p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.83D4, C42.174C23, C4⋊Q8.25C4, C4.10D8⋊39C2, C4⋊C8.208C22, C42.115(C2×C4), (C22×C4).243D4, C4⋊Q8.246C22, C4.112(C8⋊C22), C4.107(C8.C22), C42.6C4.25C2, (C2×C42).218C22, C23.186(C22⋊C4), C22.11(C4.10D4), C2.14(C23.38D4), C2.15(C23.37D4), (C2×C4⋊Q8).6C2, (C2×C4⋊C4).22C4, C4⋊C4.43(C2×C4), (C2×C4).1245(C2×D4), (C2×C4).168(C22×C4), (C22×C4).240(C2×C4), C2.20(C2×C4.10D4), (C2×C4).107(C22⋊C4), C22.232(C2×C22⋊C4), SmallGroup(128,288)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.83D4
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=bc3 >
Subgroups: 228 in 114 conjugacy classes, 48 normal (24 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, Q8, C23, C42, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C2×Q8, C8⋊C4, C22⋊C8, C4⋊C8, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C4⋊Q8, C4⋊Q8, C22×Q8, C4.10D8, C42.6C4, C2×C4⋊Q8, C42.83D4
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, C23.37D4, C23.38D4, C42.83D4
Character table of C42.83D4
| 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 | 0 | 0 | -4 | 4 | 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 | 4 | -4 | 0 | 0 | 0 | 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 | 4 | -4 | 0 | 0 | 0 | 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 |
| ρ24 | 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 | symplectic lifted from C4.10D4, 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 | 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 |
►On 64 points
Generators in S64
(1 9 57 17)(2 14 58 22)(3 11 59 19)(4 16 60 24)(5 13 61 21)(6 10 62 18)(7 15 63 23)(8 12 64 20)(25 40 46 53)(26 37 47 50)(27 34 48 55)(28 39 41 52)(29 36 42 49)(30 33 43 54)(31 38 44 51)(32 35 45 56)
(1 15 61 19)(2 20 62 16)(3 9 63 21)(4 22 64 10)(5 11 57 23)(6 24 58 12)(7 13 59 17)(8 18 60 14)(25 51 42 34)(26 35 43 52)(27 53 44 36)(28 37 45 54)(29 55 46 38)(30 39 47 56)(31 49 48 40)(32 33 41 50)
(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 15 40 61 31 19 49)(2 52 20 26 62 35 16 43)(3 46 9 38 63 29 21 55)(4 50 22 32 64 33 10 41)(5 44 11 36 57 27 23 53)(6 56 24 30 58 39 12 47)(7 42 13 34 59 25 17 51)(8 54 18 28 60 37 14 45)
G:=sub<Sym(64)| (1,9,57,17)(2,14,58,22)(3,11,59,19)(4,16,60,24)(5,13,61,21)(6,10,62,18)(7,15,63,23)(8,12,64,20)(25,40,46,53)(26,37,47,50)(27,34,48,55)(28,39,41,52)(29,36,42,49)(30,33,43,54)(31,38,44,51)(32,35,45,56), (1,15,61,19)(2,20,62,16)(3,9,63,21)(4,22,64,10)(5,11,57,23)(6,24,58,12)(7,13,59,17)(8,18,60,14)(25,51,42,34)(26,35,43,52)(27,53,44,36)(28,37,45,54)(29,55,46,38)(30,39,47,56)(31,49,48,40)(32,33,41,50), (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), (1,48,15,40,61,31,19,49)(2,52,20,26,62,35,16,43)(3,46,9,38,63,29,21,55)(4,50,22,32,64,33,10,41)(5,44,11,36,57,27,23,53)(6,56,24,30,58,39,12,47)(7,42,13,34,59,25,17,51)(8,54,18,28,60,37,14,45)>;
G:=Group( (1,9,57,17)(2,14,58,22)(3,11,59,19)(4,16,60,24)(5,13,61,21)(6,10,62,18)(7,15,63,23)(8,12,64,20)(25,40,46,53)(26,37,47,50)(27,34,48,55)(28,39,41,52)(29,36,42,49)(30,33,43,54)(31,38,44,51)(32,35,45,56), (1,15,61,19)(2,20,62,16)(3,9,63,21)(4,22,64,10)(5,11,57,23)(6,24,58,12)(7,13,59,17)(8,18,60,14)(25,51,42,34)(26,35,43,52)(27,53,44,36)(28,37,45,54)(29,55,46,38)(30,39,47,56)(31,49,48,40)(32,33,41,50), (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), (1,48,15,40,61,31,19,49)(2,52,20,26,62,35,16,43)(3,46,9,38,63,29,21,55)(4,50,22,32,64,33,10,41)(5,44,11,36,57,27,23,53)(6,56,24,30,58,39,12,47)(7,42,13,34,59,25,17,51)(8,54,18,28,60,37,14,45) );
G=PermutationGroup([[(1,9,57,17),(2,14,58,22),(3,11,59,19),(4,16,60,24),(5,13,61,21),(6,10,62,18),(7,15,63,23),(8,12,64,20),(25,40,46,53),(26,37,47,50),(27,34,48,55),(28,39,41,52),(29,36,42,49),(30,33,43,54),(31,38,44,51),(32,35,45,56)], [(1,15,61,19),(2,20,62,16),(3,9,63,21),(4,22,64,10),(5,11,57,23),(6,24,58,12),(7,13,59,17),(8,18,60,14),(25,51,42,34),(26,35,43,52),(27,53,44,36),(28,37,45,54),(29,55,46,38),(30,39,47,56),(31,49,48,40),(32,33,41,50)], [(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)], [(1,48,15,40,61,31,19,49),(2,52,20,26,62,35,16,43),(3,46,9,38,63,29,21,55),(4,50,22,32,64,33,10,41),(5,44,11,36,57,27,23,53),(6,56,24,30,58,39,12,47),(7,42,13,34,59,25,17,51),(8,54,18,28,60,37,14,45)]])
Matrix representation of C42.83D4 ►in GL8(𝔽17)
| 16 | 15 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 13 | 13 | 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 4 | 1 | 0 | 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 | 8 | 4 | 0 |
| 0 | 0 | 0 | 0 | 2 | 0 | 0 | 13 |
| 16 | 15 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 13 | 0 | 1 | 0 | 0 | 0 | 0 |
| 4 | 4 | 16 | 0 | 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 |
| 10 | 0 | 6 | 11 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 6 | 0 | 0 | 0 | 0 |
| 6 | 3 | 12 | 5 | 0 | 0 | 0 | 0 |
| 14 | 3 | 5 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 15 | 0 | 15 |
| 0 | 0 | 0 | 0 | 2 | 2 | 2 | 0 |
| 0 | 0 | 0 | 0 | 6 | 15 | 15 | 2 |
| 0 | 0 | 0 | 0 | 11 | 6 | 15 | 8 |
| 4 | 0 | 15 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 13 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 15 | 15 | 15 | 0 |
| 0 | 0 | 0 | 0 | 9 | 15 | 0 | 15 |
| 0 | 0 | 0 | 0 | 1 | 4 | 2 | 2 |
| 0 | 0 | 0 | 0 | 16 | 1 | 8 | 2 |
G:=sub<GL(8,GF(17))| [16,1,13,0,0,0,0,0,15,1,13,4,0,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,4,0,0,2,0,0,0,0,0,13,8,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,13],[16,1,0,4,0,0,0,0,15,1,13,4,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[10,12,6,14,0,0,0,0,0,0,3,3,0,0,0,0,6,0,12,5,0,0,0,0,11,6,5,12,0,0,0,0,0,0,0,0,9,2,6,11,0,0,0,0,15,2,15,6,0,0,0,0,0,2,15,15,0,0,0,0,15,0,2,8],[4,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,15,1,13,4,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,15,9,1,16,0,0,0,0,15,15,4,1,0,0,0,0,15,0,2,8,0,0,0,0,0,15,2,2] >;
C42.83D4 in GAP, Magma, Sage, TeX
C_4^2._{83}D_4 % in TeX
G:=Group("C4^2.83D4"); // GroupNames label
G:=SmallGroup(128,288);
// by ID
G=gap.SmallGroup(128,288);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,232,1430,1123,1018,248,1971,242]);
// 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*c^3>;
// generators/relations
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