p-group, metabelian, nilpotent (class 2), monomial
Aliases: C22.33C24, C23.39C23, C42.38C22, C2.72+ 1+4, C2.52- 1+4, (C4×D4)⋊11C2, C22⋊Q8⋊8C2, C4⋊D4.8C2, C42.C2⋊4C2, C42⋊2C2⋊2C2, C4⋊C4.28C22, (C2×C4).20C23, (C2×D4).66C22, C22.5(C4○D4), C22.D4⋊5C2, (C2×Q8).29C22, C22⋊C4.16C22, (C22×C4).13C22, (C2×C4⋊C4)⋊19C2, C2.16(C2×C4○D4), SmallGroup(64,220)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C22.33C24
G = < a,b,c,d,e,f | a2=b2=c2=f2=1, d2=b, e2=a, ab=ba, dcd-1=fcf=ac=ca, ede-1=ad=da, ae=ea, af=fa, ece-1=bc=cb, bd=db, be=eb, bf=fb, df=fd, ef=fe >
Subgroups: 161 in 109 conjugacy classes, 73 normal (19 characteristic)
C1, C2, C2, C4, C22, C22, C22, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×C4⋊C4, C4×D4, C4⋊D4, C22⋊Q8, C22⋊Q8, C22.D4, C42.C2, C42⋊2C2, C22.33C24
Quotients: C1, C2, C22, C23, C4○D4, C24, C2×C4○D4, 2+ 1+4, 2- 1+4, C22.33C24
Character table of C22.33C24
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | |
| ρ1 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | linear of order 2 |
| ρ17 | 2 | -2 | 2 | -2 | -2 | 2 | 0 | 0 | 2i | -2i | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ18 | 2 | -2 | 2 | -2 | -2 | 2 | 0 | 0 | -2i | 2i | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ19 | 2 | -2 | 2 | -2 | 2 | -2 | 0 | 0 | -2i | 2i | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ20 | 2 | -2 | 2 | -2 | 2 | -2 | 0 | 0 | 2i | -2i | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ21 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 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 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from 2- 1+4, Schur index 2 |
►On 32 points
(1 27)(2 28)(3 25)(4 26)(5 30)(6 31)(7 32)(8 29)(9 15)(10 16)(11 13)(12 14)(17 23)(18 24)(19 21)(20 22)
(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(13 15)(14 16)(17 19)(18 20)(21 23)(22 24)(25 27)(26 28)(29 31)(30 32)
(2 28)(4 26)(5 32)(6 8)(7 30)(9 15)(11 13)(17 21)(18 20)(19 23)(22 24)(29 31)
(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)
(1 31 27 6)(2 7 28 32)(3 29 25 8)(4 5 26 30)(9 17 15 23)(10 24 16 18)(11 19 13 21)(12 22 14 20)
(1 9)(2 10)(3 11)(4 12)(5 22)(6 23)(7 24)(8 21)(13 25)(14 26)(15 27)(16 28)(17 31)(18 32)(19 29)(20 30)
G:=sub<Sym(32)| (1,27)(2,28)(3,25)(4,26)(5,30)(6,31)(7,32)(8,29)(9,15)(10,16)(11,13)(12,14)(17,23)(18,24)(19,21)(20,22), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24)(25,27)(26,28)(29,31)(30,32), (2,28)(4,26)(5,32)(6,8)(7,30)(9,15)(11,13)(17,21)(18,20)(19,23)(22,24)(29,31), (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), (1,31,27,6)(2,7,28,32)(3,29,25,8)(4,5,26,30)(9,17,15,23)(10,24,16,18)(11,19,13,21)(12,22,14,20), (1,9)(2,10)(3,11)(4,12)(5,22)(6,23)(7,24)(8,21)(13,25)(14,26)(15,27)(16,28)(17,31)(18,32)(19,29)(20,30)>;
G:=Group( (1,27)(2,28)(3,25)(4,26)(5,30)(6,31)(7,32)(8,29)(9,15)(10,16)(11,13)(12,14)(17,23)(18,24)(19,21)(20,22), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24)(25,27)(26,28)(29,31)(30,32), (2,28)(4,26)(5,32)(6,8)(7,30)(9,15)(11,13)(17,21)(18,20)(19,23)(22,24)(29,31), (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), (1,31,27,6)(2,7,28,32)(3,29,25,8)(4,5,26,30)(9,17,15,23)(10,24,16,18)(11,19,13,21)(12,22,14,20), (1,9)(2,10)(3,11)(4,12)(5,22)(6,23)(7,24)(8,21)(13,25)(14,26)(15,27)(16,28)(17,31)(18,32)(19,29)(20,30) );
G=PermutationGroup([[(1,27),(2,28),(3,25),(4,26),(5,30),(6,31),(7,32),(8,29),(9,15),(10,16),(11,13),(12,14),(17,23),(18,24),(19,21),(20,22)], [(1,3),(2,4),(5,7),(6,8),(9,11),(10,12),(13,15),(14,16),(17,19),(18,20),(21,23),(22,24),(25,27),(26,28),(29,31),(30,32)], [(2,28),(4,26),(5,32),(6,8),(7,30),(9,15),(11,13),(17,21),(18,20),(19,23),(22,24),(29,31)], [(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)], [(1,31,27,6),(2,7,28,32),(3,29,25,8),(4,5,26,30),(9,17,15,23),(10,24,16,18),(11,19,13,21),(12,22,14,20)], [(1,9),(2,10),(3,11),(4,12),(5,22),(6,23),(7,24),(8,21),(13,25),(14,26),(15,27),(16,28),(17,31),(18,32),(19,29),(20,30)]])
C22.33C24 is a maximal subgroup of
C4⋊C4.D4 C4⋊C4.6D4 C4⋊C4.12D4 C4⋊C4.18D4 C22.44C25 C22.48C25 C22.80C25 C22.81C25 C22.94C25 C22.102C25 C23.144C24 C22.110C25 C22.122C25 C22.123C25 C22.124C25 C22.125C25 C22.127C25 C22.128C25 C22.130C25 C22.142C25 C22.149C25 C22.151C25 C22.153C25 C22.155C25 C22.156C25
C2p.2- 1+4: C22.50C25 C22.96C25 C22.104C25 C22.141C25 C22.143C25 C22.146C25 C22.148C25 C22.152C25 ...
C22.33C24 is a maximal quotient of
C23.195C24 C24.547C23 C24.195C23 C24.198C23 C23.211C24 C23.214C24 C24.203C23 C24.204C23 C23.218C24 C24.563C23 C24.258C23 C24.269C23 C23.349C24 C24.276C23 C24.283C23 C24.286C23 C24.573C23 C23.388C24 C24.577C23 C23.417C24 C24.313C23 C24.326C23 C24.332C23 C23.473C24 C24.339C23 C24.341C23 C23.485C24 C23.488C24 C23.490C24 C23.494C24 C23.496C24 C24.587C23 C24.589C23 C23.527C24 C23.530C24 C42⋊30D4 C24.374C23 C24.592C23 C23.543C24 C23.546C24 C24.375C23 C23.550C24 C23.551C24 C24.376C23 C23.553C24 C23.554C24 C23.555C24 C24.378C23 C42⋊11Q8 C23.567C24 C23.580C24 C23.581C24 C24.394C23 C23.589C24 C23.591C24 C23.593C24 C23.595C24 C24.403C23 C23.605C24 C23.606C24 C23.608C24 C23.618C24 C24.418C23 C23.624C24 C23.627C24 C24.420C23 C23.632C24 C23.637C24 C24.426C23 C24.427C23 C23.640C24 C23.643C24 C24.430C23 C23.645C24 C24.432C23 C23.647C24 C24.435C23 C24.438C23 C24.443C23 C23.667C24 C23.669C24 C24.445C23 C23.671C24 C23.673C24 C23.675C24 C23.676C24 C23.677C24 C23.681C24 C23.682C24 C23.683C24 C23.685C24 C23.686C24 C23.687C24 C23.689C24 C24.454C23 C23.691C24 C23.692C24 C23.693C24 C23.694C24 C23.695C24 C23.696C24 C23.698C24 C23.702C24 C23.705C24 C23.707C24 C23.708C24 C23.709C24 C23.710C24 C23.724C24 C23.726C24 C23.727C24 C23.734C24 C23.735C24 C23.736C24 C23.737C24 C23.738C24 C23.739C24
C42.D2p: C42.190D4 C42.198D4 C42.118D6 C42.150D6 C42.161D6 C42.118D10 C42.150D10 C42.161D10 ...
C4⋊C4.D2p: C23.360C24 C24.572C23 C23.456C24 C23.458C24 C23.590C24 C24.401C23 C23.607C24 C23.611C24 ...
Matrix representation of C22.33C24 ►in GL6(𝔽5)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 2 | 0 | 0 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(6,GF(5))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,4],[2,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4,0,0,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,1,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;
C22.33C24 in GAP, Magma, Sage, TeX
C_2^2._{33}C_2^4 % in TeX
G:=Group("C2^2.33C2^4"); // GroupNames label
G:=SmallGroup(64,220);
// by ID
G=gap.SmallGroup(64,220);
# by ID
G:=PCGroup([6,-2,2,2,2,-2,2,217,199,650,188,86,579]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=f^2=1,d^2=b,e^2=a,a*b=b*a,d*c*d^-1=f*c*f=a*c=c*a,e*d*e^-1=a*d=d*a,a*e=e*a,a*f=f*a,e*c*e^-1=b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,d*f=f*d,e*f=f*e>;
// generators/relations
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