metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C24⋊5Dic3, C3⋊3C2≀C4, (C23×C6)⋊3C4, (C2×D4).7D6, (C2×C12).3D4, C22≀C2.2S3, C12.D4⋊2C2, C22⋊C4⋊1Dic3, (C6×D4).5C22, (C22×C6).14D4, C6.19(C23⋊C4), C23.7D6⋊2C2, C23.5(C3⋊D4), C23.6(C2×Dic3), C2.4(C23.7D6), C22.12(C6.D4), (C3×C22⋊C4)⋊1C4, (C2×C4).5(C3⋊D4), (C3×C22≀C2).1C2, (C22×C6).13(C2×C4), (C2×C6).94(C22⋊C4), SmallGroup(192,95)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24⋊5Dic3
G = < a,b,c,d,e,f | a2=b2=c2=d2=e6=1, f2=e3, ab=ba, eae-1=ac=ca, ad=da, faf-1=abcd, bc=cb, ebe-1=bd=db, fbf-1=bcd, fcf-1=cd=dc, ce=ec, de=ed, df=fd, fef-1=e-1 >
Subgroups: 304 in 94 conjugacy classes, 23 normal (all characteristic)
C1, C2, C2, C3, C4, C22, C22, C6, C6, C8, C2×C4, C2×C4, D4, C23, C23, Dic3, C12, C2×C6, C2×C6, C22⋊C4, C22⋊C4, M4(2), C2×D4, C2×D4, C24, C3⋊C8, C2×Dic3, C2×C12, C2×C12, C3×D4, C22×C6, C22×C6, C23⋊C4, C4.D4, C22≀C2, C4.Dic3, C6.D4, C3×C22⋊C4, C3×C22⋊C4, C6×D4, C6×D4, C23×C6, C2≀C4, C12.D4, C23.7D6, C3×C22≀C2, C24⋊5Dic3
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Dic3, D6, C22⋊C4, C2×Dic3, C3⋊D4, C23⋊C4, C6.D4, C2≀C4, C23.7D6, C24⋊5Dic3
Character table of C24⋊5Dic3
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3 | 4A | 4B | 4C | 4D | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 6I | 6J | 8A | 8B | 12A | 12B | 12C | |
| size | 1 | 1 | 2 | 4 | 4 | 4 | 4 | 2 | 4 | 8 | 24 | 24 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 24 | 24 | 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 | -i | i | 1 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | -1 | i | -i | 1 | 1 | -1 | linear of order 4 |
| ρ6 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | i | -i | 1 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | -1 | -i | i | 1 | 1 | -1 | linear of order 4 |
| ρ7 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | -i | i | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -i | i | -1 | -1 | -1 | linear of order 4 |
| ρ8 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | i | -i | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | i | -i | -1 | -1 | -1 | linear of order 4 |
| ρ9 | 2 | 2 | 2 | 2 | -2 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 2 | 2 | 2 | 0 | -2 | 0 | 0 | -2 | 0 | 2 | 0 | 0 | 0 | 0 | -2 | orthogonal lifted from D4 |
| ρ10 | 2 | 2 | 2 | 2 | 2 | -2 | -2 | -1 | 2 | -2 | 0 | 0 | -1 | -1 | -1 | 1 | -1 | 1 | 1 | -1 | 1 | -1 | 0 | 0 | 1 | 1 | -1 | orthogonal lifted from D6 |
| ρ11 | 2 | 2 | 2 | -2 | -2 | 0 | 0 | 2 | 2 | 0 | 0 | 0 | 2 | 2 | 2 | 0 | -2 | 0 | 0 | -2 | 0 | -2 | 0 | 0 | 0 | 0 | 2 | orthogonal lifted from D4 |
| ρ12 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | -1 | 2 | 2 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 0 | 0 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ13 | 2 | 2 | 2 | -2 | 2 | 2 | 2 | -1 | -2 | -2 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 0 | 0 | 1 | 1 | 1 | symplectic lifted from Dic3, Schur index 2 |
| ρ14 | 2 | 2 | 2 | -2 | 2 | -2 | -2 | -1 | -2 | 2 | 0 | 0 | -1 | -1 | -1 | 1 | -1 | 1 | 1 | -1 | 1 | 1 | 0 | 0 | -1 | -1 | 1 | symplectic lifted from Dic3, Schur index 2 |
| ρ15 | 2 | 2 | 2 | 2 | -2 | 0 | 0 | -1 | -2 | 0 | 0 | 0 | -1 | -1 | -1 | √-3 | 1 | √-3 | -√-3 | 1 | -√-3 | -1 | 0 | 0 | √-3 | -√-3 | 1 | complex lifted from C3⋊D4 |
| ρ16 | 2 | 2 | 2 | -2 | -2 | 0 | 0 | -1 | 2 | 0 | 0 | 0 | -1 | -1 | -1 | √-3 | 1 | √-3 | -√-3 | 1 | -√-3 | 1 | 0 | 0 | -√-3 | √-3 | -1 | complex lifted from C3⋊D4 |
| ρ17 | 2 | 2 | 2 | -2 | -2 | 0 | 0 | -1 | 2 | 0 | 0 | 0 | -1 | -1 | -1 | -√-3 | 1 | -√-3 | √-3 | 1 | √-3 | 1 | 0 | 0 | √-3 | -√-3 | -1 | complex lifted from C3⋊D4 |
| ρ18 | 2 | 2 | 2 | 2 | -2 | 0 | 0 | -1 | -2 | 0 | 0 | 0 | -1 | -1 | -1 | -√-3 | 1 | -√-3 | √-3 | 1 | √-3 | -1 | 0 | 0 | -√-3 | √-3 | 1 | complex lifted from C3⋊D4 |
| ρ19 | 4 | -4 | 0 | 0 | 0 | -2 | 2 | 4 | 0 | 0 | 0 | 0 | 0 | -4 | 0 | 2 | 0 | -2 | 2 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2≀C4 |
| ρ20 | 4 | -4 | 0 | 0 | 0 | 2 | -2 | 4 | 0 | 0 | 0 | 0 | 0 | -4 | 0 | -2 | 0 | 2 | -2 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2≀C4 |
| ρ21 | 4 | 4 | -4 | 0 | 0 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | -4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C23⋊C4 |
| ρ22 | 4 | 4 | -4 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 2 | -2 | 2 | 0 | -2√-3 | 0 | 0 | 2√-3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C23.7D6 |
| ρ23 | 4 | -4 | 0 | 0 | 0 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | -2√-3 | 2 | 2√-3 | -1-√-3 | 0 | 1+√-3 | -1+√-3 | 0 | 1-√-3 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ24 | 4 | -4 | 0 | 0 | 0 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 2√-3 | 2 | -2√-3 | 1-√-3 | 0 | -1+√-3 | 1+√-3 | 0 | -1-√-3 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ25 | 4 | 4 | -4 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 2 | -2 | 2 | 0 | 2√-3 | 0 | 0 | -2√-3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C23.7D6 |
| ρ26 | 4 | -4 | 0 | 0 | 0 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | -2√-3 | 2 | 2√-3 | 1+√-3 | 0 | -1-√-3 | 1-√-3 | 0 | -1+√-3 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ27 | 4 | -4 | 0 | 0 | 0 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 2√-3 | 2 | -2√-3 | -1+√-3 | 0 | 1-√-3 | -1-√-3 | 0 | 1+√-3 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
►On 24 points - transitive group 24T289
Generators in S24
(1 15)(2 16)(3 17)(4 18)(5 13)(6 14)(8 22)(10 24)(12 20)
(2 16)(4 18)(6 14)(7 21)(9 23)(11 19)
(7 21)(8 22)(9 23)(10 24)(11 19)(12 20)
(1 15)(2 16)(3 17)(4 18)(5 13)(6 14)(7 21)(8 22)(9 23)(10 24)(11 19)(12 20)
(1 2 3 4 5 6)(7 8 9 10 11 12)(13 14 15 16 17 18)(19 20 21 22 23 24)
(1 21 4 24)(2 20 5 23)(3 19 6 22)(7 18 10 15)(8 17 11 14)(9 16 12 13)
G:=sub<Sym(24)| (1,15)(2,16)(3,17)(4,18)(5,13)(6,14)(8,22)(10,24)(12,20), (2,16)(4,18)(6,14)(7,21)(9,23)(11,19), (7,21)(8,22)(9,23)(10,24)(11,19)(12,20), (1,15)(2,16)(3,17)(4,18)(5,13)(6,14)(7,21)(8,22)(9,23)(10,24)(11,19)(12,20), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24), (1,21,4,24)(2,20,5,23)(3,19,6,22)(7,18,10,15)(8,17,11,14)(9,16,12,13)>;
G:=Group( (1,15)(2,16)(3,17)(4,18)(5,13)(6,14)(8,22)(10,24)(12,20), (2,16)(4,18)(6,14)(7,21)(9,23)(11,19), (7,21)(8,22)(9,23)(10,24)(11,19)(12,20), (1,15)(2,16)(3,17)(4,18)(5,13)(6,14)(7,21)(8,22)(9,23)(10,24)(11,19)(12,20), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24), (1,21,4,24)(2,20,5,23)(3,19,6,22)(7,18,10,15)(8,17,11,14)(9,16,12,13) );
G=PermutationGroup([[(1,15),(2,16),(3,17),(4,18),(5,13),(6,14),(8,22),(10,24),(12,20)], [(2,16),(4,18),(6,14),(7,21),(9,23),(11,19)], [(7,21),(8,22),(9,23),(10,24),(11,19),(12,20)], [(1,15),(2,16),(3,17),(4,18),(5,13),(6,14),(7,21),(8,22),(9,23),(10,24),(11,19),(12,20)], [(1,2,3,4,5,6),(7,8,9,10,11,12),(13,14,15,16,17,18),(19,20,21,22,23,24)], [(1,21,4,24),(2,20,5,23),(3,19,6,22),(7,18,10,15),(8,17,11,14),(9,16,12,13)]])
G:=TransitiveGroup(24,289);
►On 24 points - transitive group 24T356
Generators in S24
(1 19)(2 23)(3 21)(4 8)(5 9)(6 7)(10 22)(11 20)(12 24)(13 16)(14 17)(15 18)
(1 22)(2 20)(3 24)(4 17)(5 15)(6 13)(7 16)(8 14)(9 18)(10 19)(11 23)(12 21)
(1 10)(2 11)(3 12)(19 22)(20 23)(21 24)
(1 10)(2 11)(3 12)(4 8)(5 9)(6 7)(13 16)(14 17)(15 18)(19 22)(20 23)(21 24)
(1 2 3)(4 5 6)(7 8 9)(10 11 12)(13 14 15 16 17 18)(19 20 21 22 23 24)
(1 9)(2 8)(3 7)(4 11)(5 10)(6 12)(13 24 16 21)(14 23 17 20)(15 22 18 19)
G:=sub<Sym(24)| (1,19)(2,23)(3,21)(4,8)(5,9)(6,7)(10,22)(11,20)(12,24)(13,16)(14,17)(15,18), (1,22)(2,20)(3,24)(4,17)(5,15)(6,13)(7,16)(8,14)(9,18)(10,19)(11,23)(12,21), (1,10)(2,11)(3,12)(19,22)(20,23)(21,24), (1,10)(2,11)(3,12)(4,8)(5,9)(6,7)(13,16)(14,17)(15,18)(19,22)(20,23)(21,24), (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24), (1,9)(2,8)(3,7)(4,11)(5,10)(6,12)(13,24,16,21)(14,23,17,20)(15,22,18,19)>;
G:=Group( (1,19)(2,23)(3,21)(4,8)(5,9)(6,7)(10,22)(11,20)(12,24)(13,16)(14,17)(15,18), (1,22)(2,20)(3,24)(4,17)(5,15)(6,13)(7,16)(8,14)(9,18)(10,19)(11,23)(12,21), (1,10)(2,11)(3,12)(19,22)(20,23)(21,24), (1,10)(2,11)(3,12)(4,8)(5,9)(6,7)(13,16)(14,17)(15,18)(19,22)(20,23)(21,24), (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24), (1,9)(2,8)(3,7)(4,11)(5,10)(6,12)(13,24,16,21)(14,23,17,20)(15,22,18,19) );
G=PermutationGroup([[(1,19),(2,23),(3,21),(4,8),(5,9),(6,7),(10,22),(11,20),(12,24),(13,16),(14,17),(15,18)], [(1,22),(2,20),(3,24),(4,17),(5,15),(6,13),(7,16),(8,14),(9,18),(10,19),(11,23),(12,21)], [(1,10),(2,11),(3,12),(19,22),(20,23),(21,24)], [(1,10),(2,11),(3,12),(4,8),(5,9),(6,7),(13,16),(14,17),(15,18),(19,22),(20,23),(21,24)], [(1,2,3),(4,5,6),(7,8,9),(10,11,12),(13,14,15,16,17,18),(19,20,21,22,23,24)], [(1,9),(2,8),(3,7),(4,11),(5,10),(6,12),(13,24,16,21),(14,23,17,20),(15,22,18,19)]])
G:=TransitiveGroup(24,356);
Matrix representation of C24⋊5Dic3 ►in GL4(𝔽7) generated by
| 1 | 0 | 4 | 0 |
| 0 | 1 | 5 | 0 |
| 0 | 0 | 6 | 0 |
| 0 | 0 | 0 | 1 |
| 2 | 6 | 0 | 4 |
| 2 | 6 | 4 | 1 |
| 0 | 0 | 6 | 0 |
| 5 | 2 | 1 | 0 |
| 0 | 6 | 3 | 2 |
| 6 | 0 | 4 | 2 |
| 0 | 0 | 6 | 0 |
| 0 | 0 | 0 | 1 |
| 6 | 0 | 0 | 0 |
| 0 | 6 | 0 | 0 |
| 0 | 0 | 6 | 0 |
| 0 | 0 | 0 | 6 |
| 3 | 5 | 4 | 1 |
| 6 | 4 | 4 | 6 |
| 5 | 5 | 5 | 4 |
| 0 | 0 | 0 | 2 |
| 2 | 3 | 2 | 1 |
| 4 | 2 | 1 | 1 |
| 4 | 3 | 5 | 5 |
| 1 | 1 | 6 | 5 |
G:=sub<GL(4,GF(7))| [1,0,0,0,0,1,0,0,4,5,6,0,0,0,0,1],[2,2,0,5,6,6,0,2,0,4,6,1,4,1,0,0],[0,6,0,0,6,0,0,0,3,4,6,0,2,2,0,1],[6,0,0,0,0,6,0,0,0,0,6,0,0,0,0,6],[3,6,5,0,5,4,5,0,4,4,5,0,1,6,4,2],[2,4,4,1,3,2,3,1,2,1,5,6,1,1,5,5] >;
C24⋊5Dic3 in GAP, Magma, Sage, TeX
C_2^4\rtimes_5{\rm Dic}_3 % in TeX
G:=Group("C2^4:5Dic3"); // GroupNames label
G:=SmallGroup(192,95);
// by ID
G=gap.SmallGroup(192,95);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,28,141,219,675,297,1684,6278]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=e^6=1,f^2=e^3,a*b=b*a,e*a*e^-1=a*c=c*a,a*d=d*a,f*a*f^-1=a*b*c*d,b*c=c*b,e*b*e^-1=b*d=d*b,f*b*f^-1=b*c*d,f*c*f^-1=c*d=d*c,c*e=e*c,d*e=e*d,d*f=f*d,f*e*f^-1=e^-1>;
// generators/relations
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