35,42,39,46),(41,43,45,47),(50,52,54,56),(58,64,62,60)], [(1,14,58,24),(2,17,59,15),(3,16,60,18),(4,19,61,9),(5,10,62,20),(6,21,63,11),(7,12,64,22),(8,23,57,13),(25,40,56,41),(26,42,49,33),(27,34,50,43),(28,44,51,35),(29,36,52,45),(30,46,53,37),(31,38,54,47),(32,48,55,39)], [(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)], [(1,48,14,55,58,39,24,32),(2,50,17,43,59,27,15,34),(3,46,16,53,60,37,18,30),(4,56,19,41,61,25,9,40),(5,44,10,51,62,35,20,28),(6,54,21,47,63,31,11,38),(7,42,12,49,64,33,22,26),(8,52,23,45,57,29,13,36)]])
Matrix representation of C42.86D4 ►in GL8(𝔽17)
| 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 | 1 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 14 | 2 | 1 | 15 |
| 0 | 0 | 0 | 0 | 15 | 3 | 1 | 16 |
| 1 | 15 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 16 | 0 | 0 | 0 | 0 | 0 | 0 |
| 16 | 1 | 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 1 | 1 | 0 | 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 | 16 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 16 |
| 1 | 4 | 0 | 0 | 0 | 0 | 0 | 0 |
| 3 | 16 | 0 | 0 | 0 | 0 | 0 | 0 |
| 15 | 15 | 9 | 5 | 0 | 0 | 0 | 0 |
| 6 | 15 | 5 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 15 | 14 | 12 | 6 |
| 0 | 0 | 0 | 0 | 15 | 6 | 15 | 5 |
| 0 | 0 | 0 | 0 | 14 | 11 | 11 | 14 |
| 0 | 0 | 0 | 0 | 7 | 3 | 15 | 2 |
| 3 | 0 | 16 | 7 | 0 | 0 | 0 | 0 |
| 8 | 0 | 13 | 3 | 0 | 0 | 0 | 0 |
| 15 | 15 | 9 | 5 | 0 | 0 | 0 | 0 |
| 1 | 14 | 9 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 10 | 8 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 9 | 11 | 14 |
| 0 | 0 | 0 | 0 | 1 | 15 | 1 | 6 |
G:=sub<GL(8,GF(17))| [4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,1,14,15,0,0,0,0,15,16,2,3,0,0,0,0,0,0,1,1,0,0,0,0,0,0,15,16],[1,1,16,0,0,0,0,0,15,16,1,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16],[1,3,15,6,0,0,0,0,4,16,15,15,0,0,0,0,0,0,9,5,0,0,0,0,0,0,5,8,0,0,0,0,0,0,0,0,15,15,14,7,0,0,0,0,14,6,11,3,0,0,0,0,12,15,11,15,0,0,0,0,6,5,14,2],[3,8,15,1,0,0,0,0,0,0,15,14,0,0,0,0,16,13,9,9,0,0,0,0,7,3,5,5,0,0,0,0,0,0,0,0,9,10,5,1,0,0,0,0,2,8,9,15,0,0,0,0,0,0,11,1,0,0,0,0,0,0,14,6] >;
C42.86D4 in GAP, Magma, Sage, TeX
C_4^2._{86}D_4 % in TeX
G:=Group("C4^2.86D4"); // GroupNames label
G:=SmallGroup(128,291);
// by ID
G=gap.SmallGroup(128,291);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,456,1430,184,1123,1018,248,1971,242]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^4=a^2,d^2=b,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1*b^2,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=b^-1*c^3>;
// generators/relations
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