p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.71D4, C42.152C23, C42.93(C2×C4), (C22×C4).229D4, C42.C2.11C4, C8⋊C4.87C22, C42.6C4.21C2, (C2×C42).196C22, C42.C2.98C22, C23.180(C22⋊C4), C42.2C22⋊10C2, C2.34(C42⋊C22), C22.10(C4.10D4), (C2×C4⋊C4).18C4, C4⋊C4.29(C2×C4), (C2×C4).1180(C2×D4), (C2×C42.C2).4C2, (C2×C4).97(C22⋊C4), (C2×C4).146(C22×C4), (C22×C4).218(C2×C4), C2.13(C2×C4.10D4), C22.210(C2×C22⋊C4), SmallGroup(128,266)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.71D4
G = < a,b,c,d | a4=b4=1, c4=a2b2, d2=b, ab=ba, cac-1=ab2, dad-1=a-1b2, cbc-1=a2b-1, bd=db, dcd-1=a2bc3 >
Subgroups: 180 in 98 conjugacy classes, 44 normal (12 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, C23, C42, C42, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C22×C4, C8⋊C4, C22⋊C8, C4⋊C8, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C42.C2, C42.C2, C42.2C22, C42.6C4, C2×C42.C2, C42.71D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C4.10D4, C2×C22⋊C4, C2×C4.10D4, C42⋊C22, C42.71D4
Character table of C42.71D4
| 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 | symplectic lifted from C4.10D4, Schur index 2 |
| 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 | symplectic lifted from C4.10D4, Schur index 2 |
| ρ23 | 4 | 4 | -4 | -4 | 0 | 0 | 4i | -4i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C42⋊C22 |
| ρ24 | 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 C42⋊C22 |
| ρ25 | 4 | 4 | -4 | -4 | 0 | 0 | -4i | 4i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C42⋊C22 |
| 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 C42⋊C22 |
►On 64 points
Generators in S64
(1 58 20 12)(2 9 21 63)(3 60 22 14)(4 11 23 57)(5 62 24 16)(6 13 17 59)(7 64 18 10)(8 15 19 61)(25 34 52 44)(26 41 53 39)(27 36 54 46)(28 43 55 33)(29 38 56 48)(30 45 49 35)(31 40 50 42)(32 47 51 37)
(1 10 24 60)(2 15 17 57)(3 12 18 62)(4 9 19 59)(5 14 20 64)(6 11 21 61)(7 16 22 58)(8 13 23 63)(25 40 56 46)(26 37 49 43)(27 34 50 48)(28 39 51 45)(29 36 52 42)(30 33 53 47)(31 38 54 44)(32 35 55 41)
(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 38 10 54 24 44 60 31)(2 26 15 37 17 49 57 43)(3 46 12 25 18 40 62 56)(4 51 9 45 19 28 59 39)(5 34 14 50 20 48 64 27)(6 30 11 33 21 53 61 47)(7 42 16 29 22 36 58 52)(8 55 13 41 23 32 63 35)
G:=sub<Sym(64)| (1,58,20,12)(2,9,21,63)(3,60,22,14)(4,11,23,57)(5,62,24,16)(6,13,17,59)(7,64,18,10)(8,15,19,61)(25,34,52,44)(26,41,53,39)(27,36,54,46)(28,43,55,33)(29,38,56,48)(30,45,49,35)(31,40,50,42)(32,47,51,37), (1,10,24,60)(2,15,17,57)(3,12,18,62)(4,9,19,59)(5,14,20,64)(6,11,21,61)(7,16,22,58)(8,13,23,63)(25,40,56,46)(26,37,49,43)(27,34,50,48)(28,39,51,45)(29,36,52,42)(30,33,53,47)(31,38,54,44)(32,35,55,41), (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), (1,38,10,54,24,44,60,31)(2,26,15,37,17,49,57,43)(3,46,12,25,18,40,62,56)(4,51,9,45,19,28,59,39)(5,34,14,50,20,48,64,27)(6,30,11,33,21,53,61,47)(7,42,16,29,22,36,58,52)(8,55,13,41,23,32,63,35)>;
G:=Group( (1,58,20,12)(2,9,21,63)(3,60,22,14)(4,11,23,57)(5,62,24,16)(6,13,17,59)(7,64,18,10)(8,15,19,61)(25,34,52,44)(26,41,53,39)(27,36,54,46)(28,43,55,33)(29,38,56,48)(30,45,49,35)(31,40,50,42)(32,47,51,37), (1,10,24,60)(2,15,17,57)(3,12,18,62)(4,9,19,59)(5,14,20,64)(6,11,21,61)(7,16,22,58)(8,13,23,63)(25,40,56,46)(26,37,49,43)(27,34,50,48)(28,39,51,45)(29,36,52,42)(30,33,53,47)(31,38,54,44)(32,35,55,41), (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), (1,38,10,54,24,44,60,31)(2,26,15,37,17,49,57,43)(3,46,12,25,18,40,62,56)(4,51,9,45,19,28,59,39)(5,34,14,50,20,48,64,27)(6,30,11,33,21,53,61,47)(7,42,16,29,22,36,58,52)(8,55,13,41,23,32,63,35) );
G=PermutationGroup([[(1,58,20,12),(2,9,21,63),(3,60,22,14),(4,11,23,57),(5,62,24,16),(6,13,17,59),(7,64,18,10),(8,15,19,61),(25,34,52,44),(26,41,53,39),(27,36,54,46),(28,43,55,33),(29,38,56,48),(30,45,49,35),(31,40,50,42),(32,47,51,37)], [(1,10,24,60),(2,15,17,57),(3,12,18,62),(4,9,19,59),(5,14,20,64),(6,11,21,61),(7,16,22,58),(8,13,23,63),(25,40,56,46),(26,37,49,43),(27,34,50,48),(28,39,51,45),(29,36,52,42),(30,33,53,47),(31,38,54,44),(32,35,55,41)], [(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)], [(1,38,10,54,24,44,60,31),(2,26,15,37,17,49,57,43),(3,46,12,25,18,40,62,56),(4,51,9,45,19,28,59,39),(5,34,14,50,20,48,64,27),(6,30,11,33,21,53,61,47),(7,42,16,29,22,36,58,52),(8,55,13,41,23,32,63,35)]])
Matrix representation of C42.71D4 ►in GL8(𝔽17)
| 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 |
| 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 13 | 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 | 0 | 16 | 16 |
| 0 | 0 | 0 | 0 | 0 | 0 | 2 | 1 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 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 | 4 |
| 0 | 0 | 4 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 10 | 13 | 0 | 0 | 0 | 0 |
| 1 | 6 | 0 | 0 | 0 | 0 | 0 | 0 |
| 11 | 16 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 7 | 12 |
| 0 | 0 | 0 | 0 | 0 | 0 | 10 | 10 |
| 0 | 0 | 0 | 0 | 0 | 14 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 | 0 | 0 |
G:=sub<GL(8,GF(17))| [0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,0,16,2,0,0,0,0,0,0,16,1,0,0,0,0,0,0,0,0,16,2,0,0,0,0,0,0,16,1],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[0,0,1,11,0,0,0,0,0,0,6,16,0,0,0,0,4,10,0,0,0,0,0,0,7,13,0,0,0,0,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,14,0,0,0,0,0,7,10,0,0,0,0,0,0,12,10,0,0],[0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0] >;
C42.71D4 in GAP, Magma, Sage, TeX
C_4^2._{71}D_4 % in TeX
G:=Group("C4^2.71D4"); // GroupNames label
G:=SmallGroup(128,266);
// by ID
G=gap.SmallGroup(128,266);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,232,1430,1123,1018,248,1971,102]);
// 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*b^2,d*a*d^-1=a^-1*b^2,c*b*c^-1=a^2*b^-1,b*d=d*b,d*c*d^-1=a^2*b*c^3>;
// generators/relations
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