p-group, metabelian, nilpotent (class 2), monomial
Aliases: C23.16C23, C42.41C22, C22.36C24, C2.72- 1+4, C2.92+ 1+4, C4⋊Q8⋊11C2, (C4×D4)⋊13C2, (C4×Q8)⋊10C2, C22⋊Q8⋊9C2, C4.4D4⋊9C2, C4⋊D4.9C2, C42⋊2C2⋊3C2, C4.22(C4○D4), C4⋊C4.31C22, (C2×C4).23C23, C42⋊C2⋊13C2, (C2×D4).67C22, C22.D4⋊7C2, C22⋊C4.4C22, (C2×Q8).30C22, (C22×C4).65C22, C2.19(C2×C4○D4), SmallGroup(64,223)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C22.36C24
G = < a,b,c,d,e,f | a2=b2=c2=e2=1, d2=ba=ab, f2=a, dcd-1=fcf-1=ac=ca, ede=ad=da, ae=ea, af=fa, ece=bc=cb, bd=db, be=eb, bf=fb, df=fd, ef=fe >
Subgroups: 161 in 108 conjugacy classes, 73 normal (31 characteristic)
C1, C2, C2, C4, C4, C22, C22, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C42⋊C2, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C22⋊Q8, C22.D4, C4.4D4, C4.4D4, C42⋊2C2, C4⋊Q8, C22.36C24
Quotients: C1, C2, C22, C23, C4○D4, C24, C2×C4○D4, 2+ 1+4, 2- 1+4, C22.36C24
Character table of C22.36C24
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 2 | 2 | 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 | 0 | 0 | 0 | -2i | -2 | -2i | 2 | 2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ18 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | -2i | 2 | 2i | -2 | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ19 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 2i | -2 | 2i | 2 | -2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ20 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 2i | 2 | -2i | -2 | 2i | -2i | 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 5)(2 6)(3 7)(4 8)(9 22)(10 23)(11 24)(12 21)(13 17)(14 18)(15 19)(16 20)(25 30)(26 31)(27 32)(28 29)
(1 7)(2 8)(3 5)(4 6)(9 24)(10 21)(11 22)(12 23)(13 19)(14 20)(15 17)(16 18)(25 32)(26 29)(27 30)(28 31)
(1 26)(2 32)(3 28)(4 30)(5 31)(6 27)(7 29)(8 25)(9 15)(10 20)(11 13)(12 18)(14 21)(16 23)(17 24)(19 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 6)(4 8)(9 24)(10 12)(11 22)(14 18)(16 20)(21 23)(25 27)(26 29)(28 31)(30 32)
(1 13 5 17)(2 14 6 18)(3 15 7 19)(4 16 8 20)(9 28 22 29)(10 25 23 30)(11 26 24 31)(12 27 21 32)
G:=sub<Sym(32)| (1,5)(2,6)(3,7)(4,8)(9,22)(10,23)(11,24)(12,21)(13,17)(14,18)(15,19)(16,20)(25,30)(26,31)(27,32)(28,29), (1,7)(2,8)(3,5)(4,6)(9,24)(10,21)(11,22)(12,23)(13,19)(14,20)(15,17)(16,18)(25,32)(26,29)(27,30)(28,31), (1,26)(2,32)(3,28)(4,30)(5,31)(6,27)(7,29)(8,25)(9,15)(10,20)(11,13)(12,18)(14,21)(16,23)(17,24)(19,22), (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), (2,6)(4,8)(9,24)(10,12)(11,22)(14,18)(16,20)(21,23)(25,27)(26,29)(28,31)(30,32), (1,13,5,17)(2,14,6,18)(3,15,7,19)(4,16,8,20)(9,28,22,29)(10,25,23,30)(11,26,24,31)(12,27,21,32)>;
G:=Group( (1,5)(2,6)(3,7)(4,8)(9,22)(10,23)(11,24)(12,21)(13,17)(14,18)(15,19)(16,20)(25,30)(26,31)(27,32)(28,29), (1,7)(2,8)(3,5)(4,6)(9,24)(10,21)(11,22)(12,23)(13,19)(14,20)(15,17)(16,18)(25,32)(26,29)(27,30)(28,31), (1,26)(2,32)(3,28)(4,30)(5,31)(6,27)(7,29)(8,25)(9,15)(10,20)(11,13)(12,18)(14,21)(16,23)(17,24)(19,22), (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), (2,6)(4,8)(9,24)(10,12)(11,22)(14,18)(16,20)(21,23)(25,27)(26,29)(28,31)(30,32), (1,13,5,17)(2,14,6,18)(3,15,7,19)(4,16,8,20)(9,28,22,29)(10,25,23,30)(11,26,24,31)(12,27,21,32) );
G=PermutationGroup([[(1,5),(2,6),(3,7),(4,8),(9,22),(10,23),(11,24),(12,21),(13,17),(14,18),(15,19),(16,20),(25,30),(26,31),(27,32),(28,29)], [(1,7),(2,8),(3,5),(4,6),(9,24),(10,21),(11,22),(12,23),(13,19),(14,20),(15,17),(16,18),(25,32),(26,29),(27,30),(28,31)], [(1,26),(2,32),(3,28),(4,30),(5,31),(6,27),(7,29),(8,25),(9,15),(10,20),(11,13),(12,18),(14,21),(16,23),(17,24),(19,22)], [(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)], [(2,6),(4,8),(9,24),(10,12),(11,22),(14,18),(16,20),(21,23),(25,27),(26,29),(28,31),(30,32)], [(1,13,5,17),(2,14,6,18),(3,15,7,19),(4,16,8,20),(9,28,22,29),(10,25,23,30),(11,26,24,31),(12,27,21,32)]])
C22.36C24 is a maximal subgroup of
C42.352C23 C42.355C23 C42.358C23 C42.359C23 C42.385C23 C42.387C23 C42.390C23 C42.391C23 C42.410C23 C42.411C23 C22.44C25 C22.49C25 C22.83C25 C22.84C25 C22.99C25 C22.102C25 C22.103C25 C22.105C25 C23.144C24 C22.110C25 C22.113C25 C22.122C25 C22.124C25 C22.125C25 C22.129C25 C22.130C25 C22.131C25 C22.134C25 C22.135C25 C22.147C25 C22.149C25 C22.150C25 C22.153C25 C22.155C25 C22.157C25
C2p.2- 1+4: C42.425C23 C42.426C23 C4.2- 1+4 C42.25C23 C42.29C23 C42.30C23 C22.50C25 C22.100C25 ...
C22.36C24 is a maximal quotient of
C24.192C23 C23.201C24 C23.202C24 C42⋊13D4 C42⋊4Q8 C23.214C24 C24.203C23 C24.204C23 C24.205C23 C23.322C24 C23.323C24 C23.327C24 C24.271C23 C23.348C24 C23.352C24 C24.282C23 C23.368C24 C24.289C23 C23.374C24 C23.377C24 C23.379C24 C24.304C23 C23.395C24 C23.408C24 C23.409C24 C23.411C24 C23.412C24 C24.309C23 C23.420C24 C24.311C23 C23.425C24 C24.315C23 C23.429C24 C23.430C24 C23.432C24 C42⋊28D4 C23.524C24 C23.525C24 C23.530C24 C24.374C23 C23.544C24 C23.545C24 C42.39Q8 C24.375C23 C23.550C24 C23.551C24 C24.376C23 C23.553C24 C23.554C24 C23.555C24 C42⋊32D4 C24.378C23 C24.379C23 C42⋊11Q8 C23.572C24 C24.393C23 C24.394C23 C23.591C24 C23.592C24 C24.405C23 C23.600C24 C24.407C23 C23.602C24 C23.605C24 C24.412C23 C23.612C24 C23.615C24 C23.617C24 C23.618C24 C24.418C23 C24.421C23 C23.630C24 C23.631C24 C23.637C24 C24.426C23 C24.427C23 C23.641C24 C24.428C23 C23.645C24 C24.432C23 C23.647C24 C23.651C24 C23.654C24 C23.655C24 C24.438C23 C23.658C24 C23.659C24 C24.440C23 C23.662C24 C23.663C24 C23.664C24 C24.443C23 C23.666C24 C23.667C24 C24.445C23 C23.672C24 C23.673C24 C23.674C24 C23.675C24 C23.677C24 C23.679C24 C24.454C23 C23.693C24 C23.695C24 C23.696C24 C23.698C24 C23.699C24 C23.700C24 C23.701C24 C23.729C24 C23.730C24 C23.731C24 C23.732C24 C23.737C24 C23.738C24 C23.739C24
C42.D2p: C42.159D4 C42.160D4 C42.187D4 C42.189D4 C42.192D4 C42.193D4 C42.99D6 C42.115D6 ...
C4⋊C4.D2p: C24.259C23 C23.351C24 C24.279C23 C24.285C23 C23.391C24 C23.392C24 C23.574C24 C23.616C24 ...
Matrix representation of C22.36C24 ►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 |
| 4 | 3 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 4 | 1 | 4 | 2 |
| 0 | 0 | 4 | 1 | 0 | 1 |
| 3 | 0 | 0 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 4 | 1 | 3 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 4 | 1 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 4 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 1 | 4 | 0 | 4 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 1 | 4 | 1 | 3 |
| 0 | 0 | 1 | 0 | 1 | 4 |
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],[4,0,0,0,0,0,3,1,0,0,0,0,0,0,0,1,4,4,0,0,1,0,1,1,0,0,0,0,4,0,0,0,0,0,2,1],[3,0,0,0,0,0,0,3,0,0,0,0,0,0,0,1,4,4,0,0,0,4,0,1,0,0,1,1,0,0,0,0,0,3,0,1],[1,4,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,1,0,0,0,1,0,4,0,0,0,0,4,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,4,1,1,0,0,1,0,4,0,0,0,0,0,1,1,0,0,0,0,3,4] >;
C22.36C24 in GAP, Magma, Sage, TeX
C_2^2._{36}C_2^4 % in TeX
G:=Group("C2^2.36C2^4"); // GroupNames label
G:=SmallGroup(64,223);
// by ID
G=gap.SmallGroup(64,223);
# by ID
G:=PCGroup([6,-2,2,2,2,-2,2,217,295,650,188,579,69]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=e^2=1,d^2=b*a=a*b,f^2=a,d*c*d^-1=f*c*f^-1=a*c=c*a,e*d*e=a*d=d*a,a*e=e*a,a*f=f*a,e*c*e=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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