metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic3⋊3D4, C23.14D6, (C2×C6)⋊3D4, (C2×D4)⋊5S3, (C6×D4)⋊9C2, D6⋊C4⋊15C2, C3⋊5(C4⋊D4), (C2×C4).19D6, C6.51(C2×D4), C2.27(S3×D4), Dic3⋊C4⋊15C2, C6.32(C4○D4), C22⋊2(C3⋊D4), (C2×C6).54C23, C6.D4⋊12C2, (C2×C12).62C22, (C22×Dic3)⋊6C2, C2.18(D4⋊2S3), C22.61(C22×S3), (C22×C6).21C22, (C22×S3).11C22, (C2×Dic3).38C22, (C2×C3⋊D4)⋊6C2, C2.15(C2×C3⋊D4), SmallGroup(96,146)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C23.14D6
G = < a,b,c,d,e | a2=b2=c2=d6=1, e2=c, ab=ba, dad-1=ac=ca, eae-1=abc, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=cd-1 >
Subgroups: 226 in 94 conjugacy classes, 35 normal (29 characteristic)
C1, C2, C2, C3, C4, C22, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, C23, C23, Dic3, Dic3, C12, D6, C2×C6, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×D4, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C22×S3, C22×C6, C4⋊D4, Dic3⋊C4, D6⋊C4, C6.D4, C22×Dic3, C2×C3⋊D4, C6×D4, C23.14D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, C3⋊D4, C22×S3, C4⋊D4, S3×D4, D4⋊2S3, C2×C3⋊D4, C23.14D6
Character table of C23.14D6
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 12A | 12B | |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 12 | 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 | -2 | -2 | 2 | 0 | -1 | -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 | 2 | 0 | -1 | 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 | 2 | -2 | 0 | -1 | -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 | -2 | 0 | -1 | 2 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | orthogonal lifted from D6 |
| ρ13 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 2 | 0 | -2 | 0 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ14 | 2 | 2 | -2 | -2 | 2 | -2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | -2 | 0 | 2 | -2 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ15 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | -2 | 0 | 2 | 0 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ16 | 2 | 2 | -2 | -2 | -2 | 2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | -2 | 0 | -2 | 2 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ17 | 2 | 2 | -2 | -2 | -2 | 2 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | -√-3 | 1 | -1 | √-3 | √-3 | -√-3 | complex lifted from C3⋊D4 |
| ρ18 | 2 | 2 | -2 | -2 | 2 | -2 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | √-3 | -1 | 1 | -√-3 | √-3 | -√-3 | complex lifted from C3⋊D4 |
| ρ19 | 2 | 2 | -2 | -2 | -2 | 2 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | √-3 | 1 | -1 | -√-3 | -√-3 | √-3 | complex lifted from C3⋊D4 |
| ρ20 | 2 | 2 | -2 | -2 | 2 | -2 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | -√-3 | -1 | 1 | √-3 | -√-3 | √-3 | complex lifted from C3⋊D4 |
| ρ21 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 2 | 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 | 0 | 2 | 0 | 2i | 0 | -2i | 0 | 0 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ23 | 4 | -4 | 4 | -4 | 0 | 0 | 0 | 0 | -2 | 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 | 0 | -2 | 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 44)(3 15)(4 46)(5 17)(6 48)(7 33)(8 24)(9 35)(10 20)(11 31)(12 22)(14 38)(16 40)(18 42)(19 29)(21 25)(23 27)(26 32)(28 34)(30 36)(37 43)(39 45)(41 47)
(1 33)(2 34)(3 35)(4 36)(5 31)(6 32)(7 13)(8 14)(9 15)(10 16)(11 17)(12 18)(19 39)(20 40)(21 41)(22 42)(23 37)(24 38)(25 47)(26 48)(27 43)(28 44)(29 45)(30 46)
(1 37)(2 38)(3 39)(4 40)(5 41)(6 42)(7 27)(8 28)(9 29)(10 30)(11 25)(12 26)(13 43)(14 44)(15 45)(16 46)(17 47)(18 48)(19 35)(20 36)(21 31)(22 32)(23 33)(24 34)
(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 18 27 48)(8 47 28 17)(9 16 29 46)(10 45 30 15)(11 14 25 44)(12 43 26 13)(19 20 35 36)(21 24 31 34)(22 33 32 23)
G:=sub<Sym(48)| (1,13)(2,44)(3,15)(4,46)(5,17)(6,48)(7,33)(8,24)(9,35)(10,20)(11,31)(12,22)(14,38)(16,40)(18,42)(19,29)(21,25)(23,27)(26,32)(28,34)(30,36)(37,43)(39,45)(41,47), (1,33)(2,34)(3,35)(4,36)(5,31)(6,32)(7,13)(8,14)(9,15)(10,16)(11,17)(12,18)(19,39)(20,40)(21,41)(22,42)(23,37)(24,38)(25,47)(26,48)(27,43)(28,44)(29,45)(30,46), (1,37)(2,38)(3,39)(4,40)(5,41)(6,42)(7,27)(8,28)(9,29)(10,30)(11,25)(12,26)(13,43)(14,44)(15,45)(16,46)(17,47)(18,48)(19,35)(20,36)(21,31)(22,32)(23,33)(24,34), (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)(7,18,27,48)(8,47,28,17)(9,16,29,46)(10,45,30,15)(11,14,25,44)(12,43,26,13)(19,20,35,36)(21,24,31,34)(22,33,32,23)>;
G:=Group( (1,13)(2,44)(3,15)(4,46)(5,17)(6,48)(7,33)(8,24)(9,35)(10,20)(11,31)(12,22)(14,38)(16,40)(18,42)(19,29)(21,25)(23,27)(26,32)(28,34)(30,36)(37,43)(39,45)(41,47), (1,33)(2,34)(3,35)(4,36)(5,31)(6,32)(7,13)(8,14)(9,15)(10,16)(11,17)(12,18)(19,39)(20,40)(21,41)(22,42)(23,37)(24,38)(25,47)(26,48)(27,43)(28,44)(29,45)(30,46), (1,37)(2,38)(3,39)(4,40)(5,41)(6,42)(7,27)(8,28)(9,29)(10,30)(11,25)(12,26)(13,43)(14,44)(15,45)(16,46)(17,47)(18,48)(19,35)(20,36)(21,31)(22,32)(23,33)(24,34), (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)(7,18,27,48)(8,47,28,17)(9,16,29,46)(10,45,30,15)(11,14,25,44)(12,43,26,13)(19,20,35,36)(21,24,31,34)(22,33,32,23) );
G=PermutationGroup([[(1,13),(2,44),(3,15),(4,46),(5,17),(6,48),(7,33),(8,24),(9,35),(10,20),(11,31),(12,22),(14,38),(16,40),(18,42),(19,29),(21,25),(23,27),(26,32),(28,34),(30,36),(37,43),(39,45),(41,47)], [(1,33),(2,34),(3,35),(4,36),(5,31),(6,32),(7,13),(8,14),(9,15),(10,16),(11,17),(12,18),(19,39),(20,40),(21,41),(22,42),(23,37),(24,38),(25,47),(26,48),(27,43),(28,44),(29,45),(30,46)], [(1,37),(2,38),(3,39),(4,40),(5,41),(6,42),(7,27),(8,28),(9,29),(10,30),(11,25),(12,26),(13,43),(14,44),(15,45),(16,46),(17,47),(18,48),(19,35),(20,36),(21,31),(22,32),(23,33),(24,34)], [(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),(7,18,27,48),(8,47,28,17),(9,16,29,46),(10,45,30,15),(11,14,25,44),(12,43,26,13),(19,20,35,36),(21,24,31,34),(22,33,32,23)]])
C23.14D6 is a maximal subgroup of
C42.102D6 C42.104D6 C42⋊14D6 Dic6⋊23D4 C42⋊19D6 C42.118D6 C42.119D6 C24.67D6 C24⋊8D6 C24.44D6 C24.45D6 C24.46D6 C24.47D6 C12⋊(C4○D4) C6.322+ 1+4 Dic6⋊19D4 Dic6⋊20D4 C6.342+ 1+4 S3×C4⋊D4 C6.372+ 1+4 C6.722- 1+4 C6.402+ 1+4 C6.422+ 1+4 C6.442+ 1+4 C6.452+ 1+4 C6.462+ 1+4 C6.1152+ 1+4 C6.472+ 1+4 C6.482+ 1+4 C6.492+ 1+4 C4⋊C4.197D6 C6.1212+ 1+4 C6.822- 1+4 C6.642+ 1+4 C6.652+ 1+4 C6.662+ 1+4 C6.852- 1+4 C6.692+ 1+4 C42.137D6 C42.138D6 C42⋊22D6 C42.145D6 C42⋊28D6 Dic6⋊11D4 C42.168D6 C42⋊30D6 D4×C3⋊D4 C24⋊12D6 C24.53D6 C6.1042- 1+4 C6.1452+ 1+4 (C2×C12)⋊17D4 C6.1082- 1+4 Dic9⋊D4 C62.55C23 Dic3⋊D12 C62.112C23 C62.113C23 C62⋊6D4 C62⋊7D4 C62⋊14D4 Dic3⋊S4 Dic15⋊D4 Dic3⋊D20 (C6×D5)⋊D4 Dic15⋊3D4 (C2×C6)⋊D20 Dic15⋊18D4 Dic15⋊12D4
C23.14D6 is a maximal quotient of
C24.55D6 C24.15D6 C24.17D6 C24.20D6 C24.24D6 C24.60D6 C24.25D6 C24.27D6 Dic3⋊(C4⋊C4) (C2×C12).54D4 D6⋊C4⋊7C4 (C2×C12).56D4 C3⋊C8⋊22D4 C4⋊D4⋊S3 C3⋊C8⋊23D4 C3⋊C8⋊5D4 C3⋊C8⋊24D4 C3⋊C8⋊6D4 C3⋊C8.29D4 C3⋊C8.6D4 Dic3⋊D8 (C6×D8).C2 Dic3⋊3SD16 Dic3⋊5SD16 (C3×D4).D4 (C3×Q8).D4 Dic3⋊3Q16 (C2×Q16)⋊S3 M4(2).D6 M4(2).13D6 M4(2).15D6 M4(2).16D6 C24.29D6 C24.31D6 C24.32D6 Dic9⋊D4 C62.55C23 Dic3⋊D12 C62.112C23 C62.113C23 C62⋊6D4 C62⋊7D4 C62⋊14D4 Dic15⋊D4 Dic3⋊D20 (C6×D5)⋊D4 Dic15⋊3D4 (C2×C6)⋊D20 Dic15⋊18D4 Dic15⋊12D4
Matrix representation of C23.14D6 ►in GL4(𝔽13) generated by
| 1 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 11 | 9 |
| 0 | 0 | 4 | 2 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 |
| 12 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 12 | 12 |
| 0 | 0 | 1 | 0 |
| 0 | 12 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 12 | 12 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(13))| [1,0,0,0,0,12,0,0,0,0,11,4,0,0,9,2],[1,0,0,0,0,1,0,0,0,0,12,0,0,0,0,12],[12,0,0,0,0,12,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,12,1,0,0,12,0],[0,1,0,0,12,0,0,0,0,0,12,0,0,0,12,1] >;
C23.14D6 in GAP, Magma, Sage, TeX
C_2^3._{14}D_6 % in TeX
G:=Group("C2^3.14D6"); // GroupNames label
G:=SmallGroup(96,146);
// by ID
G=gap.SmallGroup(96,146);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-3,217,218,188,2309]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^6=1,e^2=c,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=c*d^-1>;
// generators/relations
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