metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C23.21D6, C22.4D12, D6⋊C4⋊7C2, C6.6(C2×D4), (C2×C4).9D6, (C2×C6).4D4, C22⋊C4⋊6S3, C4⋊Dic3⋊5C2, C2.8(C2×D12), C6.23(C4○D4), (C2×C6).27C23, (C2×C12).3C22, (C22×Dic3)⋊2C2, C3⋊2(C22.D4), C2.10(D4⋊2S3), (C22×S3).5C22, C22.45(C22×S3), (C22×C6).16C22, (C2×Dic3).28C22, (C3×C22⋊C4)⋊4C2, (C2×C3⋊D4).5C2, SmallGroup(96,93)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C23.21D6
G = < a,b,c,d | a2=b2=c12=1, d2=b, cac-1=ab=ba, ad=da, bc=cb, bd=db, dcd-1=bc-1 >
Subgroups: 186 in 78 conjugacy classes, 33 normal (15 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, S3, C6, C6, C6, C2×C4, C2×C4, D4, C23, C23, Dic3, C12, D6, C2×C6, C2×C6, C2×C6, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Dic3, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C22×S3, C22×C6, C22.D4, C4⋊Dic3, D6⋊C4, C3×C22⋊C4, C22×Dic3, C2×C3⋊D4, C23.21D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, D12, C22×S3, C22.D4, C2×D12, D4⋊2S3, C23.21D6
Character table of C23.21D6
| 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 | 2 | 2 | 12 | 2 | 4 | 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 | 2 | 2 | 0 | -1 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ10 | 2 | 2 | 2 | 2 | -2 | -2 | 0 | -1 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | orthogonal lifted from D6 |
| ρ11 | 2 | 2 | 2 | 2 | -2 | -2 | 0 | -1 | 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 | -2 | 2 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ13 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | -1 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ14 | 2 | -2 | -2 | 2 | 2 | -2 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ15 | 2 | -2 | -2 | 2 | -2 | 2 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | -1 | 1 | √3 | -√3 | -√3 | √3 | orthogonal lifted from D12 |
| ρ16 | 2 | -2 | -2 | 2 | 2 | -2 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | 1 | -1 | √3 | -√3 | √3 | -√3 | orthogonal lifted from D12 |
| ρ17 | 2 | -2 | -2 | 2 | 2 | -2 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | 1 | -1 | -√3 | √3 | -√3 | √3 | orthogonal lifted from D12 |
| ρ18 | 2 | -2 | -2 | 2 | -2 | 2 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | -1 | 1 | -√3 | √3 | √3 | -√3 | orthogonal lifted from D12 |
| ρ19 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 2 | 0 | 0 | -2i | 0 | 2i | 0 | 0 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ20 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 2i | 0 | -2i | 0 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ21 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 2 | 0 | 0 | 2i | 0 | -2i | 0 | 0 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ22 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | -2i | 0 | 2i | 0 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| 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 | symplectic lifted from D4⋊2S3, Schur index 2 |
| 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 25)(2 44)(3 27)(4 46)(5 29)(6 48)(7 31)(8 38)(9 33)(10 40)(11 35)(12 42)(13 36)(14 43)(15 26)(16 45)(17 28)(18 47)(19 30)(20 37)(21 32)(22 39)(23 34)(24 41)
(1 14)(2 15)(3 16)(4 17)(5 18)(6 19)(7 20)(8 21)(9 22)(10 23)(11 24)(12 13)(25 43)(26 44)(27 45)(28 46)(29 47)(30 48)(31 37)(32 38)(33 39)(34 40)(35 41)(36 42)
(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 13 14 12)(2 11 15 24)(3 23 16 10)(4 9 17 22)(5 21 18 8)(6 7 19 20)(25 36 43 42)(26 41 44 35)(27 34 45 40)(28 39 46 33)(29 32 47 38)(30 37 48 31)
G:=sub<Sym(48)| (1,25)(2,44)(3,27)(4,46)(5,29)(6,48)(7,31)(8,38)(9,33)(10,40)(11,35)(12,42)(13,36)(14,43)(15,26)(16,45)(17,28)(18,47)(19,30)(20,37)(21,32)(22,39)(23,34)(24,41), (1,14)(2,15)(3,16)(4,17)(5,18)(6,19)(7,20)(8,21)(9,22)(10,23)(11,24)(12,13)(25,43)(26,44)(27,45)(28,46)(29,47)(30,48)(31,37)(32,38)(33,39)(34,40)(35,41)(36,42), (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), (1,13,14,12)(2,11,15,24)(3,23,16,10)(4,9,17,22)(5,21,18,8)(6,7,19,20)(25,36,43,42)(26,41,44,35)(27,34,45,40)(28,39,46,33)(29,32,47,38)(30,37,48,31)>;
G:=Group( (1,25)(2,44)(3,27)(4,46)(5,29)(6,48)(7,31)(8,38)(9,33)(10,40)(11,35)(12,42)(13,36)(14,43)(15,26)(16,45)(17,28)(18,47)(19,30)(20,37)(21,32)(22,39)(23,34)(24,41), (1,14)(2,15)(3,16)(4,17)(5,18)(6,19)(7,20)(8,21)(9,22)(10,23)(11,24)(12,13)(25,43)(26,44)(27,45)(28,46)(29,47)(30,48)(31,37)(32,38)(33,39)(34,40)(35,41)(36,42), (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), (1,13,14,12)(2,11,15,24)(3,23,16,10)(4,9,17,22)(5,21,18,8)(6,7,19,20)(25,36,43,42)(26,41,44,35)(27,34,45,40)(28,39,46,33)(29,32,47,38)(30,37,48,31) );
G=PermutationGroup([[(1,25),(2,44),(3,27),(4,46),(5,29),(6,48),(7,31),(8,38),(9,33),(10,40),(11,35),(12,42),(13,36),(14,43),(15,26),(16,45),(17,28),(18,47),(19,30),(20,37),(21,32),(22,39),(23,34),(24,41)], [(1,14),(2,15),(3,16),(4,17),(5,18),(6,19),(7,20),(8,21),(9,22),(10,23),(11,24),(12,13),(25,43),(26,44),(27,45),(28,46),(29,47),(30,48),(31,37),(32,38),(33,39),(34,40),(35,41),(36,42)], [(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)], [(1,13,14,12),(2,11,15,24),(3,23,16,10),(4,9,17,22),(5,21,18,8),(6,7,19,20),(25,36,43,42),(26,41,44,35),(27,34,45,40),(28,39,46,33),(29,32,47,38),(30,37,48,31)]])
C23.21D6 is a maximal subgroup of
C23.5D12 C23⋊4D12 C24.42D6 C42⋊10D6 C42.92D6 C42.96D6 C42.102D6 D4⋊5D12 D4⋊6D12 C42.118D6 C24.67D6 C24⋊7D6 C24.44D6 C6.462+ 1+4 C6.1152+ 1+4 C6.472+ 1+4 C6.482+ 1+4 C4⋊C4.187D6 C6.532+ 1+4 C6.772- 1+4 C6.782- 1+4 C6.792- 1+4 S3×C22.D4 C6.822- 1+4 C6.1222+ 1+4 C6.662+ 1+4 C6.852- 1+4 C6.692+ 1+4 C42⋊22D6 C42.143D6 C42.144D6 C42.145D6 C42.161D6 C42.163D6 C42.164D6 C42.165D6 C22.4D36 D6.D12 D6.9D12 C62.57D4 C62.60D4 C62.69D4 D10.16D12 D10.17D12 C10.(C2×D12) (C2×C10).D12 C22.D60
C23.21D6 is a maximal quotient of
C2.(C4×D12) (C2×C4).17D12 (C22×C4).85D6 D6⋊C4⋊3C4 (C2×C4).21D12 (C2×C12).33D4 C23.39D12 C23.40D12 C23.15D12 C23.43D12 C22.D24 C23.18D12 C24.56D6 C24.58D6 C24.21D6 C24.60D6 C24.27D6 C22.4D36 D6.D12 D6.9D12 C62.57D4 C62.60D4 C62.69D4 D10.16D12 D10.17D12 C10.(C2×D12) (C2×C10).D12 C22.D60
Matrix representation of C23.21D6 ►in GL6(𝔽13)
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 5 |
| 0 | 0 | 0 | 0 | 8 | 0 |
| 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 |
| 9 | 3 | 0 | 0 | 0 | 0 |
| 3 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 9 | 3 | 0 | 0 | 0 | 0 |
| 8 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 12 | 0 |
G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,8,0,0,0,0,5,0],[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],[9,3,0,0,0,0,3,4,0,0,0,0,0,0,1,12,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[9,8,0,0,0,0,3,4,0,0,0,0,0,0,1,0,0,0,0,0,1,12,0,0,0,0,0,0,0,12,0,0,0,0,1,0] >;
C23.21D6 in GAP, Magma, Sage, TeX
C_2^3._{21}D_6 % in TeX
G:=Group("C2^3.21D6"); // GroupNames label
G:=SmallGroup(96,93);
// by ID
G=gap.SmallGroup(96,93);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-3,217,218,188,122,2309]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^2=c^12=1,d^2=b,c*a*c^-1=a*b=b*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=b*c^-1>;
// generators/relations
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