metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic3.1D4, C23.11D6, D6⋊C4⋊6C2, (C2×C4).8D6, C22⋊C4⋊5S3, C6.21(C2×D4), C2.10(S3×D4), (C2×Dic6)⋊3C2, C3⋊2(C4.4D4), (C4×Dic3)⋊12C2, C6.10(C4○D4), C6.D4⋊5C2, (C2×C6).26C23, C2.12(C4○D12), C2.9(D4⋊2S3), (C2×C12).54C22, (C22×S3).4C22, (C22×C6).15C22, C22.44(C22×S3), (C2×Dic3).27C22, (C3×C22⋊C4)⋊7C2, (C2×C3⋊D4).4C2, SmallGroup(96,92)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C23.11D6
G = < a,b,c,d,e | a2=b2=c2=1, d6=b, e2=cb=bc, eae-1=ab=ba, dad-1=ac=ca, bd=db, be=eb, cd=dc, ce=ec, ede-1=cd5 >
Subgroups: 186 in 76 conjugacy classes, 31 normal (29 characteristic)
C1, C2, C2, C3, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, Q8, C23, C23, Dic3, Dic3, C12, D6, C2×C6, C2×C6, C42, C22⋊C4, C22⋊C4, C2×D4, C2×Q8, Dic6, C2×Dic3, C3⋊D4, C2×C12, C22×S3, C22×C6, C4.4D4, C4×Dic3, D6⋊C4, C6.D4, C3×C22⋊C4, C2×Dic6, C2×C3⋊D4, C23.11D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, C22×S3, C4.4D4, C4○D12, S3×D4, D4⋊2S3, C23.11D6
Character table of C23.11D6
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | 6B | 6C | 6D | 6E | 12A | 12B | 12C | 12D | |
| size | 1 | 1 | 1 | 1 | 4 | 12 | 2 | 2 | 2 | 4 | 6 | 6 | 6 | 6 | 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 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | -2 | 0 | 2 | 0 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | 2 | 2 | 2 | 2 | 0 | -1 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ11 | 2 | 2 | 2 | 2 | -2 | 0 | -1 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | orthogonal lifted from D6 |
| ρ12 | 2 | -2 | -2 | 2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 2 | 0 | -2 | 0 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ13 | 2 | 2 | 2 | 2 | -2 | 0 | -1 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | orthogonal lifted from D6 |
| ρ14 | 2 | 2 | 2 | 2 | 2 | 0 | -1 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ15 | 2 | 2 | -2 | -2 | 0 | 0 | 2 | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 2i | -2i | complex lifted from C4○D4 |
| ρ16 | 2 | -2 | 2 | -2 | 0 | 0 | 2 | 0 | 0 | 0 | 2i | 0 | -2i | 0 | 0 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ17 | 2 | 2 | -2 | -2 | 0 | 0 | 2 | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | -2i | 2i | complex lifted from C4○D4 |
| ρ18 | 2 | -2 | 2 | -2 | 0 | 0 | 2 | 0 | 0 | 0 | -2i | 0 | 2i | 0 | 0 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ19 | 2 | 2 | -2 | -2 | 0 | 0 | -1 | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | -√-3 | √-3 | -√3 | √3 | i | -i | complex lifted from C4○D12 |
| ρ20 | 2 | 2 | -2 | -2 | 0 | 0 | -1 | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | √-3 | -√-3 | √3 | -√3 | i | -i | complex lifted from C4○D12 |
| ρ21 | 2 | 2 | -2 | -2 | 0 | 0 | -1 | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | -√-3 | √-3 | √3 | -√3 | -i | i | complex lifted from C4○D12 |
| ρ22 | 2 | 2 | -2 | -2 | 0 | 0 | -1 | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | √-3 | -√-3 | -√3 | √3 | -i | i | complex lifted from C4○D12 |
| ρ23 | 4 | -4 | -4 | 4 | 0 | 0 | -2 | 0 | 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 | -2 | 0 | 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 13)(2 28)(3 15)(4 30)(5 17)(6 32)(7 19)(8 34)(9 21)(10 36)(11 23)(12 26)(14 44)(16 46)(18 48)(20 38)(22 40)(24 42)(25 41)(27 43)(29 45)(31 47)(33 37)(35 39)
(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 43)(2 44)(3 45)(4 46)(5 47)(6 48)(7 37)(8 38)(9 39)(10 40)(11 41)(12 42)(13 27)(14 28)(15 29)(16 30)(17 31)(18 32)(19 33)(20 34)(21 35)(22 36)(23 25)(24 26)
(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 37 42)(2 41 38 5)(3 4 39 40)(7 12 43 48)(8 47 44 11)(9 10 45 46)(13 26 33 18)(14 17 34 25)(15 36 35 16)(19 32 27 24)(20 23 28 31)(21 30 29 22)
G:=sub<Sym(48)| (1,13)(2,28)(3,15)(4,30)(5,17)(6,32)(7,19)(8,34)(9,21)(10,36)(11,23)(12,26)(14,44)(16,46)(18,48)(20,38)(22,40)(24,42)(25,41)(27,43)(29,45)(31,47)(33,37)(35,39), (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), (1,43)(2,44)(3,45)(4,46)(5,47)(6,48)(7,37)(8,38)(9,39)(10,40)(11,41)(12,42)(13,27)(14,28)(15,29)(16,30)(17,31)(18,32)(19,33)(20,34)(21,35)(22,36)(23,25)(24,26), (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), (1,6,37,42)(2,41,38,5)(3,4,39,40)(7,12,43,48)(8,47,44,11)(9,10,45,46)(13,26,33,18)(14,17,34,25)(15,36,35,16)(19,32,27,24)(20,23,28,31)(21,30,29,22)>;
G:=Group( (1,13)(2,28)(3,15)(4,30)(5,17)(6,32)(7,19)(8,34)(9,21)(10,36)(11,23)(12,26)(14,44)(16,46)(18,48)(20,38)(22,40)(24,42)(25,41)(27,43)(29,45)(31,47)(33,37)(35,39), (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), (1,43)(2,44)(3,45)(4,46)(5,47)(6,48)(7,37)(8,38)(9,39)(10,40)(11,41)(12,42)(13,27)(14,28)(15,29)(16,30)(17,31)(18,32)(19,33)(20,34)(21,35)(22,36)(23,25)(24,26), (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), (1,6,37,42)(2,41,38,5)(3,4,39,40)(7,12,43,48)(8,47,44,11)(9,10,45,46)(13,26,33,18)(14,17,34,25)(15,36,35,16)(19,32,27,24)(20,23,28,31)(21,30,29,22) );
G=PermutationGroup([[(1,13),(2,28),(3,15),(4,30),(5,17),(6,32),(7,19),(8,34),(9,21),(10,36),(11,23),(12,26),(14,44),(16,46),(18,48),(20,38),(22,40),(24,42),(25,41),(27,43),(29,45),(31,47),(33,37),(35,39)], [(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)], [(1,43),(2,44),(3,45),(4,46),(5,47),(6,48),(7,37),(8,38),(9,39),(10,40),(11,41),(12,42),(13,27),(14,28),(15,29),(16,30),(17,31),(18,32),(19,33),(20,34),(21,35),(22,36),(23,25),(24,26)], [(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)], [(1,6,37,42),(2,41,38,5),(3,4,39,40),(7,12,43,48),(8,47,44,11),(9,10,45,46),(13,26,33,18),(14,17,34,25),(15,36,35,16),(19,32,27,24),(20,23,28,31),(21,30,29,22)]])
C23.11D6 is a maximal subgroup of
C24.38D6 C24.41D6 C24.42D6 C42.93D6 C42.97D6 C42.98D6 C42.99D6 C42.102D6 C42.228D6 Dic6⋊23D4 C42⋊18D6 C42.114D6 C42⋊19D6 C42.115D6 C42.117D6 C24.44D6 C24.45D6 C24.46D6 C24⋊9D6 C12⋊(C4○D4) Dic6⋊19D4 C6.382+ 1+4 C6.402+ 1+4 C6.422+ 1+4 C6.452+ 1+4 C6.462+ 1+4 C6.492+ 1+4 C6.162- 1+4 Dic6⋊22D4 C6.222- 1+4 C6.232- 1+4 C6.242- 1+4 C6.252- 1+4 C6.592+ 1+4 C6.792- 1+4 C4⋊C4.197D6 C6.1212+ 1+4 C6.612+ 1+4 C6.1222+ 1+4 C6.652+ 1+4 C6.672+ 1+4 C6.692+ 1+4 C42.137D6 C42.138D6 S3×C4.4D4 Dic6⋊10D4 C42.143D6 C42.144D6 C42⋊24D6 C42.160D6 C42⋊25D6 C42⋊26D6 C42.189D6 C42.164D6 C42.165D6 Dic9.D4 Dic3.D12 C62.77C23 C62.83C23 C62.85C23 C62.95C23 C62.101C23 C62.229C23 Dic3.D20 (C2×Dic6)⋊D5 Dic15.10D4 Dic15.31D4 C23.D5⋊S3 Dic15.19D4 C23.11D30
C23.11D6 is a maximal quotient of
(C2×C12)⋊Q8 C3⋊(C42⋊8C4) (C2×Dic3).9D4 (C2×C4).Dic6 D6⋊C4⋊5C4 D6⋊C4⋊3C4 (C22×S3)⋊Q8 C6.(C4⋊D4) Dic3.SD16 C4⋊C4.D6 C12⋊Q8⋊C2 (C2×C8).200D6 Dic3.1Q16 (C2×C8).D6 (C2×Q8).36D6 Q8⋊C4⋊S3 C24.14D6 C23⋊2Dic6 C24.19D6 C24.20D6 C24.24D6 C24.25D6 C24.27D6 Dic9.D4 Dic3.D12 C62.77C23 C62.83C23 C62.85C23 C62.95C23 C62.101C23 C62.229C23 Dic3.D20 (C2×Dic6)⋊D5 Dic15.10D4 Dic15.31D4 C23.D5⋊S3 Dic15.19D4 C23.11D30
Matrix representation of C23.11D6 ►in GL6(𝔽13)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 11 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 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 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 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 | 0 | 1 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 0 | 0 | 0 |
| 0 | 0 | 5 | 8 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 0 | 0 | 0 | 0 | 12 | 0 |
G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,11,12,0,0,0,0,0,0,1,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[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],[1,12,0,0,0,0,1,0,0,0,0,0,0,0,5,0,0,0,0,0,0,5,0,0,0,0,0,0,0,12,0,0,0,0,1,0],[1,0,0,0,0,0,1,12,0,0,0,0,0,0,5,5,0,0,0,0,0,8,0,0,0,0,0,0,0,12,0,0,0,0,12,0] >;
C23.11D6 in GAP, Magma, Sage, TeX
C_2^3._{11}D_6 % in TeX
G:=Group("C2^3.11D6"); // GroupNames label
G:=SmallGroup(96,92);
// by ID
G=gap.SmallGroup(96,92);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-3,217,55,506,188,2309]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=1,d^6=b,e^2=c*b=b*c,e*a*e^-1=a*b=b*a,d*a*d^-1=a*c=c*a,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=c*d^5>;
// generators/relations
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