p-group, metabelian, nilpotent (class 3), monomial
Aliases: C4.142+ 1+4, C8⋊7D4⋊5C2, C8⋊2D4⋊13C2, C22⋊D8⋊20C2, C2.D8⋊9C22, (C2×D4).154D4, C2.28(D4○D8), C4⋊D4⋊6C22, (C2×Q8).130D4, C23.89(C2×D4), C4.Q8⋊21C22, C4⋊C4.138C23, C22⋊C8⋊17C22, (C2×C4).397C24, (C2×C8).155C23, (C22×C8)⋊10C22, (C2×D8).21C22, D4⋊C4⋊30C22, (C2×D4).148C23, (C22×D4)⋊25C22, C42⋊C2⋊18C22, C22.29C24⋊15C2, C23.19D4⋊25C2, C2.78(C23⋊3D4), (C2×M4(2))⋊17C22, (C22×C4).300C23, C22.657(C22×D4), (C2×C4).534(C2×D4), (C22×C8)⋊C2⋊13C2, (C2×C4○D4).165C22, SmallGroup(128,1931)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C4.142+ 1+4
G = < a,b,c,d,e | a4=c2=1, b4=e2=a2, d2=ab2, dbd-1=ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc=ab3, be=eb, dcd-1=ece-1=a2c, ede-1=a-1b2d >
Subgroups: 588 in 221 conjugacy classes, 84 normal (16 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C23, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), D8, C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C2×Q8, C4○D4, C24, C22⋊C8, D4⋊C4, C4.Q8, C2.D8, C42⋊C2, C22≀C2, C4⋊D4, C4.4D4, C4⋊1D4, C22×C8, C2×M4(2), C2×D8, C22×D4, C2×C4○D4, (C22×C8)⋊C2, C22⋊D8, C8⋊7D4, C8⋊2D4, C23.19D4, C22.29C24, C4.142+ 1+4
Quotients: C1, C2, C22, D4, C23, C2×D4, C24, C22×D4, 2+ 1+4, C23⋊3D4, D4○D8, C4.142+ 1+4
Character table of C4.142+ 1+4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 8A | 8B | 8C | 8D | 8E | 8F | |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 2 | 2 | 4 | 4 | 4 | 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 | 0 | -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 | -2 | -2 | 0 | 0 | 0 | 0 | -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 | -2 | 2 | 0 | 0 | 0 | 0 | -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 | 2 | 2 | 0 | 0 | 0 | 0 | -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 | 0 | 0 | 0 | 0 | -4 | 4 | 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 | 0 | 4 | -4 | 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 | orthogonal lifted from D4○D8 |
| 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 | orthogonal lifted from D4○D8 |
►On 32 points
(1 25 5 29)(2 26 6 30)(3 27 7 31)(4 28 8 32)(9 23 13 19)(10 24 14 20)(11 17 15 21)(12 18 16 22)
(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 28)(4 30)(6 32)(8 26)(9 13)(10 22)(11 15)(12 24)(14 18)(16 20)(17 21)(19 23)
(1 19 27 11)(2 10 28 18)(3 17 29 9)(4 16 30 24)(5 23 31 15)(6 14 32 22)(7 21 25 13)(8 12 26 20)
(1 9 5 13)(2 10 6 14)(3 11 7 15)(4 12 8 16)(17 31 21 27)(18 32 22 28)(19 25 23 29)(20 26 24 30)
G:=sub<Sym(32)| (1,25,5,29)(2,26,6,30)(3,27,7,31)(4,28,8,32)(9,23,13,19)(10,24,14,20)(11,17,15,21)(12,18,16,22), (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), (2,28)(4,30)(6,32)(8,26)(9,13)(10,22)(11,15)(12,24)(14,18)(16,20)(17,21)(19,23), (1,19,27,11)(2,10,28,18)(3,17,29,9)(4,16,30,24)(5,23,31,15)(6,14,32,22)(7,21,25,13)(8,12,26,20), (1,9,5,13)(2,10,6,14)(3,11,7,15)(4,12,8,16)(17,31,21,27)(18,32,22,28)(19,25,23,29)(20,26,24,30)>;
G:=Group( (1,25,5,29)(2,26,6,30)(3,27,7,31)(4,28,8,32)(9,23,13,19)(10,24,14,20)(11,17,15,21)(12,18,16,22), (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), (2,28)(4,30)(6,32)(8,26)(9,13)(10,22)(11,15)(12,24)(14,18)(16,20)(17,21)(19,23), (1,19,27,11)(2,10,28,18)(3,17,29,9)(4,16,30,24)(5,23,31,15)(6,14,32,22)(7,21,25,13)(8,12,26,20), (1,9,5,13)(2,10,6,14)(3,11,7,15)(4,12,8,16)(17,31,21,27)(18,32,22,28)(19,25,23,29)(20,26,24,30) );
G=PermutationGroup([[(1,25,5,29),(2,26,6,30),(3,27,7,31),(4,28,8,32),(9,23,13,19),(10,24,14,20),(11,17,15,21),(12,18,16,22)], [(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)], [(2,28),(4,30),(6,32),(8,26),(9,13),(10,22),(11,15),(12,24),(14,18),(16,20),(17,21),(19,23)], [(1,19,27,11),(2,10,28,18),(3,17,29,9),(4,16,30,24),(5,23,31,15),(6,14,32,22),(7,21,25,13),(8,12,26,20)], [(1,9,5,13),(2,10,6,14),(3,11,7,15),(4,12,8,16),(17,31,21,27),(18,32,22,28),(19,25,23,29),(20,26,24,30)]])
Matrix representation of C4.142+ 1+4 ►in GL8(𝔽17)
| 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 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 15 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 16 | 15 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 |
| 0 | 0 | 16 | 15 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 | 0 | 0 |
| 16 | 15 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 6 | 6 |
| 0 | 0 | 0 | 0 | 0 | 0 | 14 | 0 |
| 0 | 0 | 0 | 0 | 11 | 11 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 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 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 16 |
| 1 | 2 | 0 | 0 | 0 | 0 | 0 | 0 |
| 16 | 16 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 16 | 15 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 | 0 | 0 |
| 16 | 15 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 | 0 | 0 |
G:=sub<GL(8,GF(17))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,1,0,0,0,0,0,0,15,1,0,0,0,0,0,0,0,0,16,1,0,0,0,0,0,0,15,1],[0,0,16,1,0,0,0,0,0,0,15,1,0,0,0,0,16,1,0,0,0,0,0,0,15,1,0,0,0,0,0,0,0,0,0,0,0,0,11,3,0,0,0,0,0,0,11,0,0,0,0,0,6,14,0,0,0,0,0,0,6,0,0,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16],[1,16,0,0,0,0,0,0,2,16,0,0,0,0,0,0,0,0,1,16,0,0,0,0,0,0,2,16,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,2,16,0,0,0,0,16,0,0,0,0,0,0,0,15,1,0,0],[16,0,0,0,0,0,0,0,15,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,2,16,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0] >;
C4.142+ 1+4 in GAP, Magma, Sage, TeX
C_4._{14}2_+^{1+4} % in TeX
G:=Group("C4.14ES+(2,2)"); // GroupNames label
G:=SmallGroup(128,1931);
// by ID
G=gap.SmallGroup(128,1931);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,456,758,219,675,1018,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=c^2=1,b^4=e^2=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*b^3,b*e=e*b,d*c*d^-1=e*c*e^-1=a^2*c,e*d*e^-1=a^-1*b^2*d>;
// generators/relations
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