non-abelian, soluble, monomial
Aliases: C24.4D6, C23.11D12, C23.2Dic6, C4⋊(A4⋊C4), (C4×A4)⋊1C4, A4⋊2(C4⋊C4), C2.1(C4⋊S4), (C2×C4).13S4, (C2×A4).4D4, (C2×A4).2Q8, C2.2(A4⋊Q8), C22⋊(C4⋊Dic3), (C23×C4).5S3, C22.15(C2×S4), (C22×C4)⋊2Dic3, C23.4(C2×Dic3), (C22×A4).5C22, (C2×C4×A4).2C2, C2.4(C2×A4⋊C4), (C2×A4⋊C4).2C2, (C2×A4).8(C2×C4), SmallGroup(192,971)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24.4D6
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e6=b, f2=a, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, fcf-1=ede-1=cd=dc, ece-1=d, df=fd, fef-1=be5 >
Subgroups: 394 in 109 conjugacy classes, 27 normal (19 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C2×C4, C2×C4, C23, C23, Dic3, C12, A4, C2×C6, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C24, C2×Dic3, C2×C12, C2×A4, C2.C42, C2×C22⋊C4, C2×C4⋊C4, C23×C4, C4⋊Dic3, A4⋊C4, C4×A4, C22×A4, C23.7Q8, C2×A4⋊C4, C2×C4×A4, C24.4D6
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, Dic3, D6, C4⋊C4, Dic6, D12, C2×Dic3, S4, C4⋊Dic3, A4⋊C4, C2×S4, A4⋊Q8, C4⋊S4, C2×A4⋊C4, C24.4D6
Character table of C24.4D6
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 6A | 6B | 6C | 12A | 12B | 12C | 12D | |
| size | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 8 | 2 | 2 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 8 | 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 | 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 | i | i | -i | -i | -i | -i | i | i | -1 | -1 | 1 | -1 | 1 | 1 | -1 | linear of order 4 |
| ρ6 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -i | -i | i | i | i | i | -i | -i | -1 | -1 | 1 | -1 | 1 | 1 | -1 | linear of order 4 |
| ρ7 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | 1 | 1 | -1 | 1 | -1 | -i | i | -i | -i | i | i | -i | i | -1 | -1 | 1 | 1 | -1 | -1 | 1 | linear of order 4 |
| ρ8 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | 1 | 1 | -1 | 1 | -1 | i | -i | i | i | -i | -i | i | -i | -1 | -1 | 1 | 1 | -1 | -1 | 1 | linear of order 4 |
| ρ9 | 2 | -2 | 2 | -2 | 2 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | -1 | -2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ11 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | -1 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ12 | 2 | -2 | 2 | -2 | 2 | -2 | 2 | -2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | √3 | -√3 | √3 | -√3 | orthogonal lifted from D12 |
| ρ13 | 2 | -2 | 2 | -2 | 2 | -2 | 2 | -2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | -√3 | √3 | -√3 | √3 | orthogonal lifted from D12 |
| ρ14 | 2 | 2 | -2 | -2 | -2 | 2 | 2 | -2 | -1 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | symplectic lifted from Dic3, Schur index 2 |
| ρ15 | 2 | -2 | -2 | 2 | -2 | -2 | 2 | 2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | √3 | √3 | -√3 | -√3 | symplectic lifted from Dic6, Schur index 2 |
| ρ16 | 2 | -2 | -2 | 2 | -2 | -2 | 2 | 2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | -√3 | -√3 | √3 | √3 | symplectic lifted from Dic6, Schur index 2 |
| ρ17 | 2 | -2 | -2 | 2 | -2 | -2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ18 | 2 | 2 | -2 | -2 | -2 | 2 | 2 | -2 | -1 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | symplectic lifted from Dic3, Schur index 2 |
| ρ19 | 3 | 3 | 3 | 3 | -1 | -1 | -1 | -1 | 0 | -3 | -3 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×S4 |
| ρ20 | 3 | 3 | 3 | 3 | -1 | -1 | -1 | -1 | 0 | 3 | 3 | -1 | -1 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S4 |
| ρ21 | 3 | 3 | 3 | 3 | -1 | -1 | -1 | -1 | 0 | -3 | -3 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | 1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×S4 |
| ρ22 | 3 | 3 | 3 | 3 | -1 | -1 | -1 | -1 | 0 | 3 | 3 | -1 | -1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S4 |
| ρ23 | 3 | 3 | -3 | -3 | 1 | -1 | -1 | 1 | 0 | 3 | -3 | -1 | 1 | -i | i | i | -i | -i | i | i | -i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from A4⋊C4 |
| ρ24 | 3 | 3 | -3 | -3 | 1 | -1 | -1 | 1 | 0 | 3 | -3 | -1 | 1 | i | -i | -i | i | i | -i | -i | i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from A4⋊C4 |
| ρ25 | 3 | 3 | -3 | -3 | 1 | -1 | -1 | 1 | 0 | -3 | 3 | 1 | -1 | -i | -i | -i | i | -i | i | i | i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from A4⋊C4 |
| ρ26 | 3 | 3 | -3 | -3 | 1 | -1 | -1 | 1 | 0 | -3 | 3 | 1 | -1 | i | i | i | -i | i | -i | -i | -i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from A4⋊C4 |
| ρ27 | 6 | -6 | 6 | -6 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C4⋊S4 |
| ρ28 | 6 | -6 | -6 | 6 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from A4⋊Q8, Schur index 2 |
►On 48 points
Generators in S48
(1 19)(2 20)(3 21)(4 22)(5 23)(6 24)(7 13)(8 14)(9 15)(10 16)(11 17)(12 18)(25 39)(26 40)(27 41)(28 42)(29 43)(30 44)(31 45)(32 46)(33 47)(34 48)(35 37)(36 38)
(1 7)(2 8)(3 9)(4 10)(5 11)(6 12)(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)(25 31)(26 32)(27 33)(28 34)(29 35)(30 36)(37 43)(38 44)(39 45)(40 46)(41 47)(42 48)
(1 42)(2 29)(3 21)(4 45)(5 32)(6 24)(7 48)(8 35)(9 15)(10 39)(11 26)(12 18)(13 34)(14 37)(16 25)(17 40)(19 28)(20 43)(22 31)(23 46)(27 41)(30 44)(33 47)(36 38)
(1 28)(2 20)(3 44)(4 31)(5 23)(6 47)(7 34)(8 14)(9 38)(10 25)(11 17)(12 41)(13 48)(15 36)(16 39)(18 27)(19 42)(21 30)(22 45)(24 33)(26 40)(29 43)(32 46)(35 37)
(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 38 19 36)(2 37 20 35)(3 48 21 34)(4 47 22 33)(5 46 23 32)(6 45 24 31)(7 44 13 30)(8 43 14 29)(9 42 15 28)(10 41 16 27)(11 40 17 26)(12 39 18 25)
G:=sub<Sym(48)| (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,13)(8,14)(9,15)(10,16)(11,17)(12,18)(25,39)(26,40)(27,41)(28,42)(29,43)(30,44)(31,45)(32,46)(33,47)(34,48)(35,37)(36,38), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24)(25,31)(26,32)(27,33)(28,34)(29,35)(30,36)(37,43)(38,44)(39,45)(40,46)(41,47)(42,48), (1,42)(2,29)(3,21)(4,45)(5,32)(6,24)(7,48)(8,35)(9,15)(10,39)(11,26)(12,18)(13,34)(14,37)(16,25)(17,40)(19,28)(20,43)(22,31)(23,46)(27,41)(30,44)(33,47)(36,38), (1,28)(2,20)(3,44)(4,31)(5,23)(6,47)(7,34)(8,14)(9,38)(10,25)(11,17)(12,41)(13,48)(15,36)(16,39)(18,27)(19,42)(21,30)(22,45)(24,33)(26,40)(29,43)(32,46)(35,37), (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), (1,38,19,36)(2,37,20,35)(3,48,21,34)(4,47,22,33)(5,46,23,32)(6,45,24,31)(7,44,13,30)(8,43,14,29)(9,42,15,28)(10,41,16,27)(11,40,17,26)(12,39,18,25)>;
G:=Group( (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,13)(8,14)(9,15)(10,16)(11,17)(12,18)(25,39)(26,40)(27,41)(28,42)(29,43)(30,44)(31,45)(32,46)(33,47)(34,48)(35,37)(36,38), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24)(25,31)(26,32)(27,33)(28,34)(29,35)(30,36)(37,43)(38,44)(39,45)(40,46)(41,47)(42,48), (1,42)(2,29)(3,21)(4,45)(5,32)(6,24)(7,48)(8,35)(9,15)(10,39)(11,26)(12,18)(13,34)(14,37)(16,25)(17,40)(19,28)(20,43)(22,31)(23,46)(27,41)(30,44)(33,47)(36,38), (1,28)(2,20)(3,44)(4,31)(5,23)(6,47)(7,34)(8,14)(9,38)(10,25)(11,17)(12,41)(13,48)(15,36)(16,39)(18,27)(19,42)(21,30)(22,45)(24,33)(26,40)(29,43)(32,46)(35,37), (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), (1,38,19,36)(2,37,20,35)(3,48,21,34)(4,47,22,33)(5,46,23,32)(6,45,24,31)(7,44,13,30)(8,43,14,29)(9,42,15,28)(10,41,16,27)(11,40,17,26)(12,39,18,25) );
G=PermutationGroup([[(1,19),(2,20),(3,21),(4,22),(5,23),(6,24),(7,13),(8,14),(9,15),(10,16),(11,17),(12,18),(25,39),(26,40),(27,41),(28,42),(29,43),(30,44),(31,45),(32,46),(33,47),(34,48),(35,37),(36,38)], [(1,7),(2,8),(3,9),(4,10),(5,11),(6,12),(13,19),(14,20),(15,21),(16,22),(17,23),(18,24),(25,31),(26,32),(27,33),(28,34),(29,35),(30,36),(37,43),(38,44),(39,45),(40,46),(41,47),(42,48)], [(1,42),(2,29),(3,21),(4,45),(5,32),(6,24),(7,48),(8,35),(9,15),(10,39),(11,26),(12,18),(13,34),(14,37),(16,25),(17,40),(19,28),(20,43),(22,31),(23,46),(27,41),(30,44),(33,47),(36,38)], [(1,28),(2,20),(3,44),(4,31),(5,23),(6,47),(7,34),(8,14),(9,38),(10,25),(11,17),(12,41),(13,48),(15,36),(16,39),(18,27),(19,42),(21,30),(22,45),(24,33),(26,40),(29,43),(32,46),(35,37)], [(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)], [(1,38,19,36),(2,37,20,35),(3,48,21,34),(4,47,22,33),(5,46,23,32),(6,45,24,31),(7,44,13,30),(8,43,14,29),(9,42,15,28),(10,41,16,27),(11,40,17,26),(12,39,18,25)]])
Matrix representation of C24.4D6 ►in GL7(𝔽13)
| 12 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 12 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 12 |
| 7 | 3 | 0 | 0 | 0 | 0 | 0 |
| 10 | 10 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 8 | 5 | 0 | 0 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 5 | 0 | 0 | 0 |
| 0 | 0 | 0 | 8 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 12 |
| 0 | 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 | 0 |
G:=sub<GL(7,GF(13))| [12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12],[7,10,0,0,0,0,0,3,10,0,0,0,0,0,0,0,1,12,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0],[8,0,0,0,0,0,0,5,5,0,0,0,0,0,0,0,5,0,0,0,0,0,0,5,8,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12,0,0] >;
C24.4D6 in GAP, Magma, Sage, TeX
C_2^4._4D_6
% in TeX
G:=Group("C2^4.4D6"); // GroupNames label
G:=SmallGroup(192,971);
// by ID
G=gap.SmallGroup(192,971);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-2,2,28,141,64,1124,4037,285,2358,475]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^6=b,f^2=a,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,f*c*f^-1=e*d*e^-1=c*d=d*c,e*c*e^-1=d,d*f=f*d,f*e*f^-1=b*e^5>;
// generators/relations
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