p-group, metabelian, nilpotent (class 3), monomial
Aliases: C4.132+ 1+4, C8⋊8D4⋊3C2, C8⋊7D4⋊18C2, C8⋊D4⋊16C2, C8⋊2D4⋊12C2, C22⋊D8⋊19C2, D4⋊D4⋊23C2, C2.D8⋊8C22, (C2×D4).153D4, C22⋊SD16⋊9C2, C2.27(D4○D8), C4⋊D4⋊5C22, (C2×Q8).129D4, C4.Q8⋊20C22, C22⋊Q8⋊5C22, C4⋊C4.137C23, C22⋊C8⋊16C22, (C2×C8).154C23, (C2×C4).396C24, (C22×C8)⋊17C22, Q8⋊C4⋊4C22, (C2×D8).68C22, C23.280(C2×D4), D4⋊C4⋊29C22, C2.41(D4○SD16), (C2×D4).147C23, C22.D8⋊21C2, (C2×Q8).135C23, C22.29C24⋊14C2, C23.20D4⋊24C2, C23.19D4⋊24C2, C23.46D4⋊10C2, C2.77(C23⋊3D4), (C2×M4(2))⋊16C22, (C22×C4).299C23, (C2×SD16).80C22, C22.656(C22×D4), C22.31C24⋊8C2, (C22×D4).384C22, C42⋊C2.154C22, (C2×C4⋊C4)⋊54C22, (C2×C4).150(C2×D4), (C2×C4○D4)⋊10C22, (C22×C8)⋊C2⋊12C2, SmallGroup(128,1930)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C4.2+ 1+4
G = < a,b,c,d,e | a4=c2=e2=1, b4=a2, d2=ab2, dbd-1=ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc=a-1b3, be=eb, dcd-1=ece=a2c, ede=ab2d >
Subgroups: 516 in 211 conjugacy classes, 84 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), D8, SD16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C24, C22⋊C8, D4⋊C4, Q8⋊C4, C4.Q8, C2.D8, C2×C4⋊C4, C42⋊C2, C22≀C2, C4⋊D4, C4⋊D4, C22⋊Q8, C22⋊Q8, C4.4D4, C4⋊1D4, C22×C8, C2×M4(2), C2×D8, C2×SD16, C22×D4, C2×C4○D4, (C22×C8)⋊C2, C22⋊D8, D4⋊D4, C22⋊SD16, C8⋊8D4, C8⋊7D4, C8⋊D4, C8⋊2D4, C22.D8, C23.46D4, C23.19D4, C23.20D4, C22.29C24, C22.31C24, C4.2+ 1+4
Quotients: C1, C2, C22, D4, C23, C2×D4, C24, C22×D4, 2+ 1+4, C23⋊3D4, D4○D8, D4○SD16, C4.2+ 1+4
Character table of C4.2+ 1+4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 8A | 8B | 8C | 8D | 8E | 8F | |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 8 | 8 | 8 | 2 | 2 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 4 | 4 | 4 | 4 | 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 | -1 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | linear of order 2 |
| ρ10 | 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 |
| ρ11 | 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 |
| ρ12 | 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 |
| ρ13 | 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 |
| ρ14 | 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 |
| ρ15 | 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 |
| ρ16 | 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 |
| ρ17 | 2 | 2 | 2 | 2 | 2 | -2 | 2 | 0 | 0 | 0 | -2 | -2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ18 | 2 | 2 | 2 | 2 | -2 | -2 | -2 | 0 | 0 | 0 | -2 | -2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ19 | 2 | 2 | 2 | 2 | -2 | 2 | 2 | 0 | 0 | 0 | -2 | -2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ20 | 2 | 2 | 2 | 2 | 2 | 2 | -2 | 0 | 0 | 0 | -2 | -2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ21 | 4 | -4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from 2+ 1+4 |
| ρ22 | 4 | -4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from 2+ 1+4 |
| ρ23 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2√2 | 0 | 2√2 | 0 | 0 | orthogonal lifted from D4○D8 |
| ρ24 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2√2 | 0 | -2√2 | 0 | 0 | orthogonal lifted from D4○D8 |
| ρ25 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2√-2 | 0 | -2√-2 | 0 | 0 | 0 | complex lifted from D4○SD16 |
| ρ26 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2√-2 | 0 | 2√-2 | 0 | 0 | 0 | complex lifted from D4○SD16 |
►On 32 points
(1 30 5 26)(2 31 6 27)(3 32 7 28)(4 25 8 29)(9 22 13 18)(10 23 14 19)(11 24 15 20)(12 17 16 21)
(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)
(2 29)(4 31)(6 25)(8 27)(9 24)(10 14)(11 18)(12 16)(13 20)(15 22)(17 21)(19 23)
(1 23 32 16)(2 15 25 22)(3 21 26 14)(4 13 27 20)(5 19 28 12)(6 11 29 18)(7 17 30 10)(8 9 31 24)
(1 14)(2 15)(3 16)(4 9)(5 10)(6 11)(7 12)(8 13)(17 28)(18 29)(19 30)(20 31)(21 32)(22 25)(23 26)(24 27)
G:=sub<Sym(32)| (1,30,5,26)(2,31,6,27)(3,32,7,28)(4,25,8,29)(9,22,13,18)(10,23,14,19)(11,24,15,20)(12,17,16,21), (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), (2,29)(4,31)(6,25)(8,27)(9,24)(10,14)(11,18)(12,16)(13,20)(15,22)(17,21)(19,23), (1,23,32,16)(2,15,25,22)(3,21,26,14)(4,13,27,20)(5,19,28,12)(6,11,29,18)(7,17,30,10)(8,9,31,24), (1,14)(2,15)(3,16)(4,9)(5,10)(6,11)(7,12)(8,13)(17,28)(18,29)(19,30)(20,31)(21,32)(22,25)(23,26)(24,27)>;
G:=Group( (1,30,5,26)(2,31,6,27)(3,32,7,28)(4,25,8,29)(9,22,13,18)(10,23,14,19)(11,24,15,20)(12,17,16,21), (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), (2,29)(4,31)(6,25)(8,27)(9,24)(10,14)(11,18)(12,16)(13,20)(15,22)(17,21)(19,23), (1,23,32,16)(2,15,25,22)(3,21,26,14)(4,13,27,20)(5,19,28,12)(6,11,29,18)(7,17,30,10)(8,9,31,24), (1,14)(2,15)(3,16)(4,9)(5,10)(6,11)(7,12)(8,13)(17,28)(18,29)(19,30)(20,31)(21,32)(22,25)(23,26)(24,27) );
G=PermutationGroup([[(1,30,5,26),(2,31,6,27),(3,32,7,28),(4,25,8,29),(9,22,13,18),(10,23,14,19),(11,24,15,20),(12,17,16,21)], [(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)], [(2,29),(4,31),(6,25),(8,27),(9,24),(10,14),(11,18),(12,16),(13,20),(15,22),(17,21),(19,23)], [(1,23,32,16),(2,15,25,22),(3,21,26,14),(4,13,27,20),(5,19,28,12),(6,11,29,18),(7,17,30,10),(8,9,31,24)], [(1,14),(2,15),(3,16),(4,9),(5,10),(6,11),(7,12),(8,13),(17,28),(18,29),(19,30),(20,31),(21,32),(22,25),(23,26),(24,27)]])
Matrix representation of C4.2+ 1+4 ►in GL8(𝔽17)
| 1 | 2 | 0 | 0 | 0 | 0 | 0 | 0 |
| 16 | 16 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 16 | 0 | 0 | 0 | 0 |
| 16 | 16 | 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 |
| 0 | 10 | 0 | 0 | 0 | 0 | 0 | 0 |
| 12 | 7 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 5 | 12 | 12 | 0 | 0 | 0 | 0 |
| 12 | 12 | 5 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 0 | 13 | 13 |
| 0 | 0 | 0 | 0 | 7 | 0 | 13 | 15 |
| 0 | 0 | 0 | 0 | 11 | 9 | 8 | 8 |
| 0 | 0 | 0 | 0 | 4 | 8 | 4 | 4 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 16 | 0 | 0 | 0 | 0 | 0 |
| 16 | 0 | 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 13 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 15 | 0 | 0 | 16 |
| 16 | 0 | 0 | 15 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 | 0 | 0 |
| 16 | 16 | 0 | 16 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 13 | 16 | 16 |
| 0 | 0 | 0 | 0 | 6 | 15 | 2 | 1 |
| 1 | 0 | 15 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 15 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 4 | 16 | 16 |
| 0 | 0 | 0 | 0 | 0 | 2 | 0 | 1 |
G:=sub<GL(8,GF(17))| [1,16,0,16,0,0,0,0,2,16,1,16,0,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,12,0,12,0,0,0,0,10,7,5,12,0,0,0,0,0,0,12,5,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,5,7,11,4,0,0,0,0,0,0,9,8,0,0,0,0,13,13,8,4,0,0,0,0,13,15,8,4],[1,0,1,16,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,13,15,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16],[16,0,16,1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,15,1,16,1,0,0,0,0,0,0,0,0,16,16,16,6,0,0,0,0,2,1,13,15,0,0,0,0,0,0,16,2,0,0,0,0,0,0,16,1],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,15,1,16,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,12,0,0,0,0,0,15,16,4,2,0,0,0,0,0,0,16,0,0,0,0,0,0,0,16,1] >;
C4.2+ 1+4 in GAP, Magma, Sage, TeX
C_4.2_+^{1+4} % in TeX
G:=Group("C4.ES+(2,2)"); // GroupNames label
G:=SmallGroup(128,1930);
// by ID
G=gap.SmallGroup(128,1930);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,758,219,675,1018,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=c^2=e^2=1,b^4=a^2,d^2=a*b^2,d*b*d^-1=a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,a*e=e*a,c*b*c=a^-1*b^3,b*e=e*b,d*c*d^-1=e*c*e=a^2*c,e*d*e=a*b^2*d>;
// generators/relations
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