p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.254C23, C4⋊C4.76D4, C8⋊2C8⋊21C2, (C2×D4).66D4, C8⋊6D4.9C2, (C2×C8).192D4, D4⋊Q8.9C2, C4⋊Q8.75C22, C4.109(C4○D8), C4.10D8⋊31C2, C4⋊C8.192C22, C4.94(C8⋊C22), (C4×C8).287C22, D4⋊2Q8.12C2, C4.SD16⋊33C2, C4.6Q16⋊10C2, (C4×D4).52C22, C2.10(C8.D4), C4.46(C8.C22), C2.22(D4.3D4), C2.15(D4.2D4), C22.215(C4⋊D4), (C2×C4).39(C4○D4), (C2×C4).1289(C2×D4), SmallGroup(128,435)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.254C23
G = < a,b,c,d,e | a4=b4=e2=1, c2=a2, d2=a-1b2, ab=ba, cac-1=a-1, ad=da, ae=ea, cbc-1=ebe=b-1, bd=db, dcd-1=a-1c, ece=bc, ede=a2d >
Subgroups: 176 in 77 conjugacy classes, 32 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, D4, Q8, C23, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), C22×C4, C2×D4, C2×Q8, C4×C8, C22⋊C8, D4⋊C4, Q8⋊C4, C4⋊C8, C4.Q8, C2.D8, C4×D4, C4⋊Q8, C2×M4(2), C4.10D8, C4.6Q16, C8⋊2C8, C8⋊6D4, D4⋊Q8, D4⋊2Q8, C4.SD16, C42.254C23
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C4⋊D4, C4○D8, C8⋊C22, C8.C22, D4.2D4, C8.D4, D4.3D4, C42.254C23
Character table of C42.254C23
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 8I | 8J | |
| size | 1 | 1 | 1 | 1 | 8 | 2 | 2 | 2 | 2 | 4 | 8 | 16 | 16 | 4 | 4 | 4 | 4 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | linear of order 2 |
| ρ9 | 2 | 2 | 2 | 2 | 0 | 2 | -2 | 2 | -2 | -2 | 0 | 0 | 0 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ10 | 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 |
| ρ11 | 2 | 2 | 2 | 2 | 0 | 2 | -2 | 2 | -2 | -2 | 0 | 0 | 0 | -2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ12 | 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 |
| ρ13 | 2 | 2 | 2 | 2 | 0 | -2 | -2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2i | -2i | 0 | complex lifted from C4○D4 |
| ρ14 | 2 | 2 | 2 | 2 | 0 | -2 | -2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2i | 2i | 0 | complex lifted from C4○D4 |
| ρ15 | 2 | -2 | -2 | 2 | 0 | -2 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 2i | -2i | 0 | √-2 | -√2 | -√-2 | 0 | 0 | √2 | complex lifted from C4○D8 |
| ρ16 | 2 | -2 | -2 | 2 | 0 | -2 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2i | 2i | 0 | -√-2 | -√2 | √-2 | 0 | 0 | √2 | complex lifted from C4○D8 |
| ρ17 | 2 | -2 | -2 | 2 | 0 | -2 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 2i | -2i | 0 | -√-2 | √2 | √-2 | 0 | 0 | -√2 | complex lifted from C4○D8 |
| ρ18 | 2 | -2 | -2 | 2 | 0 | -2 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2i | 2i | 0 | √-2 | √2 | -√-2 | 0 | 0 | -√2 | complex lifted from C4○D8 |
| ρ19 | 4 | -4 | -4 | 4 | 0 | 4 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C8⋊C22 |
| ρ20 | 4 | -4 | 4 | -4 | 0 | 0 | 4 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from C8.C22, Schur index 2 |
| ρ21 | 4 | -4 | 4 | -4 | 0 | 0 | -4 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from C8.C22, Schur index 2 |
| ρ22 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2√-2 | 0 | 0 | -2√-2 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from D4.3D4 |
| ρ23 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2√-2 | 0 | 0 | 2√-2 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from D4.3D4 |
►On 64 points
Generators in S64
(1 47 5 43)(2 48 6 44)(3 41 7 45)(4 42 8 46)(9 31 13 27)(10 32 14 28)(11 25 15 29)(12 26 16 30)(17 64 21 60)(18 57 22 61)(19 58 23 62)(20 59 24 63)(33 54 37 50)(34 55 38 51)(35 56 39 52)(36 49 40 53)
(1 50 41 35)(2 51 42 36)(3 52 43 37)(4 53 44 38)(5 54 45 39)(6 55 46 40)(7 56 47 33)(8 49 48 34)(9 21 25 62)(10 22 26 63)(11 23 27 64)(12 24 28 57)(13 17 29 58)(14 18 30 59)(15 19 31 60)(16 20 32 61)
(1 64 5 60)(2 22 6 18)(3 62 7 58)(4 20 8 24)(9 33 13 37)(10 55 14 51)(11 39 15 35)(12 53 16 49)(17 43 21 47)(19 41 23 45)(25 56 29 52)(26 40 30 36)(27 54 31 50)(28 38 32 34)(42 63 46 59)(44 61 48 57)
(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)
(2 6)(4 8)(9 21)(10 18)(11 23)(12 20)(13 17)(14 22)(15 19)(16 24)(25 62)(26 59)(27 64)(28 61)(29 58)(30 63)(31 60)(32 57)(33 56)(34 53)(35 50)(36 55)(37 52)(38 49)(39 54)(40 51)(42 46)(44 48)
G:=sub<Sym(64)| (1,47,5,43)(2,48,6,44)(3,41,7,45)(4,42,8,46)(9,31,13,27)(10,32,14,28)(11,25,15,29)(12,26,16,30)(17,64,21,60)(18,57,22,61)(19,58,23,62)(20,59,24,63)(33,54,37,50)(34,55,38,51)(35,56,39,52)(36,49,40,53), (1,50,41,35)(2,51,42,36)(3,52,43,37)(4,53,44,38)(5,54,45,39)(6,55,46,40)(7,56,47,33)(8,49,48,34)(9,21,25,62)(10,22,26,63)(11,23,27,64)(12,24,28,57)(13,17,29,58)(14,18,30,59)(15,19,31,60)(16,20,32,61), (1,64,5,60)(2,22,6,18)(3,62,7,58)(4,20,8,24)(9,33,13,37)(10,55,14,51)(11,39,15,35)(12,53,16,49)(17,43,21,47)(19,41,23,45)(25,56,29,52)(26,40,30,36)(27,54,31,50)(28,38,32,34)(42,63,46,59)(44,61,48,57), (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), (2,6)(4,8)(9,21)(10,18)(11,23)(12,20)(13,17)(14,22)(15,19)(16,24)(25,62)(26,59)(27,64)(28,61)(29,58)(30,63)(31,60)(32,57)(33,56)(34,53)(35,50)(36,55)(37,52)(38,49)(39,54)(40,51)(42,46)(44,48)>;
G:=Group( (1,47,5,43)(2,48,6,44)(3,41,7,45)(4,42,8,46)(9,31,13,27)(10,32,14,28)(11,25,15,29)(12,26,16,30)(17,64,21,60)(18,57,22,61)(19,58,23,62)(20,59,24,63)(33,54,37,50)(34,55,38,51)(35,56,39,52)(36,49,40,53), (1,50,41,35)(2,51,42,36)(3,52,43,37)(4,53,44,38)(5,54,45,39)(6,55,46,40)(7,56,47,33)(8,49,48,34)(9,21,25,62)(10,22,26,63)(11,23,27,64)(12,24,28,57)(13,17,29,58)(14,18,30,59)(15,19,31,60)(16,20,32,61), (1,64,5,60)(2,22,6,18)(3,62,7,58)(4,20,8,24)(9,33,13,37)(10,55,14,51)(11,39,15,35)(12,53,16,49)(17,43,21,47)(19,41,23,45)(25,56,29,52)(26,40,30,36)(27,54,31,50)(28,38,32,34)(42,63,46,59)(44,61,48,57), (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), (2,6)(4,8)(9,21)(10,18)(11,23)(12,20)(13,17)(14,22)(15,19)(16,24)(25,62)(26,59)(27,64)(28,61)(29,58)(30,63)(31,60)(32,57)(33,56)(34,53)(35,50)(36,55)(37,52)(38,49)(39,54)(40,51)(42,46)(44,48) );
G=PermutationGroup([[(1,47,5,43),(2,48,6,44),(3,41,7,45),(4,42,8,46),(9,31,13,27),(10,32,14,28),(11,25,15,29),(12,26,16,30),(17,64,21,60),(18,57,22,61),(19,58,23,62),(20,59,24,63),(33,54,37,50),(34,55,38,51),(35,56,39,52),(36,49,40,53)], [(1,50,41,35),(2,51,42,36),(3,52,43,37),(4,53,44,38),(5,54,45,39),(6,55,46,40),(7,56,47,33),(8,49,48,34),(9,21,25,62),(10,22,26,63),(11,23,27,64),(12,24,28,57),(13,17,29,58),(14,18,30,59),(15,19,31,60),(16,20,32,61)], [(1,64,5,60),(2,22,6,18),(3,62,7,58),(4,20,8,24),(9,33,13,37),(10,55,14,51),(11,39,15,35),(12,53,16,49),(17,43,21,47),(19,41,23,45),(25,56,29,52),(26,40,30,36),(27,54,31,50),(28,38,32,34),(42,63,46,59),(44,61,48,57)], [(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)], [(2,6),(4,8),(9,21),(10,18),(11,23),(12,20),(13,17),(14,22),(15,19),(16,24),(25,62),(26,59),(27,64),(28,61),(29,58),(30,63),(31,60),(32,57),(33,56),(34,53),(35,50),(36,55),(37,52),(38,49),(39,54),(40,51),(42,46),(44,48)]])
Matrix representation of C42.254C23 ►in GL6(𝔽17)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 15 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 15 |
| 0 | 0 | 0 | 0 | 1 | 1 |
| 3 | 14 | 0 | 0 | 0 | 0 |
| 14 | 14 | 0 | 0 | 0 | 0 |
| 0 | 0 | 10 | 10 | 0 | 0 |
| 0 | 0 | 12 | 7 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 7 |
| 0 | 0 | 0 | 0 | 5 | 10 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 7 | 12 | 7 |
| 0 | 0 | 5 | 5 | 5 | 5 |
| 0 | 0 | 5 | 10 | 12 | 7 |
| 0 | 0 | 12 | 12 | 5 | 5 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 16 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 16 | 16 |
G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,1,0,0,0,0,0,0,1,0,0],[0,16,0,0,0,0,1,0,0,0,0,0,0,0,16,1,0,0,0,0,15,1,0,0,0,0,0,0,16,1,0,0,0,0,15,1],[3,14,0,0,0,0,14,14,0,0,0,0,0,0,10,12,0,0,0,0,10,7,0,0,0,0,0,0,7,5,0,0,0,0,7,10],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,12,5,5,12,0,0,7,5,10,12,0,0,12,5,12,5,0,0,7,5,7,5],[1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,16,0,0,0,0,0,16,0,0,0,0,0,0,1,16,0,0,0,0,0,16] >;
C42.254C23 in GAP, Magma, Sage, TeX
C_4^2._{254}C_2^3 % in TeX
G:=Group("C4^2.254C2^3"); // GroupNames label
G:=SmallGroup(128,435);
// by ID
G=gap.SmallGroup(128,435);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,2,224,141,736,422,387,1123,136,2804,718,172]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=e^2=1,c^2=a^2,d^2=a^-1*b^2,a*b=b*a,c*a*c^-1=a^-1,a*d=d*a,a*e=e*a,c*b*c^-1=e*b*e=b^-1,b*d=d*b,d*c*d^-1=a^-1*c,e*c*e=b*c,e*d*e=a^2*d>;
// generators/relations
Export