metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C23.7D6, C23⋊2Dic3, (C2×C4)⋊Dic3, (C2×C12)⋊1C4, (C2×C6).2D4, C3⋊2(C23⋊C4), (C22×C6)⋊2C4, (C6×D4).6C2, (C2×D4).3S3, C6.D4⋊2C2, C6.15(C22⋊C4), C22.2(C3⋊D4), (C22×C6).6C22, C22.3(C2×Dic3), C2.5(C6.D4), (C2×C6).29(C2×C4), SmallGroup(96,41)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C23.7D6
G = < a,b,c,d,e | a2=b2=c2=d6=1, e2=ba=ab, dad-1=eae-1=ac=ca, ebe-1=bc=cb, bd=db, cd=dc, ce=ec, ede-1=bcd-1 >
Character table of C23.7D6
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 12A | 12B | |
| size | 1 | 1 | 2 | 2 | 2 | 4 | 2 | 4 | 12 | 12 | 12 | 12 | 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 | trivial |
| ρ2 | 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 | 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 | linear of order 2 |
| ρ5 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | i | i | -i | -i | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 4 |
| ρ6 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | -i | i | i | -i | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | linear of order 4 |
| ρ7 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | i | -i | -i | i | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | linear of order 4 |
| ρ8 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | -i | -i | i | i | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 4 |
| ρ9 | 2 | 2 | -2 | -2 | 2 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | -2 | 0 | 0 | 2 | -2 | 0 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | 2 | 2 | 2 | 2 | 2 | -1 | 2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ11 | 2 | 2 | 2 | 2 | 2 | -2 | -1 | -2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | orthogonal lifted from D6 |
| ρ12 | 2 | 2 | 2 | -2 | -2 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | -2 | 0 | 0 | -2 | 2 | 0 | 0 | orthogonal lifted from D4 |
| ρ13 | 2 | 2 | -2 | 2 | -2 | 2 | -1 | -2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | symplectic lifted from Dic3, Schur index 2 |
| ρ14 | 2 | 2 | -2 | 2 | -2 | -2 | -1 | 2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | symplectic lifted from Dic3, Schur index 2 |
| ρ15 | 2 | 2 | -2 | -2 | 2 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | √-3 | -√-3 | -1 | 1 | -√-3 | √-3 | complex lifted from C3⋊D4 |
| ρ16 | 2 | 2 | 2 | -2 | -2 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | -√-3 | √-3 | 1 | -1 | -√-3 | √-3 | complex lifted from C3⋊D4 |
| ρ17 | 2 | 2 | 2 | -2 | -2 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | √-3 | -√-3 | 1 | -1 | √-3 | -√-3 | complex lifted from C3⋊D4 |
| ρ18 | 2 | 2 | -2 | -2 | 2 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | -√-3 | √-3 | -1 | 1 | √-3 | -√-3 | complex lifted from C3⋊D4 |
| ρ19 | 4 | -4 | 0 | 0 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C23⋊C4 |
| ρ20 | 4 | -4 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | -2√-3 | 2 | 2√-3 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ21 | 4 | -4 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 2√-3 | 2 | -2√-3 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
►On 24 points - transitive group 24T96
Generators in S24
(2 13)(4 15)(6 17)(8 21)(10 23)(12 19)
(7 20)(8 21)(9 22)(10 23)(11 24)(12 19)
(1 18)(2 13)(3 14)(4 15)(5 16)(6 17)(7 20)(8 21)(9 22)(10 23)(11 24)(12 19)
(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 19)(2 11 13 24)(3 23)(4 9 15 22)(5 21)(6 7 17 20)(8 16)(10 14)(12 18)
G:=sub<Sym(24)| (2,13)(4,15)(6,17)(8,21)(10,23)(12,19), (7,20)(8,21)(9,22)(10,23)(11,24)(12,19), (1,18)(2,13)(3,14)(4,15)(5,16)(6,17)(7,20)(8,21)(9,22)(10,23)(11,24)(12,19), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24), (1,19)(2,11,13,24)(3,23)(4,9,15,22)(5,21)(6,7,17,20)(8,16)(10,14)(12,18)>;
G:=Group( (2,13)(4,15)(6,17)(8,21)(10,23)(12,19), (7,20)(8,21)(9,22)(10,23)(11,24)(12,19), (1,18)(2,13)(3,14)(4,15)(5,16)(6,17)(7,20)(8,21)(9,22)(10,23)(11,24)(12,19), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24), (1,19)(2,11,13,24)(3,23)(4,9,15,22)(5,21)(6,7,17,20)(8,16)(10,14)(12,18) );
G=PermutationGroup([[(2,13),(4,15),(6,17),(8,21),(10,23),(12,19)], [(7,20),(8,21),(9,22),(10,23),(11,24),(12,19)], [(1,18),(2,13),(3,14),(4,15),(5,16),(6,17),(7,20),(8,21),(9,22),(10,23),(11,24),(12,19)], [(1,2,3,4,5,6),(7,8,9,10,11,12),(13,14,15,16,17,18),(19,20,21,22,23,24)], [(1,19),(2,11,13,24),(3,23),(4,9,15,22),(5,21),(6,7,17,20),(8,16),(10,14),(12,18)]])
G:=TransitiveGroup(24,96);
►On 24 points - transitive group 24T98
Generators in S24
(1 21)(2 19)(3 23)(4 15)(5 13)(6 17)(7 24)(8 22)(9 20)(10 14)(11 18)(12 16)
(1 7)(2 8)(3 9)(19 22)(20 23)(21 24)
(1 7)(2 8)(3 9)(4 11)(5 12)(6 10)(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 12 24 16)(2 11 22 18)(3 10 20 14)(4 19 15 8)(5 21 13 7)(6 23 17 9)
G:=sub<Sym(24)| (1,21)(2,19)(3,23)(4,15)(5,13)(6,17)(7,24)(8,22)(9,20)(10,14)(11,18)(12,16), (1,7)(2,8)(3,9)(19,22)(20,23)(21,24), (1,7)(2,8)(3,9)(4,11)(5,12)(6,10)(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), (1,12,24,16)(2,11,22,18)(3,10,20,14)(4,19,15,8)(5,21,13,7)(6,23,17,9)>;
G:=Group( (1,21)(2,19)(3,23)(4,15)(5,13)(6,17)(7,24)(8,22)(9,20)(10,14)(11,18)(12,16), (1,7)(2,8)(3,9)(19,22)(20,23)(21,24), (1,7)(2,8)(3,9)(4,11)(5,12)(6,10)(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), (1,12,24,16)(2,11,22,18)(3,10,20,14)(4,19,15,8)(5,21,13,7)(6,23,17,9) );
G=PermutationGroup([[(1,21),(2,19),(3,23),(4,15),(5,13),(6,17),(7,24),(8,22),(9,20),(10,14),(11,18),(12,16)], [(1,7),(2,8),(3,9),(19,22),(20,23),(21,24)], [(1,7),(2,8),(3,9),(4,11),(5,12),(6,10),(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)], [(1,12,24,16),(2,11,22,18),(3,10,20,14),(4,19,15,8),(5,21,13,7),(6,23,17,9)]])
G:=TransitiveGroup(24,98);
►On 24 points - transitive group 24T111
Generators in S24
(2 20)(4 22)(6 24)(8 16)(10 18)(12 14)
(1 7)(2 8)(3 9)(4 10)(5 11)(6 12)(13 23)(14 24)(15 19)(16 20)(17 21)(18 22)
(1 19)(2 20)(3 21)(4 22)(5 23)(6 24)(7 15)(8 16)(9 17)(10 18)(11 13)(12 14)
(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 6 7 14)(2 11 16 5)(3 4 9 18)(8 23 20 13)(10 21 22 17)(12 19 24 15)
G:=sub<Sym(24)| (2,20)(4,22)(6,24)(8,16)(10,18)(12,14), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,23)(14,24)(15,19)(16,20)(17,21)(18,22), (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,15)(8,16)(9,17)(10,18)(11,13)(12,14), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24), (1,6,7,14)(2,11,16,5)(3,4,9,18)(8,23,20,13)(10,21,22,17)(12,19,24,15)>;
G:=Group( (2,20)(4,22)(6,24)(8,16)(10,18)(12,14), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,23)(14,24)(15,19)(16,20)(17,21)(18,22), (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,15)(8,16)(9,17)(10,18)(11,13)(12,14), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24), (1,6,7,14)(2,11,16,5)(3,4,9,18)(8,23,20,13)(10,21,22,17)(12,19,24,15) );
G=PermutationGroup([[(2,20),(4,22),(6,24),(8,16),(10,18),(12,14)], [(1,7),(2,8),(3,9),(4,10),(5,11),(6,12),(13,23),(14,24),(15,19),(16,20),(17,21),(18,22)], [(1,19),(2,20),(3,21),(4,22),(5,23),(6,24),(7,15),(8,16),(9,17),(10,18),(11,13),(12,14)], [(1,2,3,4,5,6),(7,8,9,10,11,12),(13,14,15,16,17,18),(19,20,21,22,23,24)], [(1,6,7,14),(2,11,16,5),(3,4,9,18),(8,23,20,13),(10,21,22,17),(12,19,24,15)]])
G:=TransitiveGroup(24,111);
C23.7D6 is a maximal subgroup of
C23.D12 C23.2D12 C23.3D12 C23.4D12 C24⋊5Dic3 (C22×C12)⋊C4 C42⋊4Dic3 C42⋊5Dic3 C23⋊C4⋊5S3 S3×C23⋊C4 C24⋊6D6 C22⋊C4⋊D6 (C6×D4)⋊10C4 2+ 1+4.5S3 2+ 1+4⋊7S3 C23⋊2Dic9 C62.31D4 C62.38D4 (C2×C6).D20 C23.7D30 (C2×C60)⋊C4 C3⋊(C23⋊F5)
C23.7D6 is a maximal quotient of
C24.3Dic3 C24.12D6 (C2×C12)⋊C8 C24⋊5Dic3 (C6×D4)⋊C4 (C6×Q8)⋊C4 (C22×C12)⋊C4 C42⋊4Dic3 C42.Dic3 C42⋊5Dic3 C42.3Dic3 C23⋊2Dic9 C62.31D4 C62.38D4 (C2×C6).D20 C23.7D30 (C2×C60)⋊C4 C3⋊(C23⋊F5)
Matrix representation of C23.7D6 ►in GL4(𝔽7) generated by
| 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 | 1 | 4 | 1 |
| 3 | 3 | 0 | 0 |
| 1 | 6 | 3 | 3 |
| 4 | 4 | 3 | 6 |
G:=sub<GL(4,GF(7))| [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],[2,3,1,4,1,3,6,4,4,0,3,3,1,0,3,6] >;
C23.7D6 in GAP, Magma, Sage, TeX
C_2^3._7D_6
% in TeX
G:=Group("C2^3.7D6"); // GroupNames label
G:=SmallGroup(96,41);
// by ID
G=gap.SmallGroup(96,41);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-3,24,121,188,579,2309]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^6=1,e^2=b*a=a*b,d*a*d^-1=e*a*e^-1=a*c=c*a,e*b*e^-1=b*c=c*b,b*d=d*b,c*d=d*c,c*e=e*c,e*d*e^-1=b*c*d^-1>;
// generators/relations
Export
Subgroup lattice of C23.7D6 in TeX
Character table of C23.7D6 in TeX