metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C23.5D12, C23⋊C4⋊4S3, C22⋊C4⋊2D6, (C2×C4).5D12, (C2×C12).7D4, (C2×D4).12D6, C6.14C22≀C2, D4⋊6D6.1C2, (C2×Dic3).1D4, (C22×S3).1D4, (C6×D4).9C22, C22.9(C2×D12), (C22×C6).18D4, C22.25(S3×D4), C23.6D6⋊2C2, C3⋊1(C23.7D4), (C22×C6).3C23, C2.17(D6⋊D4), C23.23D6⋊1C2, C6.D4⋊2C22, C23.21D6⋊1C2, C23.13(C22×S3), (C22×Dic3)⋊1C22, (C3×C23⋊C4)⋊5C2, (C2×C6).18(C2×D4), (C3×C22⋊C4)⋊2C22, (C2×C3⋊D4).3C22, SmallGroup(192,301)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C23.5D12
G = < a,b,c,d,e | a2=b2=c2=d12=1, e2=c, ab=ba, ac=ca, dad-1=eae-1=abc, dbd-1=bc=cb, be=eb, cd=dc, ce=ec, ede-1=cd-1 >
Subgroups: 544 in 160 conjugacy classes, 39 normal (21 characteristic)
C1, C2, C2, C3, C4, C22, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, Q8, C23, C23, Dic3, C12, D6, C2×C6, C2×C6, C2×C6, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×D4, C4○D4, Dic6, C4×S3, D12, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×D4, C22×S3, C22×S3, C22×C6, C23⋊C4, C23⋊C4, C22.D4, 2+ 1+4, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C6.D4, C6.D4, C3×C22⋊C4, C4○D12, S3×D4, D4⋊2S3, C22×Dic3, C2×C3⋊D4, C2×C3⋊D4, C6×D4, C23.7D4, C23.6D6, C3×C23⋊C4, C23.21D6, C23.23D6, D4⋊6D6, C23.5D12
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, D12, C22×S3, C22≀C2, C2×D12, S3×D4, C23.7D4, D6⋊D4, C23.5D12
Character table of C23.5D12
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | 6B | 6C | 6D | 6E | 12A | 12B | 12C | 12D | 12E | |
| size | 1 | 1 | 2 | 2 | 2 | 4 | 12 | 12 | 2 | 4 | 8 | 8 | 12 | 12 | 12 | 12 | 24 | 2 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | linear of order 2 |
| ρ9 | 2 | 2 | -2 | -2 | 2 | 0 | -2 | 0 | 2 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | 2 | -2 | 2 | -2 | 0 | 0 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ11 | 2 | 2 | -2 | -2 | 2 | 0 | 2 | 0 | 2 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ12 | 2 | 2 | 2 | -2 | -2 | -2 | 0 | 0 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 2 | -2 | -2 | 0 | 0 | 0 | 2 | 0 | orthogonal lifted from D4 |
| ρ13 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ14 | 2 | 2 | 2 | 2 | 2 | -2 | 0 | 0 | -1 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | 1 | 1 | -1 | 1 | 1 | -1 | orthogonal lifted from D6 |
| ρ15 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | -1 | 1 | orthogonal lifted from D6 |
| ρ16 | 2 | 2 | -2 | 2 | -2 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ17 | 2 | 2 | 2 | -2 | -2 | 2 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | -2 | 0 | orthogonal lifted from D4 |
| ρ18 | 2 | 2 | 2 | 2 | 2 | -2 | 0 | 0 | -1 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | orthogonal lifted from D6 |
| ρ19 | 2 | 2 | 2 | -2 | -2 | 2 | 0 | 0 | -1 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | -1 | 1 | -1 | -√3 | -√3 | √3 | 1 | √3 | orthogonal lifted from D12 |
| ρ20 | 2 | 2 | 2 | -2 | -2 | 2 | 0 | 0 | -1 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | -1 | 1 | -1 | √3 | √3 | -√3 | 1 | -√3 | orthogonal lifted from D12 |
| ρ21 | 2 | 2 | 2 | -2 | -2 | -2 | 0 | 0 | -1 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | -1 | 1 | 1 | -√3 | √3 | √3 | -1 | -√3 | orthogonal lifted from D12 |
| ρ22 | 2 | 2 | 2 | -2 | -2 | -2 | 0 | 0 | -1 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | -1 | 1 | 1 | √3 | -√3 | -√3 | -1 | √3 | orthogonal lifted from D12 |
| ρ23 | 4 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S3×D4 |
| ρ24 | 4 | 4 | -4 | 4 | -4 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S3×D4 |
| ρ25 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | -2i | 2i | 0 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C23.7D4 |
| ρ26 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 2i | -2i | 0 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C23.7D4 |
| ρ27 | 8 | -8 | 0 | 0 | 0 | 0 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
►On 48 points
Generators in S48
(2 20)(3 44)(4 30)(6 24)(7 48)(8 34)(10 16)(11 40)(12 26)(13 33)(14 37)(17 25)(18 41)(21 29)(22 45)(28 43)(32 47)(36 39)
(1 27)(2 20)(3 29)(4 22)(5 31)(6 24)(7 33)(8 14)(9 35)(10 16)(11 25)(12 18)(13 48)(15 38)(17 40)(19 42)(21 44)(23 46)(26 41)(28 43)(30 45)(32 47)(34 37)(36 39)
(1 42)(2 43)(3 44)(4 45)(5 46)(6 47)(7 48)(8 37)(9 38)(10 39)(11 40)(12 41)(13 33)(14 34)(15 35)(16 36)(17 25)(18 26)(19 27)(20 28)(21 29)(22 30)(23 31)(24 32)
(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 33 34 35 36)(37 38 39 40 41 42 43 44 45 46 47 48)
(1 41 42 12)(2 11 43 40)(3 39 44 10)(4 9 45 38)(5 37 46 8)(6 7 47 48)(13 24 33 32)(14 31 34 23)(15 22 35 30)(16 29 36 21)(17 20 25 28)(18 27 26 19)
G:=sub<Sym(48)| (2,20)(3,44)(4,30)(6,24)(7,48)(8,34)(10,16)(11,40)(12,26)(13,33)(14,37)(17,25)(18,41)(21,29)(22,45)(28,43)(32,47)(36,39), (1,27)(2,20)(3,29)(4,22)(5,31)(6,24)(7,33)(8,14)(9,35)(10,16)(11,25)(12,18)(13,48)(15,38)(17,40)(19,42)(21,44)(23,46)(26,41)(28,43)(30,45)(32,47)(34,37)(36,39), (1,42)(2,43)(3,44)(4,45)(5,46)(6,47)(7,48)(8,37)(9,38)(10,39)(11,40)(12,41)(13,33)(14,34)(15,35)(16,36)(17,25)(18,26)(19,27)(20,28)(21,29)(22,30)(23,31)(24,32), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45,46,47,48), (1,41,42,12)(2,11,43,40)(3,39,44,10)(4,9,45,38)(5,37,46,8)(6,7,47,48)(13,24,33,32)(14,31,34,23)(15,22,35,30)(16,29,36,21)(17,20,25,28)(18,27,26,19)>;
G:=Group( (2,20)(3,44)(4,30)(6,24)(7,48)(8,34)(10,16)(11,40)(12,26)(13,33)(14,37)(17,25)(18,41)(21,29)(22,45)(28,43)(32,47)(36,39), (1,27)(2,20)(3,29)(4,22)(5,31)(6,24)(7,33)(8,14)(9,35)(10,16)(11,25)(12,18)(13,48)(15,38)(17,40)(19,42)(21,44)(23,46)(26,41)(28,43)(30,45)(32,47)(34,37)(36,39), (1,42)(2,43)(3,44)(4,45)(5,46)(6,47)(7,48)(8,37)(9,38)(10,39)(11,40)(12,41)(13,33)(14,34)(15,35)(16,36)(17,25)(18,26)(19,27)(20,28)(21,29)(22,30)(23,31)(24,32), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45,46,47,48), (1,41,42,12)(2,11,43,40)(3,39,44,10)(4,9,45,38)(5,37,46,8)(6,7,47,48)(13,24,33,32)(14,31,34,23)(15,22,35,30)(16,29,36,21)(17,20,25,28)(18,27,26,19) );
G=PermutationGroup([[(2,20),(3,44),(4,30),(6,24),(7,48),(8,34),(10,16),(11,40),(12,26),(13,33),(14,37),(17,25),(18,41),(21,29),(22,45),(28,43),(32,47),(36,39)], [(1,27),(2,20),(3,29),(4,22),(5,31),(6,24),(7,33),(8,14),(9,35),(10,16),(11,25),(12,18),(13,48),(15,38),(17,40),(19,42),(21,44),(23,46),(26,41),(28,43),(30,45),(32,47),(34,37),(36,39)], [(1,42),(2,43),(3,44),(4,45),(5,46),(6,47),(7,48),(8,37),(9,38),(10,39),(11,40),(12,41),(13,33),(14,34),(15,35),(16,36),(17,25),(18,26),(19,27),(20,28),(21,29),(22,30),(23,31),(24,32)], [(1,2,3,4,5,6,7,8,9,10,11,12),(13,14,15,16,17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32,33,34,35,36),(37,38,39,40,41,42,43,44,45,46,47,48)], [(1,41,42,12),(2,11,43,40),(3,39,44,10),(4,9,45,38),(5,37,46,8),(6,7,47,48),(13,24,33,32),(14,31,34,23),(15,22,35,30),(16,29,36,21),(17,20,25,28),(18,27,26,19)]])
Matrix representation of C23.5D12 ►in GL8(𝔽13)
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 4 | 4 | 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 4 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 12 |
| 12 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 10 | 12 | 7 |
| 0 | 0 | 0 | 0 | 3 | 6 | 3 | 6 |
| 0 | 0 | 0 | 0 | 0 | 0 | 7 | 8 |
| 0 | 0 | 0 | 0 | 0 | 0 | 7 | 6 |
| 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 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 12 |
| 3 | 10 | 5 | 3 | 0 | 0 | 0 | 0 |
| 3 | 6 | 5 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 10 | 10 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 11 | 2 | 4 |
| 0 | 0 | 0 | 0 | 0 | 8 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 4 | 7 | 3 |
| 0 | 0 | 0 | 0 | 2 | 9 | 11 | 8 |
| 3 | 10 | 5 | 3 | 0 | 0 | 0 | 0 |
| 7 | 10 | 3 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 10 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 6 | 3 | 0 |
| 0 | 0 | 0 | 0 | 0 | 5 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 12 | 11 | 0 |
| 0 | 0 | 0 | 0 | 11 | 4 | 2 | 5 |
G:=sub<GL(8,GF(13))| [1,0,4,0,0,0,0,0,0,1,4,4,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,4,1,0,12],[12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,7,3,0,0,0,0,0,0,10,6,0,0,0,0,0,0,12,3,7,7,0,0,0,0,7,6,8,6],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12],[3,3,0,0,0,0,0,0,10,6,0,0,0,0,0,0,5,5,7,10,0,0,0,0,3,5,3,10,0,0,0,0,0,0,0,0,3,0,11,2,0,0,0,0,11,8,4,9,0,0,0,0,2,0,7,11,0,0,0,0,4,0,3,8],[3,7,0,0,0,0,0,0,10,10,0,0,0,0,0,0,5,3,3,6,0,0,0,0,3,5,3,10,0,0,0,0,0,0,0,0,2,0,7,11,0,0,0,0,6,5,12,4,0,0,0,0,3,0,11,2,0,0,0,0,0,0,0,5] >;
C23.5D12 in GAP, Magma, Sage, TeX
C_2^3._5D_{12} % in TeX
G:=Group("C2^3.5D12"); // GroupNames label
G:=SmallGroup(192,301);
// by ID
G=gap.SmallGroup(192,301);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,254,219,226,570,1684,438,6278]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^12=1,e^2=c,a*b=b*a,a*c=c*a,d*a*d^-1=e*a*e^-1=a*b*c,d*b*d^-1=b*c=c*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=c*d^-1>;
// generators/relations
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