direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C22×Dic3, C6.9C23, C23.3S3, C22.11D6, (C2×C6)⋊3C4, C6⋊2(C2×C4), C3⋊2(C22×C4), C2.2(C22×S3), (C22×C6).3C2, (C2×C6).12C22, SmallGroup(48,42)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — C22×Dic3 |
Generators and relations for C22×Dic3
G = < a,b,c,d | a2=b2=c6=1, d2=c3, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >
Subgroups: 76 in 54 conjugacy classes, 43 normal (7 characteristic)
C1, C2, C2, C3, C4, C22, C6, C6, C2×C4, C23, Dic3, C2×C6, C22×C4, C2×Dic3, C22×C6, C22×Dic3
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, Dic3, D6, C22×C4, C2×Dic3, C22×S3, C22×Dic3
Character table of C22×Dic3
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | 6B | 6C | 6D | 6E | 6F | 6G | |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | |
| 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 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | i | -i | -i | i | i | -i | -i | i | -1 | -1 | -1 | -1 | 1 | 1 | 1 | linear of order 4 |
| ρ10 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | -1 | 1 | i | i | -i | i | -i | i | -i | -i | 1 | -1 | -1 | 1 | -1 | -1 | 1 | linear of order 4 |
| ρ11 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | -1 | 1 | -i | -i | i | -i | i | -i | i | i | 1 | -1 | -1 | 1 | -1 | -1 | 1 | linear of order 4 |
| ρ12 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | -i | i | i | -i | -i | i | i | -i | -1 | -1 | -1 | -1 | 1 | 1 | 1 | linear of order 4 |
| ρ13 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | 1 | -i | -i | -i | i | -i | i | i | i | -1 | 1 | -1 | 1 | 1 | -1 | -1 | linear of order 4 |
| ρ14 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | -i | i | -i | i | i | -i | i | -i | 1 | 1 | -1 | -1 | -1 | 1 | -1 | linear of order 4 |
| ρ15 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | i | -i | i | -i | -i | i | -i | i | 1 | 1 | -1 | -1 | -1 | 1 | -1 | linear of order 4 |
| ρ16 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | 1 | i | i | i | -i | i | -i | -i | -i | -1 | 1 | -1 | 1 | 1 | -1 | -1 | linear of order 4 |
| ρ17 | 2 | 2 | -2 | -2 | -2 | -2 | 2 | 2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | orthogonal lifted from D6 |
| ρ18 | 2 | -2 | 2 | 2 | -2 | -2 | 2 | -2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | orthogonal lifted from D6 |
| ρ19 | 2 | -2 | -2 | -2 | 2 | 2 | 2 | -2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | orthogonal lifted from D6 |
| ρ20 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ21 | 2 | -2 | -2 | 2 | 2 | -2 | -2 | 2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | symplectic lifted from Dic3, Schur index 2 |
| ρ22 | 2 | 2 | 2 | -2 | 2 | -2 | -2 | -2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | -1 | 1 | 1 | -1 | symplectic lifted from Dic3, Schur index 2 |
| ρ23 | 2 | 2 | -2 | 2 | -2 | 2 | -2 | -2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | symplectic lifted from Dic3, Schur index 2 |
| ρ24 | 2 | -2 | 2 | -2 | -2 | 2 | -2 | 2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | 1 | 1 | 1 | -1 | 1 | symplectic lifted from Dic3, Schur index 2 |
►Regular action on 48 points
Generators in S48
(1 14)(2 15)(3 16)(4 17)(5 18)(6 13)(7 22)(8 23)(9 24)(10 19)(11 20)(12 21)(25 40)(26 41)(27 42)(28 37)(29 38)(30 39)(31 46)(32 47)(33 48)(34 43)(35 44)(36 45)
(1 8)(2 9)(3 10)(4 11)(5 12)(6 7)(13 22)(14 23)(15 24)(16 19)(17 20)(18 21)(25 34)(26 35)(27 36)(28 31)(29 32)(30 33)(37 46)(38 47)(39 48)(40 43)(41 44)(42 45)
(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 34 4 31)(2 33 5 36)(3 32 6 35)(7 26 10 29)(8 25 11 28)(9 30 12 27)(13 44 16 47)(14 43 17 46)(15 48 18 45)(19 38 22 41)(20 37 23 40)(21 42 24 39)
G:=sub<Sym(48)| (1,14)(2,15)(3,16)(4,17)(5,18)(6,13)(7,22)(8,23)(9,24)(10,19)(11,20)(12,21)(25,40)(26,41)(27,42)(28,37)(29,38)(30,39)(31,46)(32,47)(33,48)(34,43)(35,44)(36,45), (1,8)(2,9)(3,10)(4,11)(5,12)(6,7)(13,22)(14,23)(15,24)(16,19)(17,20)(18,21)(25,34)(26,35)(27,36)(28,31)(29,32)(30,33)(37,46)(38,47)(39,48)(40,43)(41,44)(42,45), (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), (1,34,4,31)(2,33,5,36)(3,32,6,35)(7,26,10,29)(8,25,11,28)(9,30,12,27)(13,44,16,47)(14,43,17,46)(15,48,18,45)(19,38,22,41)(20,37,23,40)(21,42,24,39)>;
G:=Group( (1,14)(2,15)(3,16)(4,17)(5,18)(6,13)(7,22)(8,23)(9,24)(10,19)(11,20)(12,21)(25,40)(26,41)(27,42)(28,37)(29,38)(30,39)(31,46)(32,47)(33,48)(34,43)(35,44)(36,45), (1,8)(2,9)(3,10)(4,11)(5,12)(6,7)(13,22)(14,23)(15,24)(16,19)(17,20)(18,21)(25,34)(26,35)(27,36)(28,31)(29,32)(30,33)(37,46)(38,47)(39,48)(40,43)(41,44)(42,45), (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), (1,34,4,31)(2,33,5,36)(3,32,6,35)(7,26,10,29)(8,25,11,28)(9,30,12,27)(13,44,16,47)(14,43,17,46)(15,48,18,45)(19,38,22,41)(20,37,23,40)(21,42,24,39) );
G=PermutationGroup([[(1,14),(2,15),(3,16),(4,17),(5,18),(6,13),(7,22),(8,23),(9,24),(10,19),(11,20),(12,21),(25,40),(26,41),(27,42),(28,37),(29,38),(30,39),(31,46),(32,47),(33,48),(34,43),(35,44),(36,45)], [(1,8),(2,9),(3,10),(4,11),(5,12),(6,7),(13,22),(14,23),(15,24),(16,19),(17,20),(18,21),(25,34),(26,35),(27,36),(28,31),(29,32),(30,33),(37,46),(38,47),(39,48),(40,43),(41,44),(42,45)], [(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)], [(1,34,4,31),(2,33,5,36),(3,32,6,35),(7,26,10,29),(8,25,11,28),(9,30,12,27),(13,44,16,47),(14,43,17,46),(15,48,18,45),(19,38,22,41),(20,37,23,40),(21,42,24,39)]])
C22×Dic3 is a maximal subgroup of
C6.C42 C23.16D6 Dic3.D4 Dic3⋊4D4 C23.21D6 C23.23D6 C23.14D6 S3×C22×C4
C22×Dic3 is a maximal quotient of
C23.26D6 D4.Dic3
Matrix representation of C22×Dic3 ►in GL4(𝔽13) generated by
| 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 | 12 | 0 |
| 0 | 0 | 0 | 12 |
| 1 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 12 |
| 0 | 0 | 1 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 |
| 0 | 0 | 5 | 0 |
| 0 | 0 | 8 | 8 |
G:=sub<GL(4,GF(13))| [1,0,0,0,0,1,0,0,0,0,12,0,0,0,0,12],[12,0,0,0,0,12,0,0,0,0,12,0,0,0,0,12],[1,0,0,0,0,12,0,0,0,0,0,1,0,0,12,1],[1,0,0,0,0,5,0,0,0,0,5,8,0,0,0,8] >;
C22×Dic3 in GAP, Magma, Sage, TeX
C_2^2\times {\rm Dic}_3 % in TeX
G:=Group("C2^2xDic3"); // GroupNames label
G:=SmallGroup(48,42);
// by ID
G=gap.SmallGroup(48,42);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-3,40,804]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^2=c^6=1,d^2=c^3,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations
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