metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C23.23D6, (C2×C6).7D4, (C2×D4).5S3, (C2×C4).18D6, C6.47(C2×D4), (C6×D4).10C2, Dic3⋊C4⋊14C2, C6.29(C4○D4), C6.D4⋊8C2, (C2×C6).50C23, (C2×C12).61C22, (C22×Dic3)⋊5C2, C3⋊5(C22.D4), C22.4(C3⋊D4), C2.15(D4⋊2S3), (C22×C6).18C22, C22.57(C22×S3), (C2×Dic3).17C22, C2.11(C2×C3⋊D4), SmallGroup(96,142)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C23.23D6
G = < a,b,c,d,e | a2=b2=c2=d6=1, e2=cb=bc, ab=ba, dad-1=eae-1=ac=ca, bd=db, be=eb, cd=dc, ce=ec, ede-1=bd-1 >
Subgroups: 162 in 78 conjugacy classes, 33 normal (17 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, C6, C6, C6, C2×C4, C2×C4, D4, C23, Dic3, C12, C2×C6, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Dic3, C2×Dic3, C2×C12, C3×D4, C22×C6, C22.D4, Dic3⋊C4, C6.D4, C6.D4, C22×Dic3, C6×D4, C23.23D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, C3⋊D4, C22×S3, C22.D4, D4⋊2S3, C2×C3⋊D4, C23.23D6
Character table of C23.23D6
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 12A | 12B | |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 2 | 4 | 6 | 6 | 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 | 2 | 2 | -1 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ10 | 2 | 2 | 2 | 2 | 2 | 2 | -2 | -1 | -2 | 0 | 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 | -2 | -1 | 2 | 0 | 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 | -1 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | orthogonal lifted from D6 |
| ρ13 | 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 |
| 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 | complex lifted from C3⋊D4 |
| 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 | complex lifted from C3⋊D4 |
| 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 | complex lifted from C3⋊D4 |
| 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 | complex lifted from C3⋊D4 |
| ρ19 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 2 | 0 | -2i | 0 | 2i | 0 | 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 | -2i | 0 | 2i | 0 | 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 | 2i | 0 | -2i | 0 | 0 | 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 13)(2 17)(3 15)(4 16)(5 14)(6 18)(7 19)(8 23)(9 21)(10 22)(11 20)(12 24)(25 37)(26 34)(27 39)(28 36)(29 41)(30 32)(31 44)(33 46)(35 48)(38 47)(40 43)(42 45)
(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 43)(26 44)(27 45)(28 46)(29 47)(30 48)(31 34)(32 35)(33 36)(37 40)(38 41)(39 42)
(1 4)(2 5)(3 6)(7 10)(8 11)(9 12)(13 16)(14 17)(15 18)(19 22)(20 23)(21 24)(25 46)(26 47)(27 48)(28 43)(29 44)(30 45)(31 41)(32 42)(33 37)(34 38)(35 39)(36 40)
(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 36 10 37)(2 32 11 39)(3 34 12 41)(4 40 7 33)(5 42 8 35)(6 38 9 31)(13 43 22 46)(14 30 23 27)(15 47 24 44)(16 28 19 25)(17 45 20 48)(18 26 21 29)
G:=sub<Sym(48)| (1,13)(2,17)(3,15)(4,16)(5,14)(6,18)(7,19)(8,23)(9,21)(10,22)(11,20)(12,24)(25,37)(26,34)(27,39)(28,36)(29,41)(30,32)(31,44)(33,46)(35,48)(38,47)(40,43)(42,45), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24)(25,43)(26,44)(27,45)(28,46)(29,47)(30,48)(31,34)(32,35)(33,36)(37,40)(38,41)(39,42), (1,4)(2,5)(3,6)(7,10)(8,11)(9,12)(13,16)(14,17)(15,18)(19,22)(20,23)(21,24)(25,46)(26,47)(27,48)(28,43)(29,44)(30,45)(31,41)(32,42)(33,37)(34,38)(35,39)(36,40), (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), (1,36,10,37)(2,32,11,39)(3,34,12,41)(4,40,7,33)(5,42,8,35)(6,38,9,31)(13,43,22,46)(14,30,23,27)(15,47,24,44)(16,28,19,25)(17,45,20,48)(18,26,21,29)>;
G:=Group( (1,13)(2,17)(3,15)(4,16)(5,14)(6,18)(7,19)(8,23)(9,21)(10,22)(11,20)(12,24)(25,37)(26,34)(27,39)(28,36)(29,41)(30,32)(31,44)(33,46)(35,48)(38,47)(40,43)(42,45), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24)(25,43)(26,44)(27,45)(28,46)(29,47)(30,48)(31,34)(32,35)(33,36)(37,40)(38,41)(39,42), (1,4)(2,5)(3,6)(7,10)(8,11)(9,12)(13,16)(14,17)(15,18)(19,22)(20,23)(21,24)(25,46)(26,47)(27,48)(28,43)(29,44)(30,45)(31,41)(32,42)(33,37)(34,38)(35,39)(36,40), (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), (1,36,10,37)(2,32,11,39)(3,34,12,41)(4,40,7,33)(5,42,8,35)(6,38,9,31)(13,43,22,46)(14,30,23,27)(15,47,24,44)(16,28,19,25)(17,45,20,48)(18,26,21,29) );
G=PermutationGroup([[(1,13),(2,17),(3,15),(4,16),(5,14),(6,18),(7,19),(8,23),(9,21),(10,22),(11,20),(12,24),(25,37),(26,34),(27,39),(28,36),(29,41),(30,32),(31,44),(33,46),(35,48),(38,47),(40,43),(42,45)], [(1,7),(2,8),(3,9),(4,10),(5,11),(6,12),(13,19),(14,20),(15,21),(16,22),(17,23),(18,24),(25,43),(26,44),(27,45),(28,46),(29,47),(30,48),(31,34),(32,35),(33,36),(37,40),(38,41),(39,42)], [(1,4),(2,5),(3,6),(7,10),(8,11),(9,12),(13,16),(14,17),(15,18),(19,22),(20,23),(21,24),(25,46),(26,47),(27,48),(28,43),(29,44),(30,45),(31,41),(32,42),(33,37),(34,38),(35,39),(36,40)], [(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)], [(1,36,10,37),(2,32,11,39),(3,34,12,41),(4,40,7,33),(5,42,8,35),(6,38,9,31),(13,43,22,46),(14,30,23,27),(15,47,24,44),(16,28,19,25),(17,45,20,48),(18,26,21,29)]])
C23.23D6 is a maximal subgroup of
(C2×D4).D6 C23.4D12 C23.5D12 2+ 1+4.5S3 C42.102D6 C42.105D6 C42⋊18D6 C42.118D6 C24.67D6 C24.43D6 C24⋊7D6 C24.46D6 C24.47D6 C4⋊C4.178D6 C6.342+ 1+4 C6.702- 1+4 C6.712- 1+4 C6.402+ 1+4 C6.732- 1+4 C6.422+ 1+4 C6.432+ 1+4 C6.442+ 1+4 C6.492+ 1+4 C6.792- 1+4 C6.802- 1+4 C6.812- 1+4 S3×C22.D4 C6.632+ 1+4 C6.672+ 1+4 C42.137D6 C42.140D6 C42⋊23D6 C42.166D6 C42.168D6 C42⋊30D6 C24⋊12D6 C24.53D6 C6.1042- 1+4 C6.1052- 1+4 (C2×D4)⋊43D6 C23.23D18 C62.54C23 C62.56D4 C62.57D4 C62.111C23 C62.72D4 (C6×D5).D4 (C2×C30).D4 C6.(C2×D20) C23.17(S3×D5) C23.22D30
C23.23D6 is a maximal quotient of
C24.56D6 C24.14D6 C24.57D6 C24.18D6 C24.20D6 C24.21D6 C6.67(C4×D4) (C2×C4).44D12 (C2×C12).55D4 (C2×C6).D8 C4⋊D4.S3 C6.Q16⋊C2 (C2×Q8).49D6 (C2×C6).Q16 (C2×Q8).51D6 C24.29D6 C24.31D6 C23.23D18 C62.54C23 C62.56D4 C62.57D4 C62.111C23 C62.72D4 (C6×D5).D4 (C2×C30).D4 C6.(C2×D20) C23.17(S3×D5) C23.22D30
Matrix representation of C23.23D6 ►in GL4(𝔽13) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 11 |
| 0 | 0 | 0 | 12 |
| 12 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 |
| 3 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 12 |
| 0 | 9 | 0 | 0 |
| 10 | 0 | 0 | 0 |
| 0 | 0 | 8 | 0 |
| 0 | 0 | 8 | 5 |
G:=sub<GL(4,GF(13))| [1,0,0,0,0,1,0,0,0,0,1,0,0,0,11,12],[12,0,0,0,0,12,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,12,0,0,0,0,12],[3,0,0,0,0,4,0,0,0,0,1,1,0,0,0,12],[0,10,0,0,9,0,0,0,0,0,8,8,0,0,0,5] >;
C23.23D6 in GAP, Magma, Sage, TeX
C_2^3._{23}D_6 % in TeX
G:=Group("C2^3.23D6"); // GroupNames label
G:=SmallGroup(96,142);
// by ID
G=gap.SmallGroup(96,142);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-3,96,218,188,2309]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^6=1,e^2=c*b=b*c,a*b=b*a,d*a*d^-1=e*a*e^-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=b*d^-1>;
// generators/relations
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