metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D6.4D4, C23.9D6, D6⋊C4⋊5C2, (C2×C4).6D6, C2.8(S3×D4), C22⋊C4⋊3S3, C4⋊Dic3⋊4C2, C6.19(C2×D4), C6.8(C4○D4), Dic3⋊C4⋊10C2, C6.D4⋊4C2, (C2×C6).24C23, C2.10(C4○D12), C2.8(D4⋊2S3), (C2×C12).52C22, C3⋊1(C22.D4), C22.42(C22×S3), (C22×C6).13C22, (C2×Dic3).6C22, (C22×S3).17C22, (S3×C2×C4)⋊10C2, (C3×C22⋊C4)⋊5C2, (C2×C3⋊D4).3C2, SmallGroup(96,90)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C23.9D6
G = < a,b,c,d,e | a2=b2=c2=1, d6=e2=b, ab=ba, dad-1=ac=ca, eae-1=abc, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=d5 >
Subgroups: 186 in 78 conjugacy classes, 31 normal (29 characteristic)
C1, C2, C2, C3, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, C23, C23, Dic3, C12, D6, D6, C2×C6, C2×C6, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C4×S3, C2×Dic3, C3⋊D4, C2×C12, C22×S3, C22×C6, C22.D4, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C6.D4, C3×C22⋊C4, S3×C2×C4, C2×C3⋊D4, C23.9D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, C22×S3, C22.D4, C4○D12, S3×D4, D4⋊2S3, C23.9D6
Character table of C23.9D6
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 6A | 6B | 6C | 6D | 6E | 12A | 12B | 12C | 12D | |
| size | 1 | 1 | 1 | 1 | 4 | 6 | 6 | 2 | 2 | 2 | 4 | 6 | 6 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | linear of order 2 |
| ρ9 | 2 | 2 | 2 | 2 | -2 | 0 | 0 | -1 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 1 | 1 | -1 | 1 | -1 | 1 | orthogonal lifted from D6 |
| ρ10 | 2 | -2 | 2 | -2 | 0 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ11 | 2 | 2 | 2 | 2 | -2 | 0 | 0 | -1 | 2 | 2 | -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 | 0 | -1 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ13 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ14 | 2 | -2 | 2 | -2 | 0 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ15 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | -2i | 2i | 0 | 0 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ16 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 2i | -2i | 0 | 0 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ17 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 2 | 2i | -2i | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | 0 | 0 | 0 | 2i | 0 | -2i | complex lifted from C4○D4 |
| ρ18 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 2 | -2i | 2i | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | 0 | 0 | 0 | -2i | 0 | 2i | complex lifted from C4○D4 |
| ρ19 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | -1 | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | -√-3 | √-3 | -√3 | -i | √3 | i | complex lifted from C4○D12 |
| ρ20 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | -1 | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | √-3 | -√-3 | √3 | -i | -√3 | i | complex lifted from C4○D12 |
| ρ21 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | -1 | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | -√-3 | √-3 | √3 | i | -√3 | -i | complex lifted from C4○D12 |
| ρ22 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | -1 | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | √-3 | -√-3 | -√3 | i | √3 | -i | complex lifted from C4○D12 |
| ρ23 | 4 | -4 | 4 | -4 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S3×D4 |
| ρ24 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from D4⋊2S3, Schur index 2 |
►On 48 points
(1 44)(2 23)(3 46)(4 13)(5 48)(6 15)(7 38)(8 17)(9 40)(10 19)(11 42)(12 21)(14 34)(16 36)(18 26)(20 28)(22 30)(24 32)(25 39)(27 41)(29 43)(31 45)(33 47)(35 37)
(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 30)(2 31)(3 32)(4 33)(5 34)(6 35)(7 36)(8 25)(9 26)(10 27)(11 28)(12 29)(13 47)(14 48)(15 37)(16 38)(17 39)(18 40)(19 41)(20 42)(21 43)(22 44)(23 45)(24 46)
(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 6 7 12)(2 11 8 5)(3 4 9 10)(13 24 19 18)(14 17 20 23)(15 22 21 16)(25 34 31 28)(26 27 32 33)(29 30 35 36)(37 44 43 38)(39 42 45 48)(40 47 46 41)
G:=sub<Sym(48)| (1,44)(2,23)(3,46)(4,13)(5,48)(6,15)(7,38)(8,17)(9,40)(10,19)(11,42)(12,21)(14,34)(16,36)(18,26)(20,28)(22,30)(24,32)(25,39)(27,41)(29,43)(31,45)(33,47)(35,37), (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), (1,30)(2,31)(3,32)(4,33)(5,34)(6,35)(7,36)(8,25)(9,26)(10,27)(11,28)(12,29)(13,47)(14,48)(15,37)(16,38)(17,39)(18,40)(19,41)(20,42)(21,43)(22,44)(23,45)(24,46), (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), (1,6,7,12)(2,11,8,5)(3,4,9,10)(13,24,19,18)(14,17,20,23)(15,22,21,16)(25,34,31,28)(26,27,32,33)(29,30,35,36)(37,44,43,38)(39,42,45,48)(40,47,46,41)>;
G:=Group( (1,44)(2,23)(3,46)(4,13)(5,48)(6,15)(7,38)(8,17)(9,40)(10,19)(11,42)(12,21)(14,34)(16,36)(18,26)(20,28)(22,30)(24,32)(25,39)(27,41)(29,43)(31,45)(33,47)(35,37), (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), (1,30)(2,31)(3,32)(4,33)(5,34)(6,35)(7,36)(8,25)(9,26)(10,27)(11,28)(12,29)(13,47)(14,48)(15,37)(16,38)(17,39)(18,40)(19,41)(20,42)(21,43)(22,44)(23,45)(24,46), (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), (1,6,7,12)(2,11,8,5)(3,4,9,10)(13,24,19,18)(14,17,20,23)(15,22,21,16)(25,34,31,28)(26,27,32,33)(29,30,35,36)(37,44,43,38)(39,42,45,48)(40,47,46,41) );
G=PermutationGroup([[(1,44),(2,23),(3,46),(4,13),(5,48),(6,15),(7,38),(8,17),(9,40),(10,19),(11,42),(12,21),(14,34),(16,36),(18,26),(20,28),(22,30),(24,32),(25,39),(27,41),(29,43),(31,45),(33,47),(35,37)], [(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)], [(1,30),(2,31),(3,32),(4,33),(5,34),(6,35),(7,36),(8,25),(9,26),(10,27),(11,28),(12,29),(13,47),(14,48),(15,37),(16,38),(17,39),(18,40),(19,41),(20,42),(21,43),(22,44),(23,45),(24,46)], [(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)], [(1,6,7,12),(2,11,8,5),(3,4,9,10),(13,24,19,18),(14,17,20,23),(15,22,21,16),(25,34,31,28),(26,27,32,33),(29,30,35,36),(37,44,43,38),(39,42,45,48),(40,47,46,41)]])
C23.9D6 is a maximal subgroup of
C24.38D6 C24.41D6 C24.42D6 C42.93D6 C42.94D6 C42.95D6 C42.99D6 C42⋊14D6 D12⋊24D4 C42⋊18D6 C42.229D6 C42.114D6 C42.116D6 C42.118D6 C42.119D6 C24⋊7D6 C24.44D6 C24.46D6 C24.47D6 C6.342+ 1+4 C6.372+ 1+4 C4⋊C4⋊21D6 C6.732- 1+4 D12⋊20D4 C6.442+ 1+4 C6.472+ 1+4 C6.492+ 1+4 C6.162- 1+4 D12⋊22D4 C6.202- 1+4 C6.212- 1+4 C6.242- 1+4 C6.252- 1+4 C6.592+ 1+4 S3×C22.D4 C6.822- 1+4 C6.1222+ 1+4 C6.622+ 1+4 C6.632+ 1+4 C6.642+ 1+4 C6.652+ 1+4 C6.662+ 1+4 C6.852- 1+4 C6.682+ 1+4 C6.692+ 1+4 C42.137D6 C42.141D6 D12⋊10D4 C42⋊22D6 C42⋊23D6 C42.234D6 C42.143D6 C42.145D6 C42⋊25D6 C42⋊26D6 C42.189D6 C42.161D6 C42.162D6 C42.165D6 C23.9D18 C62.23C23 C62.24C23 D6.9D12 C62.75C23 C62.111C23 C62.117C23 C62.227C23 D30.34D4 D30.D4 D10⋊C4⋊S3 D6.9D20 (S3×C10).D4 D30.16D4 D30.28D4
C23.9D6 is a maximal quotient of
C6.(C4×D4) C2.(C4×Dic6) C6.(C4⋊Q8) (C2×Dic3).9D4 C22.58(S3×D4) D6⋊C4⋊C4 (C2×C4).21D12 C6.(C4⋊D4) D6.D8 D6.SD16 D6⋊C8⋊11C2 C24⋊1C4⋊C2 D6.1SD16 D6.Q16 D6⋊C8.C2 C8⋊Dic3⋊C2 C24.15D6 C24.18D6 C24.19D6 C24.20D6 C24.23D6 C24.25D6 C24.27D6 C23.9D18 C62.23C23 C62.24C23 D6.9D12 C62.75C23 C62.111C23 C62.117C23 C62.227C23 D30.34D4 D30.D4 D10⋊C4⋊S3 D6.9D20 (S3×C10).D4 D30.16D4 D30.28D4
Matrix representation of C23.9D6 ►in GL6(𝔽13)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 0 | 0 | 0 |
| 0 | 0 | 0 | 5 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 0 | 0 | 0 |
| 0 | 0 | 0 | 8 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[1,12,0,0,0,0,1,0,0,0,0,0,0,0,5,0,0,0,0,0,0,5,0,0,0,0,0,0,1,0,0,0,0,0,0,12],[1,0,0,0,0,0,1,12,0,0,0,0,0,0,5,0,0,0,0,0,0,8,0,0,0,0,0,0,1,0,0,0,0,0,0,12] >;
C23.9D6 in GAP, Magma, Sage, TeX
C_2^3._9D_6
% in TeX
G:=Group("C2^3.9D6"); // GroupNames label
G:=SmallGroup(96,90);
// by ID
G=gap.SmallGroup(96,90);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-3,55,218,188,2309]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=1,d^6=e^2=b,a*b=b*a,d*a*d^-1=a*c=c*a,e*a*e^-1=a*b*c,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^5>;
// generators/relations
Export