p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.85D4, C42.176C23, C4⋊Q8.26C4, C4⋊C8.210C22, C42.117(C2×C4), C4.6Q16⋊24C2, (C22×C4).245D4, C4⋊Q8.248C22, (C22×Q8).11C4, C4.108(C8.C22), C42.6C4.26C2, (C2×C42).220C22, C23.187(C22⋊C4), C22.19(C4.D4), C2.15(C23.38D4), (C2×C4⋊Q8).7C2, (C2×Q8).34(C2×C4), (C2×C4).1247(C2×D4), C2.22(C2×C4.D4), (C22×C4).242(C2×C4), (C2×C4).170(C22×C4), (C2×C4).108(C22⋊C4), C22.234(C2×C22⋊C4), SmallGroup(128,290)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.85D4
G = < a,b,c,d | a4=b4=1, c4=a2b2, d2=b, ab=ba, cac-1=a-1b2, dad-1=ab2, cbc-1=b-1, bd=db, dcd-1=b-1c3 >
Subgroups: 228 in 114 conjugacy classes, 48 normal (12 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, Q8, C23, C42, C42, C4⋊C4, C2×C8, C22×C4, C22×C4, C22×C4, C2×Q8, C2×Q8, C8⋊C4, C22⋊C8, C4⋊C8, C2×C42, C2×C4⋊C4, C4⋊Q8, C4⋊Q8, C22×Q8, C4.6Q16, C42.6C4, C2×C4⋊Q8, C42.85D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C4.D4, C2×C22⋊C4, C8.C22, C2×C4.D4, C23.38D4, C42.85D4
Character table of C42.85D4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 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 | 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 C4.D4 |
| 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 | orthogonal lifted from C4.D4 |
| ρ23 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | -4 | 4 | 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 | -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 |
| ρ25 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 4 | -4 | 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 | -4 | 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 |
►On 64 points
Generators in S64
(1 33 62 56)(2 38 63 53)(3 35 64 50)(4 40 57 55)(5 37 58 52)(6 34 59 49)(7 39 60 54)(8 36 61 51)(9 24 32 46)(10 21 25 43)(11 18 26 48)(12 23 27 45)(13 20 28 42)(14 17 29 47)(15 22 30 44)(16 19 31 41)
(1 50 58 39)(2 40 59 51)(3 52 60 33)(4 34 61 53)(5 54 62 35)(6 36 63 55)(7 56 64 37)(8 38 57 49)(9 18 28 44)(10 45 29 19)(11 20 30 46)(12 47 31 21)(13 22 32 48)(14 41 25 23)(15 24 26 42)(16 43 27 17)
(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 50 13 58 22 39 32)(2 16 40 43 59 27 51 17)(3 46 52 11 60 20 33 30)(4 14 34 41 61 25 53 23)(5 44 54 9 62 18 35 28)(6 12 36 47 63 31 55 21)(7 42 56 15 64 24 37 26)(8 10 38 45 57 29 49 19)
G:=sub<Sym(64)| (1,33,62,56)(2,38,63,53)(3,35,64,50)(4,40,57,55)(5,37,58,52)(6,34,59,49)(7,39,60,54)(8,36,61,51)(9,24,32,46)(10,21,25,43)(11,18,26,48)(12,23,27,45)(13,20,28,42)(14,17,29,47)(15,22,30,44)(16,19,31,41), (1,50,58,39)(2,40,59,51)(3,52,60,33)(4,34,61,53)(5,54,62,35)(6,36,63,55)(7,56,64,37)(8,38,57,49)(9,18,28,44)(10,45,29,19)(11,20,30,46)(12,47,31,21)(13,22,32,48)(14,41,25,23)(15,24,26,42)(16,43,27,17), (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), (1,48,50,13,58,22,39,32)(2,16,40,43,59,27,51,17)(3,46,52,11,60,20,33,30)(4,14,34,41,61,25,53,23)(5,44,54,9,62,18,35,28)(6,12,36,47,63,31,55,21)(7,42,56,15,64,24,37,26)(8,10,38,45,57,29,49,19)>;
G:=Group( (1,33,62,56)(2,38,63,53)(3,35,64,50)(4,40,57,55)(5,37,58,52)(6,34,59,49)(7,39,60,54)(8,36,61,51)(9,24,32,46)(10,21,25,43)(11,18,26,48)(12,23,27,45)(13,20,28,42)(14,17,29,47)(15,22,30,44)(16,19,31,41), (1,50,58,39)(2,40,59,51)(3,52,60,33)(4,34,61,53)(5,54,62,35)(6,36,63,55)(7,56,64,37)(8,38,57,49)(9,18,28,44)(10,45,29,19)(11,20,30,46)(12,47,31,21)(13,22,32,48)(14,41,25,23)(15,24,26,42)(16,43,27,17), (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), (1,48,50,13,58,22,39,32)(2,16,40,43,59,27,51,17)(3,46,52,11,60,20,33,30)(4,14,34,41,61,25,53,23)(5,44,54,9,62,18,35,28)(6,12,36,47,63,31,55,21)(7,42,56,15,64,24,37,26)(8,10,38,45,57,29,49,19) );
G=PermutationGroup([[(1,33,62,56),(2,38,63,53),(3,35,64,50),(4,40,57,55),(5,37,58,52),(6,34,59,49),(7,39,60,54),(8,36,61,51),(9,24,32,46),(10,21,25,43),(11,18,26,48),(12,23,27,45),(13,20,28,42),(14,17,29,47),(15,22,30,44),(16,19,31,41)], [(1,50,58,39),(2,40,59,51),(3,52,60,33),(4,34,61,53),(5,54,62,35),(6,36,63,55),(7,56,64,37),(8,38,57,49),(9,18,28,44),(10,45,29,19),(11,20,30,46),(12,47,31,21),(13,22,32,48),(14,41,25,23),(15,24,26,42),(16,43,27,17)], [(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)], [(1,48,50,13,58,22,39,32),(2,16,40,43,59,27,51,17),(3,46,52,11,60,20,33,30),(4,14,34,41,61,25,53,23),(5,44,54,9,62,18,35,28),(6,12,36,47,63,31,55,21),(7,42,56,15,64,24,37,26),(8,10,38,45,57,29,49,19)]])
Matrix representation of C42.85D4 ►in GL8(𝔽17)
| 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 |
| 0 | 0 | 0 | 0 | 1 | 10 | 0 | 16 |
| 0 | 0 | 0 | 0 | 7 | 1 | 1 | 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 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 | 13 | 0 | 1 |
| 0 | 0 | 0 | 0 | 13 | 9 | 16 | 0 |
| 0 | 0 | 5 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 12 | 0 | 0 | 0 | 0 |
| 5 | 12 | 0 | 0 | 0 | 0 | 0 | 0 |
| 12 | 12 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 12 | 0 | 8 |
| 0 | 0 | 0 | 0 | 10 | 2 | 8 | 0 |
| 0 | 0 | 0 | 0 | 5 | 12 | 15 | 7 |
| 0 | 0 | 0 | 0 | 8 | 6 | 5 | 6 |
| 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 | 6 | 8 | 15 | 0 |
| 0 | 0 | 0 | 0 | 10 | 14 | 0 | 15 |
| 0 | 0 | 0 | 0 | 7 | 3 | 11 | 9 |
| 0 | 0 | 0 | 0 | 7 | 2 | 7 | 3 |
G:=sub<GL(8,GF(17))| [0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,16,1,7,0,0,0,0,1,0,10,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,16,8,13,0,0,0,0,1,0,13,9,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0],[0,0,5,12,0,0,0,0,0,0,12,12,0,0,0,0,5,5,0,0,0,0,0,0,5,12,0,0,0,0,0,0,0,0,0,0,11,10,5,8,0,0,0,0,12,2,12,6,0,0,0,0,0,8,15,5,0,0,0,0,8,0,7,6],[0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,6,10,7,7,0,0,0,0,8,14,3,2,0,0,0,0,15,0,11,7,0,0,0,0,0,15,9,3] >;
C42.85D4 in GAP, Magma, Sage, TeX
C_4^2._{85}D_4 % in TeX
G:=Group("C4^2.85D4"); // GroupNames label
G:=SmallGroup(128,290);
// by ID
G=gap.SmallGroup(128,290);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,456,1430,1123,1018,248,1971,242]);
// 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=a^-1*b^2,d*a*d^-1=a*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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