p-group, metabelian, nilpotent (class 3), monomial
Aliases: C4.152+ 1+4, C8⋊D4⋊17C2, C8⋊8D4⋊27C2, Q8⋊D4⋊9C2, (C2×D4).155D4, (C2×Q8).131D4, C23.90(C2×D4), C4.Q8⋊35C22, C2.D8⋊31C22, C22⋊Q8⋊6C22, C22⋊SD16⋊10C2, C4⋊C4.139C23, C22⋊C8⋊18C22, (C2×C4).398C24, (C2×C8).321C23, (C22×C8)⋊37C22, C2.42(D4○SD16), Q8⋊C4⋊32C22, (C2×D4).149C23, C4⋊D4.39C22, (C2×Q8).136C23, (C22×Q8)⋊21C22, C42⋊C2⋊19C22, C23.20D4⋊25C2, C23.19D4⋊26C2, C2.79(C23⋊3D4), (C2×M4(2))⋊18C22, (C22×C4).301C23, (C2×SD16).81C22, C22.658(C22×D4), D4⋊C4.104C22, C22.29C24.14C2, (C22×D4).385C22, C23.38C23⋊14C2, (C2×C4).535(C2×D4), (C22×C8)⋊C2⋊14C2, (C2×C4○D4).166C22, SmallGroup(128,1932)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C4.152+ 1+4
G = < a,b,c,d,e | a4=c2=1, b4=e2=a2, d2=a-1b2, dbd-1=ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc=a-1b3, be=eb, dcd-1=ece-1=a2c, ede-1=ab2d >
Subgroups: 476 in 205 conjugacy classes, 84 normal (30 characteristic)
C1, C2, 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), SD16, C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C2×Q8, C2×Q8, C2×Q8, C4○D4, C24, C22⋊C8, D4⋊C4, Q8⋊C4, C4.Q8, C2.D8, C42⋊C2, C22≀C2, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C4⋊1D4, C4⋊Q8, C22×C8, C2×M4(2), C2×SD16, C22×D4, C22×Q8, C2×C4○D4, (C22×C8)⋊C2, Q8⋊D4, C22⋊SD16, C8⋊8D4, C8⋊D4, C23.19D4, C23.20D4, C22.29C24, C23.38C23, C4.152+ 1+4
Quotients: C1, C2, C22, D4, C23, C2×D4, C24, C22×D4, 2+ 1+4, C23⋊3D4, D4○SD16, C4.152+ 1+4
Character table of C4.152+ 1+4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 8A | 8B | 8C | 8D | 8E | 8F | |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 8 | 8 | 2 | 2 | 4 | 4 | 4 | 8 | 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 | -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 | 0 | 0 | -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 | 0 | 0 | -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 | 0 | 0 | -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 | 0 | 0 | 0 | 4 | -4 | 0 | 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 | -4 | 4 | 0 | 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 | 0 | 2√-2 | -2√-2 | 0 | 0 | complex lifted from D4○SD16 |
| ρ24 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2√-2 | 2√-2 | 0 | 0 | complex lifted from D4○SD16 |
| ρ25 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2√-2 | -2√-2 | 0 | 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 | 2√-2 | 0 | 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 17 13 21)(10 18 14 22)(11 19 15 23)(12 20 16 24)
(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 13)(10 20)(11 15)(12 22)(14 24)(16 18)(17 21)(19 23)
(1 23 28 9)(2 12 29 18)(3 21 30 15)(4 10 31 24)(5 19 32 13)(6 16 25 22)(7 17 26 11)(8 14 27 20)
(1 11 5 15)(2 12 6 16)(3 13 7 9)(4 14 8 10)(17 32 21 28)(18 25 22 29)(19 26 23 30)(20 27 24 31)
G:=sub<Sym(32)| (1,30,5,26)(2,31,6,27)(3,32,7,28)(4,25,8,29)(9,17,13,21)(10,18,14,22)(11,19,15,23)(12,20,16,24), (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)(9,13)(10,20)(11,15)(12,22)(14,24)(16,18)(17,21)(19,23), (1,23,28,9)(2,12,29,18)(3,21,30,15)(4,10,31,24)(5,19,32,13)(6,16,25,22)(7,17,26,11)(8,14,27,20), (1,11,5,15)(2,12,6,16)(3,13,7,9)(4,14,8,10)(17,32,21,28)(18,25,22,29)(19,26,23,30)(20,27,24,31)>;
G:=Group( (1,30,5,26)(2,31,6,27)(3,32,7,28)(4,25,8,29)(9,17,13,21)(10,18,14,22)(11,19,15,23)(12,20,16,24), (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)(9,13)(10,20)(11,15)(12,22)(14,24)(16,18)(17,21)(19,23), (1,23,28,9)(2,12,29,18)(3,21,30,15)(4,10,31,24)(5,19,32,13)(6,16,25,22)(7,17,26,11)(8,14,27,20), (1,11,5,15)(2,12,6,16)(3,13,7,9)(4,14,8,10)(17,32,21,28)(18,25,22,29)(19,26,23,30)(20,27,24,31) );
G=PermutationGroup([[(1,30,5,26),(2,31,6,27),(3,32,7,28),(4,25,8,29),(9,17,13,21),(10,18,14,22),(11,19,15,23),(12,20,16,24)], [(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),(9,13),(10,20),(11,15),(12,22),(14,24),(16,18),(17,21),(19,23)], [(1,23,28,9),(2,12,29,18),(3,21,30,15),(4,10,31,24),(5,19,32,13),(6,16,25,22),(7,17,26,11),(8,14,27,20)], [(1,11,5,15),(2,12,6,16),(3,13,7,9),(4,14,8,10),(17,32,21,28),(18,25,22,29),(19,26,23,30),(20,27,24,31)]])
Matrix representation of C4.152+ 1+4 ►in GL8(𝔽17)
| 1 | 15 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 16 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 16 | 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 | 0 | 10 | 7 | 0 | 0 | 0 | 0 |
| 5 | 0 | 0 | 7 | 0 | 0 | 0 | 0 |
| 0 | 12 | 5 | 12 | 0 | 0 | 0 | 0 |
| 12 | 5 | 5 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 0 | 2 | 2 |
| 0 | 0 | 0 | 0 | 3 | 10 | 13 | 15 |
| 0 | 0 | 0 | 0 | 8 | 8 | 7 | 0 |
| 0 | 0 | 0 | 0 | 1 | 9 | 3 | 10 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 16 | 0 | 0 | 0 | 0 | 0 |
| 1 | 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 | 10 | 10 | 16 | 0 |
| 0 | 0 | 0 | 0 | 0 | 7 | 0 | 16 |
| 1 | 0 | 0 | 15 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 16 | 0 | 0 | 0 | 0 |
| 1 | 16 | 0 | 16 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 10 | 0 | 16 | 16 |
| 0 | 0 | 0 | 0 | 14 | 7 | 2 | 1 |
| 1 | 0 | 15 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 16 | 0 | 0 | 0 | 0 | 0 |
| 1 | 16 | 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 10 | 0 | 16 | 16 |
| 0 | 0 | 0 | 0 | 0 | 10 | 0 | 1 |
G:=sub<GL(8,GF(17))| [1,1,0,1,0,0,0,0,15,16,16,16,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,5,0,12,0,0,0,0,0,0,12,5,0,0,0,0,10,0,5,5,0,0,0,0,7,7,12,12,0,0,0,0,0,0,0,0,7,3,8,1,0,0,0,0,0,10,8,9,0,0,0,0,2,13,7,3,0,0,0,0,2,15,0,10],[1,0,1,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,10,0,0,0,0,0,0,1,10,7,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16],[1,0,1,1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,15,16,16,16,0,0,0,0,0,0,0,0,16,2,10,14,0,0,0,0,16,1,0,7,0,0,0,0,0,0,16,2,0,0,0,0,0,0,16,1],[1,0,1,1,0,0,0,0,0,0,0,16,0,0,0,0,15,16,16,16,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,10,0,0,0,0,0,1,16,0,10,0,0,0,0,0,0,16,0,0,0,0,0,0,0,16,1] >;
C4.152+ 1+4 in GAP, Magma, Sage, TeX
C_4._{15}2_+^{1+4} % in TeX
G:=Group("C4.15ES+(2,2)"); // GroupNames label
G:=SmallGroup(128,1932);
// by ID
G=gap.SmallGroup(128,1932);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,456,758,219,675,1018,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=c^2=1,b^4=e^2=a^2,d^2=a^-1*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^-1=a^2*c,e*d*e^-1=a*b^2*d>;
// generators/relations
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