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## G = C22×Dic3order 48 = 24·3

### Direct product of C22 and Dic3

Aliases: C22×Dic3, C6.9C23, C23.3S3, C22.11D6, (C2×C6)⋊3C4, C62(C2×C4), C32(C22×C4), C2.2(C22×S3), (C22×C6).3C2, (C2×C6).12C22, SmallGroup(48,42)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — C22×Dic3
 Chief series C1 — C3 — C6 — Dic3 — C2×Dic3 — C22×Dic3
 Lower central C3 — C22×Dic3
 Upper central C1 — C23

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

Smallest permutation representation of C22×Dic3
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  Dic34D4  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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