p-group, metabelian, nilpotent (class 2), monomial
Aliases: C23.36C23, C42.61C22, C22.23C24, (C4×D4)⋊8C2, (C4×Q8)⋊7C2, C4○2(C4⋊D4), (C2×C42)⋊10C2, C4○(C4.4D4), C4○2(C22⋊Q8), C22⋊Q8⋊22C2, C4○(C42.C2), C4○(C42⋊2C2), C42⋊C2⋊9C2, C4.56(C4○D4), C4.4D4⋊17C2, C4⋊D4.10C2, C4⋊C4.70C22, (C2×C4).54C23, C42.C2⋊13C2, C42⋊2C2⋊10C2, (C2×D4).62C22, C22.3(C4○D4), (C2×Q8).54C22, C4○2(C22.D4), C22.D4⋊16C2, C22⋊C4.13C22, (C22×C4).100C22, C2.12(C2×C4○D4), (C2×C4)○(C4.4D4), (C2×C4)○(C42.C2), (C2×C4)○(C42⋊2C2), (C2×C4)○(C22.D4), SmallGroup(64,210)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C23.36C23
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e2=c, f2=b, ab=ba, dad=ac=ca, ae=ea, af=fa, bc=cb, ede-1=bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, df=fd, ef=fe >
Subgroups: 161 in 117 conjugacy classes, 77 normal (31 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×C42, C42⋊C2, C4×D4, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C42.C2, C42⋊2C2, C23.36C23
Quotients: C1, C2, C22, C23, C4○D4, C24, C2×C4○D4, C23.36C23
Character table of C23.36C23
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | 4Q | 4R | 4S | 4T | |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 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 | 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 | 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 | -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 | -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 | -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 | -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 | 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 | 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 | -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 | -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 | 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 | 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 | 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 | 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 | -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 | -1 | -1 | linear of order 2 |
| ρ17 | 2 | -2 | 2 | -2 | 2 | -2 | 0 | 0 | -2i | 2i | 2i | -2i | 0 | 2i | 0 | 0 | 0 | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ18 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 2i | 2i | -2i | -2i | 2 | 0 | 0 | 0 | -2i | 0 | 0 | 0 | 2i | -2 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ19 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | -2 | 2 | -2 | 2 | 0 | 0 | -2i | -2i | 0 | 0 | 2i | 2i | 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 | -2i | 0 | 0 | 0 | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ21 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 2 | -2 | 2 | -2 | 0 | 0 | 2i | -2i | 0 | 0 | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ22 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 2 | -2 | 2 | -2 | 0 | 0 | -2i | 2i | 0 | 0 | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ23 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | -2 | 2 | -2 | 2 | 0 | 0 | 2i | 2i | 0 | 0 | -2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ24 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | -2i | -2i | 2i | 2i | 2 | 0 | 0 | 0 | 2i | 0 | 0 | 0 | -2i | -2 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ25 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 2i | 2i | -2i | -2i | -2 | 0 | 0 | 0 | 2i | 0 | 0 | 0 | -2i | 2 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ26 | 2 | -2 | 2 | -2 | -2 | 2 | 0 | 0 | 2i | -2i | -2i | 2i | 0 | 2i | 0 | 0 | 0 | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ27 | 2 | -2 | 2 | -2 | 2 | -2 | 0 | 0 | 2i | -2i | -2i | 2i | 0 | -2i | 0 | 0 | 0 | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ28 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | -2i | -2i | 2i | 2i | -2 | 0 | 0 | 0 | -2i | 0 | 0 | 0 | 2i | 2 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
►On 32 points
(1 23)(2 24)(3 21)(4 22)(5 10)(6 11)(7 12)(8 9)(13 19)(14 20)(15 17)(16 18)(25 29)(26 30)(27 31)(28 32)
(1 27)(2 28)(3 25)(4 26)(5 20)(6 17)(7 18)(8 19)(9 13)(10 14)(11 15)(12 16)(21 29)(22 30)(23 31)(24 32)
(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 18)(6 8)(7 20)(10 14)(12 16)(17 19)(21 23)(22 32)(24 30)(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 15 27 11)(2 16 28 12)(3 13 25 9)(4 14 26 10)(5 22 20 30)(6 23 17 31)(7 24 18 32)(8 21 19 29)
G:=sub<Sym(32)| (1,23)(2,24)(3,21)(4,22)(5,10)(6,11)(7,12)(8,9)(13,19)(14,20)(15,17)(16,18)(25,29)(26,30)(27,31)(28,32), (1,27)(2,28)(3,25)(4,26)(5,20)(6,17)(7,18)(8,19)(9,13)(10,14)(11,15)(12,16)(21,29)(22,30)(23,31)(24,32), (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)(5,18)(6,8)(7,20)(10,14)(12,16)(17,19)(21,23)(22,32)(24,30)(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), (1,15,27,11)(2,16,28,12)(3,13,25,9)(4,14,26,10)(5,22,20,30)(6,23,17,31)(7,24,18,32)(8,21,19,29)>;
G:=Group( (1,23)(2,24)(3,21)(4,22)(5,10)(6,11)(7,12)(8,9)(13,19)(14,20)(15,17)(16,18)(25,29)(26,30)(27,31)(28,32), (1,27)(2,28)(3,25)(4,26)(5,20)(6,17)(7,18)(8,19)(9,13)(10,14)(11,15)(12,16)(21,29)(22,30)(23,31)(24,32), (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)(5,18)(6,8)(7,20)(10,14)(12,16)(17,19)(21,23)(22,32)(24,30)(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), (1,15,27,11)(2,16,28,12)(3,13,25,9)(4,14,26,10)(5,22,20,30)(6,23,17,31)(7,24,18,32)(8,21,19,29) );
G=PermutationGroup([[(1,23),(2,24),(3,21),(4,22),(5,10),(6,11),(7,12),(8,9),(13,19),(14,20),(15,17),(16,18),(25,29),(26,30),(27,31),(28,32)], [(1,27),(2,28),(3,25),(4,26),(5,20),(6,17),(7,18),(8,19),(9,13),(10,14),(11,15),(12,16),(21,29),(22,30),(23,31),(24,32)], [(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),(5,18),(6,8),(7,20),(10,14),(12,16),(17,19),(21,23),(22,32),(24,30),(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)], [(1,15,27,11),(2,16,28,12),(3,13,25,9),(4,14,26,10),(5,22,20,30),(6,23,17,31),(7,24,18,32),(8,21,19,29)]])
C23.36C23 is a maximal subgroup of
C22.33C25 C22.44C25 C22.48C25 C22.49C25 C22.50C25 C22.64C25 C22.69C25 C22.70C25 C22.71C25 C22.80C25 C22.82C25 C22.83C25 C22.84C25 C22.94C25 C22.96C25 C22.99C25 C22.101C25 C22.102C25 C22.103C25 C22.104C25 C22.105C25 C22.106C25 C22.107C25 C22.108C25 C23.144C24 C22.110C25 C22.111C25 C23.146C24 C22.113C25 C22.118C25 C22.120C25 C22.132C25 C22.133C25 C22.134C25 C22.135C25 C22.136C25 C22.140C25 C22.141C25 C22.142C25 C22.143C25 C22.144C25 C22.146C25 C22.147C25 C22.148C25 C22.149C25 C22.150C25 C22.151C25 C22.152C25 C22.153C25 C22.154C25 C22.155C25 C22.156C25 C22.157C25
C42.D2p: C42.373D4 C42.47D4 C42.305D4 C42.52D4 C42.375D4 C42.57D4 C42.58D4 C42.63D4 ...
(C2×C4p).C23: C42.291C23 C42.292C23 C42.293C23 C42.294C23 C42.307C23 C42.308C23 C42.309C23 C42.310C23 ...
C23.36C23 is a maximal quotient of
C23.165C24 C42⋊42D4 C43⋊9C2 C42⋊14Q8 C43⋊2C2 C4×C4⋊D4 C4×C22.D4 C4×C42⋊2C2 C4×C42.C2 C42⋊15D4 C23.295C24 C23.301C24 C42.34Q8 C24.563C23 C24.254C23 C23.321C24 C24.258C23 C23.327C24 C23.344C24 C24.271C23 C23.348C24 C23.350C24 C24.278C23 C23.359C24 C24.286C23 C23.367C24 C23.368C24 C24.289C23 C24.290C23 C23.374C24 C24.293C23 C23.377C24 C24.295C23 C23.379C24 C23.388C24 C24.577C23 C24.304C23 C23.395C24 C23.405C24 C23.408C24 C23.409C24 C23.410C24 C23.413C24 C23.414C24 C24.309C23 C23.416C24 C23.417C24 C23.418C24 C23.420C24 C24.311C23 C23.422C24 C24.313C23 C23.424C24 C23.425C24 C23.426C24 C24.315C23 C23.428C24 C23.429C24 C23.430C24 C23.431C24 C23.432C24 C23.433C24 C24.326C23 C24.327C23 C23.457C24 C24.331C23 C24.332C23 C23.472C24 C23.473C24 C24.338C23 C24.339C23 C24.340C23 C24.341C23 C23.478C24 C23.485C24 C24.345C23 C23.488C24 C24.346C23 C23.490C24 C23.493C24 C23.494C24 C24.347C23 C23.496C24 C24.348C23 C23.548C24 C24.375C23 C23.550C24 C23.551C24 C24.376C23 C23.553C24 C23.554C24 C23.555C24 C42⋊46D4 C42⋊43D4 C23.753C24 C24.598C23 C24.599C23 C43⋊13C2 C42⋊15Q8 C43.18C2 C43⋊4C2 C43⋊5C2
C42.D2p: C4×C22⋊Q8 C4×C4.4D4 C42.162D4 C42.163D4 C42.439D4 C43⋊14C2 C42.277D6 C42.93D6 ...
C4⋊C4.D2p: C24.259C23 C23.353C24 C23.354C24 C24.279C23 C23.360C24 C23.369C24 C23.375C24 C24.301C23 ...
Matrix representation of C23.36C23 ►in GL4(𝔽5) generated by
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 1 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 4 |
| 0 | 2 | 0 | 0 |
| 2 | 0 | 0 | 0 |
| 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 3 |
| 2 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
G:=sub<GL(4,GF(5))| [0,1,0,0,1,0,0,0,0,0,0,1,0,0,1,0],[4,0,0,0,0,4,0,0,0,0,1,0,0,0,0,1],[4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[1,0,0,0,0,4,0,0,0,0,1,0,0,0,0,4],[0,2,0,0,2,0,0,0,0,0,3,0,0,0,0,3],[2,0,0,0,0,2,0,0,0,0,4,0,0,0,0,4] >;
C23.36C23 in GAP, Magma, Sage, TeX
C_2^3._{36}C_2^3 % in TeX
G:=Group("C2^3.36C2^3"); // GroupNames label
G:=SmallGroup(64,210);
// by ID
G=gap.SmallGroup(64,210);
# by ID
G:=PCGroup([6,-2,2,2,2,-2,2,217,199,650,69]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^2=c,f^2=b,a*b=b*a,d*a*d=a*c=c*a,a*e=e*a,a*f=f*a,b*c=c*b,e*d*e^-1=b*d=d*b,b*e=e*b,b*f=f*b,c*d=d*c,c*e=e*c,c*f=f*c,d*f=f*d,e*f=f*e>;
// generators/relations
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