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G = C22×Dic3order 48 = 24·3

Direct product of C22 and Dic3

direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary

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

C1C3 — C22×Dic3
C1C3C6Dic3C2×Dic3 — C22×Dic3
C3 — C22×Dic3
C1C23

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 [×6], C3, C4 [×4], C22 [×7], C6, C6 [×6], C2×C4 [×6], C23, Dic3 [×4], C2×C6 [×7], C22×C4, C2×Dic3 [×6], C22×C6, C22×Dic3
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], C23, Dic3 [×4], D6 [×3], C22×C4, C2×Dic3 [×6], C22×S3, C22×Dic3

Character table of C22×Dic3

 class 12A2B2C2D2E2F2G34A4B4C4D4E4F4G4H6A6B6C6D6E6F6G
 size 111111112333333332222222
ρ1111111111111111111111111    trivial
ρ21-1-1-1111-111-111-1-11-1-111-1-1-11    linear of order 2
ρ31-111-1-11-111-1-1-1111-11-11-11-1-1    linear of order 2
ρ411-1-1-1-111111-1-1-1-111-1-111-11-1    linear of order 2
ρ5111111111-1-1-1-1-1-1-1-11111111    linear of order 2
ρ61-1-1-1111-11-11-1-111-11-111-1-1-11    linear of order 2
ρ71-111-1-11-11-1111-1-1-111-11-11-1-1    linear of order 2
ρ811-1-1-1-1111-1-11111-1-1-1-111-11-1    linear of order 2
ρ91-1-111-1-111i-i-iii-i-ii-1-1-1-1111    linear of order 4
ρ10111-11-1-1-11ii-ii-ii-i-i1-1-11-1-11    linear of order 4
ρ11111-11-1-1-11-i-ii-ii-iii1-1-11-1-11    linear of order 4
ρ121-1-111-1-111-iii-i-iii-i-1-1-1-1111    linear of order 4
ρ1311-11-11-1-11-i-i-ii-iiii-11-111-1-1    linear of order 4
ρ141-11-1-11-111-ii-iii-ii-i11-1-1-11-1    linear of order 4
ρ151-11-1-11-111i-ii-i-ii-ii11-1-1-11-1    linear of order 4
ρ1611-11-11-1-11iii-ii-i-i-i-11-111-1-1    linear of order 4
ρ1722-2-2-2-222-10000000011-1-11-11    orthogonal lifted from D6
ρ182-222-2-22-2-100000000-11-11-111    orthogonal lifted from D6
ρ192-2-2-2222-2-1000000001-1-1111-1    orthogonal lifted from D6
ρ2022222222-100000000-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ212-2-222-2-22-1000000001111-1-1-1    symplectic lifted from Dic3, Schur index 2
ρ22222-22-2-2-2-100000000-111-111-1    symplectic lifted from Dic3, Schur index 2
ρ2322-22-22-2-2-1000000001-11-1-111    symplectic lifted from Dic3, Schur index 2
ρ242-22-2-22-22-100000000-1-1111-11    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)])

Matrix representation of C22×Dic3 in GL4(𝔽13) generated by

1000
0100
00120
00012
,
12000
01200
00120
00012
,
1000
01200
00012
0011
,
1000
0500
0050
0088
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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