p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.84D4, C42.175C23, (C4×D4).9C4, C4⋊D4.16C4, C4.D8⋊25C2, C4.10D8⋊40C2, C4⋊C8.209C22, C42.116(C2×C4), (C22×C4).244D4, C4⋊Q8.247C22, C4.113(C8⋊C22), C4⋊M4(2)⋊21C2, C42.6C4⋊43C2, C4.88(C8.C22), C4⋊1D4.131C22, C23.66(C22⋊C4), (C2×C42).219C22, C2.16(C23.37D4), C2.16(C23.36D4), C22.26C24.17C2, C2.20(M4(2).8C22), C4⋊C4.44(C2×C4), (C2×D4).35(C2×C4), (C2×C4).1246(C2×D4), (C2×C4).169(C22×C4), (C22×C4).241(C2×C4), (C2×C4).323(C22⋊C4), C22.233(C2×C22⋊C4), SmallGroup(128,289)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.84D4
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=a2bc3 >
Subgroups: 284 in 123 conjugacy classes, 46 normal (28 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C8, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C8⋊C4, C22⋊C8, C4⋊C8, C4⋊C8, C2×C42, C4×D4, C4×D4, C4⋊D4, C4⋊D4, C4.4D4, C4⋊1D4, C4⋊Q8, C2×M4(2), C2×C4○D4, C4.D8, C4.10D8, C4⋊M4(2), C42.6C4, C22.26C24, C42.84D4
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, C23.37D4, C42.84D4
Character table of C42.84D4
| 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 | 4 | 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 | 0 | 0 | -4 | 0 | 0 | 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 | 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 | 0 | 4 | 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 | 0 | 0 | 4i | 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 | 0 | 0 | -4i | 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 45 5 41)(2 28 6 32)(3 47 7 43)(4 30 8 26)(9 29 13 25)(10 48 14 44)(11 31 15 27)(12 42 16 46)(17 56 21 52)(18 60 22 64)(19 50 23 54)(20 62 24 58)(33 53 37 49)(34 57 38 61)(35 55 39 51)(36 59 40 63)
(1 50 11 57)(2 58 12 51)(3 52 13 59)(4 60 14 53)(5 54 15 61)(6 62 16 55)(7 56 9 63)(8 64 10 49)(17 25 40 47)(18 48 33 26)(19 27 34 41)(20 42 35 28)(21 29 36 43)(22 44 37 30)(23 31 38 45)(24 46 39 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)(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 50 10 11 49 57 8)(2 7 58 56 12 9 51 63)(3 62 52 16 13 55 59 6)(4 5 60 54 14 15 53 61)(17 28 25 20 40 42 47 35)(18 34 48 41 33 19 26 27)(21 32 29 24 36 46 43 39)(22 38 44 45 37 23 30 31)
G:=sub<Sym(64)| (1,45,5,41)(2,28,6,32)(3,47,7,43)(4,30,8,26)(9,29,13,25)(10,48,14,44)(11,31,15,27)(12,42,16,46)(17,56,21,52)(18,60,22,64)(19,50,23,54)(20,62,24,58)(33,53,37,49)(34,57,38,61)(35,55,39,51)(36,59,40,63), (1,50,11,57)(2,58,12,51)(3,52,13,59)(4,60,14,53)(5,54,15,61)(6,62,16,55)(7,56,9,63)(8,64,10,49)(17,25,40,47)(18,48,33,26)(19,27,34,41)(20,42,35,28)(21,29,36,43)(22,44,37,30)(23,31,38,45)(24,46,39,32), (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)(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), (1,64,50,10,11,49,57,8)(2,7,58,56,12,9,51,63)(3,62,52,16,13,55,59,6)(4,5,60,54,14,15,53,61)(17,28,25,20,40,42,47,35)(18,34,48,41,33,19,26,27)(21,32,29,24,36,46,43,39)(22,38,44,45,37,23,30,31)>;
G:=Group( (1,45,5,41)(2,28,6,32)(3,47,7,43)(4,30,8,26)(9,29,13,25)(10,48,14,44)(11,31,15,27)(12,42,16,46)(17,56,21,52)(18,60,22,64)(19,50,23,54)(20,62,24,58)(33,53,37,49)(34,57,38,61)(35,55,39,51)(36,59,40,63), (1,50,11,57)(2,58,12,51)(3,52,13,59)(4,60,14,53)(5,54,15,61)(6,62,16,55)(7,56,9,63)(8,64,10,49)(17,25,40,47)(18,48,33,26)(19,27,34,41)(20,42,35,28)(21,29,36,43)(22,44,37,30)(23,31,38,45)(24,46,39,32), (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)(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), (1,64,50,10,11,49,57,8)(2,7,58,56,12,9,51,63)(3,62,52,16,13,55,59,6)(4,5,60,54,14,15,53,61)(17,28,25,20,40,42,47,35)(18,34,48,41,33,19,26,27)(21,32,29,24,36,46,43,39)(22,38,44,45,37,23,30,31) );
G=PermutationGroup([[(1,45,5,41),(2,28,6,32),(3,47,7,43),(4,30,8,26),(9,29,13,25),(10,48,14,44),(11,31,15,27),(12,42,16,46),(17,56,21,52),(18,60,22,64),(19,50,23,54),(20,62,24,58),(33,53,37,49),(34,57,38,61),(35,55,39,51),(36,59,40,63)], [(1,50,11,57),(2,58,12,51),(3,52,13,59),(4,60,14,53),(5,54,15,61),(6,62,16,55),(7,56,9,63),(8,64,10,49),(17,25,40,47),(18,48,33,26),(19,27,34,41),(20,42,35,28),(21,29,36,43),(22,44,37,30),(23,31,38,45),(24,46,39,32)], [(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),(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)], [(1,64,50,10,11,49,57,8),(2,7,58,56,12,9,51,63),(3,62,52,16,13,55,59,6),(4,5,60,54,14,15,53,61),(17,28,25,20,40,42,47,35),(18,34,48,41,33,19,26,27),(21,32,29,24,36,46,43,39),(22,38,44,45,37,23,30,31)]])
Matrix representation of C42.84D4 ►in GL8(𝔽17)
| 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 | 13 | 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 | 13 |
| 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 | 16 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 16 | 15 |
| 0 | 0 | 0 | 0 | 1 | 0 | 1 | 1 |
| 0 | 0 | 3 | 14 | 0 | 0 | 0 | 0 |
| 0 | 0 | 14 | 14 | 0 | 0 | 0 | 0 |
| 14 | 3 | 0 | 0 | 0 | 0 | 0 | 0 |
| 3 | 3 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 11 | 12 |
| 0 | 0 | 0 | 0 | 10 | 10 | 7 | 12 |
| 0 | 0 | 0 | 0 | 6 | 0 | 16 | 5 |
| 0 | 0 | 0 | 0 | 10 | 10 | 7 | 7 |
| 0 | 0 | 3 | 14 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 3 | 0 | 0 | 0 | 0 |
| 14 | 3 | 0 | 0 | 0 | 0 | 0 | 0 |
| 14 | 14 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 11 | 12 |
| 0 | 0 | 0 | 0 | 5 | 7 | 5 | 15 |
| 0 | 0 | 0 | 0 | 6 | 0 | 16 | 5 |
| 0 | 0 | 0 | 0 | 0 | 7 | 0 | 10 |
G:=sub<GL(8,GF(17))| [0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,13],[0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,16,2,0,1,0,0,0,0,16,1,1,0,0,0,0,0,0,0,16,1,0,0,0,0,0,0,15,1],[0,0,14,3,0,0,0,0,0,0,3,3,0,0,0,0,3,14,0,0,0,0,0,0,14,14,0,0,0,0,0,0,0,0,0,0,1,10,6,10,0,0,0,0,0,10,0,10,0,0,0,0,11,7,16,7,0,0,0,0,12,12,5,7],[0,0,14,14,0,0,0,0,0,0,3,14,0,0,0,0,3,3,0,0,0,0,0,0,14,3,0,0,0,0,0,0,0,0,0,0,1,5,6,0,0,0,0,0,0,7,0,7,0,0,0,0,11,5,16,0,0,0,0,0,12,15,5,10] >;
C42.84D4 in GAP, Magma, Sage, TeX
C_4^2._{84}D_4 % in TeX
G:=Group("C4^2.84D4"); // GroupNames label
G:=SmallGroup(128,289);
// by ID
G=gap.SmallGroup(128,289);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,1430,184,1123,1018,248,1971,242]);
// 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=a^2*b*c^3>;
// generators/relations
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