p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.88D4, C42.179C23, C4.10D8⋊42C2, C4⋊C8.213C22, C4.88(C8⋊C22), C42.120(C2×C4), C42⋊C2.8C4, (C22×C4).248D4, C4⋊Q8.251C22, C42.C2.14C4, C4.90(C8.C22), C4.17(C4.10D4), C4⋊M4(2).19C2, C23.69(C22⋊C4), (C2×C42).223C22, C2.19(C23.36D4), C23.37C23.18C2, C4⋊C4.46(C2×C4), (C2×C4).1250(C2×D4), (C22×C4).245(C2×C4), (C2×C4).173(C22×C4), C2.21(C2×C4.10D4), (C2×C4).110(C22⋊C4), C22.237(C2×C22⋊C4), SmallGroup(128,293)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.88D4
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=bc3 >
Subgroups: 196 in 105 conjugacy classes, 48 normal (12 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, Q8, C23, C42, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, C22×C4, C2×Q8, C4⋊C8, C4⋊C8, C2×C42, C42⋊C2, C4×Q8, C22⋊Q8, C42.C2, C4⋊Q8, C2×M4(2), C4.10D8, C4⋊M4(2), C23.37C23, C42.88D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C4.10D4, C2×C22⋊C4, C8⋊C22, C8.C22, C2×C4.10D4, C23.36D4, C42.88D4
Character table of C42.88D4
| 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 | 4 | -4 | 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 | 0 | 0 | 0 | 0 | 0 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C8⋊C22 |
| ρ23 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 4 | -4 | 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 | 0 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from C4.10D4, Schur index 2 |
| ρ25 | 4 | -4 | 4 | -4 | 0 | -4 | 4 | 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 |
| ρ26 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from C4.10D4, Schur index 2 |
►On 64 points
Generators in S64
(1 35 54 63)(2 64 55 36)(3 37 56 57)(4 58 49 38)(5 39 50 59)(6 60 51 40)(7 33 52 61)(8 62 53 34)(9 23 27 43)(10 44 28 24)(11 17 29 45)(12 46 30 18)(13 19 31 47)(14 48 32 20)(15 21 25 41)(16 42 26 22)
(1 61 50 37)(2 38 51 62)(3 63 52 39)(4 40 53 64)(5 57 54 33)(6 34 55 58)(7 59 56 35)(8 36 49 60)(9 21 31 45)(10 46 32 22)(11 23 25 47)(12 48 26 24)(13 17 27 41)(14 42 28 18)(15 19 29 43)(16 44 30 20)
(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 61 26 50 24 37 12)(2 15 38 19 51 29 62 43)(3 46 63 32 52 22 39 10)(4 13 40 17 53 27 64 41)(5 44 57 30 54 20 33 16)(6 11 34 23 55 25 58 47)(7 42 59 28 56 18 35 14)(8 9 36 21 49 31 60 45)
G:=sub<Sym(64)| (1,35,54,63)(2,64,55,36)(3,37,56,57)(4,58,49,38)(5,39,50,59)(6,60,51,40)(7,33,52,61)(8,62,53,34)(9,23,27,43)(10,44,28,24)(11,17,29,45)(12,46,30,18)(13,19,31,47)(14,48,32,20)(15,21,25,41)(16,42,26,22), (1,61,50,37)(2,38,51,62)(3,63,52,39)(4,40,53,64)(5,57,54,33)(6,34,55,58)(7,59,56,35)(8,36,49,60)(9,21,31,45)(10,46,32,22)(11,23,25,47)(12,48,26,24)(13,17,27,41)(14,42,28,18)(15,19,29,43)(16,44,30,20), (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), (1,48,61,26,50,24,37,12)(2,15,38,19,51,29,62,43)(3,46,63,32,52,22,39,10)(4,13,40,17,53,27,64,41)(5,44,57,30,54,20,33,16)(6,11,34,23,55,25,58,47)(7,42,59,28,56,18,35,14)(8,9,36,21,49,31,60,45)>;
G:=Group( (1,35,54,63)(2,64,55,36)(3,37,56,57)(4,58,49,38)(5,39,50,59)(6,60,51,40)(7,33,52,61)(8,62,53,34)(9,23,27,43)(10,44,28,24)(11,17,29,45)(12,46,30,18)(13,19,31,47)(14,48,32,20)(15,21,25,41)(16,42,26,22), (1,61,50,37)(2,38,51,62)(3,63,52,39)(4,40,53,64)(5,57,54,33)(6,34,55,58)(7,59,56,35)(8,36,49,60)(9,21,31,45)(10,46,32,22)(11,23,25,47)(12,48,26,24)(13,17,27,41)(14,42,28,18)(15,19,29,43)(16,44,30,20), (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), (1,48,61,26,50,24,37,12)(2,15,38,19,51,29,62,43)(3,46,63,32,52,22,39,10)(4,13,40,17,53,27,64,41)(5,44,57,30,54,20,33,16)(6,11,34,23,55,25,58,47)(7,42,59,28,56,18,35,14)(8,9,36,21,49,31,60,45) );
G=PermutationGroup([[(1,35,54,63),(2,64,55,36),(3,37,56,57),(4,58,49,38),(5,39,50,59),(6,60,51,40),(7,33,52,61),(8,62,53,34),(9,23,27,43),(10,44,28,24),(11,17,29,45),(12,46,30,18),(13,19,31,47),(14,48,32,20),(15,21,25,41),(16,42,26,22)], [(1,61,50,37),(2,38,51,62),(3,63,52,39),(4,40,53,64),(5,57,54,33),(6,34,55,58),(7,59,56,35),(8,36,49,60),(9,21,31,45),(10,46,32,22),(11,23,25,47),(12,48,26,24),(13,17,27,41),(14,42,28,18),(15,19,29,43),(16,44,30,20)], [(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)], [(1,48,61,26,50,24,37,12),(2,15,38,19,51,29,62,43),(3,46,63,32,52,22,39,10),(4,13,40,17,53,27,64,41),(5,44,57,30,54,20,33,16),(6,11,34,23,55,25,58,47),(7,42,59,28,56,18,35,14),(8,9,36,21,49,31,60,45)]])
Matrix representation of C42.88D4 ►in GL8(𝔽17)
| 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 13 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 2 |
| 0 | 0 | 0 | 0 | 0 | 0 | 16 | 16 |
| 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 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 16 | 15 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 | 0 | 0 | 0 | 0 |
| 13 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 13 | 9 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 13 | 13 |
| 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 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 13 | 13 |
| 0 | 0 | 0 | 0 | 13 | 9 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 | 0 | 0 |
G:=sub<GL(8,GF(17))| [4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,1,16,0,0,0,0,0,0,2,16,0,0,0,0,0,0,0,0,1,16,0,0,0,0,0,0,2,16],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,16,0,0,0,0,0,0,2,16,0,0,0,0,0,0,0,0,16,1,0,0,0,0,0,0,15,1],[0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,9,4,0,0,0,0,0,0,0,0,4,13,0,0,0,0,0,0,0,13],[0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,9,4,0,0,0,0,4,13,0,0,0,0,0,0,0,13,0,0] >;
C42.88D4 in GAP, Magma, Sage, TeX
C_4^2._{88}D_4 % in TeX
G:=Group("C4^2.88D4"); // GroupNames label
G:=SmallGroup(128,293);
// by ID
G=gap.SmallGroup(128,293);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,232,1430,352,1123,1018,248,1971,242]);
// 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=b*c^3>;
// generators/relations
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