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G = C4×Dic3order 48 = 24·3

Direct product of C4 and Dic3

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

Aliases: C4×Dic3, C3⋊C42, C122C4, C22.3D6, C2.2(C4×S3), C6.7(C2×C4), (C2×C4).6S3, (C2×C12).6C2, (C2×C6).3C22, C2.2(C2×Dic3), (C2×Dic3).4C2, SmallGroup(48,11)

Series: Derived Chief Lower central Upper central

C1C3 — C4×Dic3
C1C3C6C2×C6C2×Dic3 — C4×Dic3
C3 — C4×Dic3
C1C2×C4

Generators and relations for C4×Dic3
 G = < a,b,c | a4=b6=1, c2=b3, ab=ba, ac=ca, cbc-1=b-1 >

3C4
3C4
3C4
3C4
3C2×C4
3C2×C4
3C42

Character table of C4×Dic3

 class 12A2B2C34A4B4C4D4E4F4G4H4I4J4K4L6A6B6C12A12B12C12D
 size 111121111333333332222222
ρ1111111111111111111111111    trivial
ρ211111-1-1-1-11-1-1-1-1111111-1-1-1-1    linear of order 2
ρ3111111111-1-1-1-1-1-1-1-11111111    linear of order 2
ρ411111-1-1-1-1-11111-1-1-1111-1-1-1-1    linear of order 2
ρ51-11-11i-ii-ii-11-11-ii-i-1-11-i-iii    linear of order 4
ρ611-1-11-111-1i-i-iii-i-ii1-1-11-1-11    linear of order 4
ρ711-1-11-111-1-iii-i-iii-i1-1-11-1-11    linear of order 4
ρ81-11-11i-ii-i-i1-11-1i-ii-1-11-i-iii    linear of order 4
ρ91-1-111-i-iii-1i-i-ii-111-11-1-ii-ii    linear of order 4
ρ1011-1-111-1-11iii-i-i-i-ii1-1-1-111-1    linear of order 4
ρ1111-1-111-1-11-i-i-iiiii-i1-1-1-111-1    linear of order 4
ρ121-11-11-ii-iii1-11-1-ii-i-1-11ii-i-i    linear of order 4
ρ131-1-111ii-i-i-1-iii-i-111-11-1i-ii-i    linear of order 4
ρ141-11-11-ii-ii-i-11-11i-ii-1-11ii-i-i    linear of order 4
ρ151-1-111-i-iii1-iii-i1-1-1-11-1-ii-ii    linear of order 4
ρ161-1-111ii-i-i1i-i-ii1-1-1-11-1i-ii-i    linear of order 4
ρ172222-1-2-2-2-200000000-1-1-11111    orthogonal lifted from D6
ρ182222-1222200000000-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ1922-2-2-12-2-2200000000-1111-1-11    symplectic lifted from Dic3, Schur index 2
ρ2022-2-2-1-222-200000000-111-111-1    symplectic lifted from Dic3, Schur index 2
ρ212-22-2-1-2i2i-2i2i0000000011-1-i-iii    complex lifted from C4×S3
ρ222-2-22-12i2i-2i-2i000000001-11-ii-ii    complex lifted from C4×S3
ρ232-22-2-12i-2i2i-2i0000000011-1ii-i-i    complex lifted from C4×S3
ρ242-2-22-1-2i-2i2i2i000000001-11i-ii-i    complex lifted from C4×S3

Smallest permutation representation of C4×Dic3
Regular action on 48 points
Generators in S48
(1 27 16 24)(2 28 17 19)(3 29 18 20)(4 30 13 21)(5 25 14 22)(6 26 15 23)(7 38 45 33)(8 39 46 34)(9 40 47 35)(10 41 48 36)(11 42 43 31)(12 37 44 32)
(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 38 4 41)(2 37 5 40)(3 42 6 39)(7 21 10 24)(8 20 11 23)(9 19 12 22)(13 36 16 33)(14 35 17 32)(15 34 18 31)(25 47 28 44)(26 46 29 43)(27 45 30 48)

G:=sub<Sym(48)| (1,27,16,24)(2,28,17,19)(3,29,18,20)(4,30,13,21)(5,25,14,22)(6,26,15,23)(7,38,45,33)(8,39,46,34)(9,40,47,35)(10,41,48,36)(11,42,43,31)(12,37,44,32), (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), (1,38,4,41)(2,37,5,40)(3,42,6,39)(7,21,10,24)(8,20,11,23)(9,19,12,22)(13,36,16,33)(14,35,17,32)(15,34,18,31)(25,47,28,44)(26,46,29,43)(27,45,30,48)>;

G:=Group( (1,27,16,24)(2,28,17,19)(3,29,18,20)(4,30,13,21)(5,25,14,22)(6,26,15,23)(7,38,45,33)(8,39,46,34)(9,40,47,35)(10,41,48,36)(11,42,43,31)(12,37,44,32), (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), (1,38,4,41)(2,37,5,40)(3,42,6,39)(7,21,10,24)(8,20,11,23)(9,19,12,22)(13,36,16,33)(14,35,17,32)(15,34,18,31)(25,47,28,44)(26,46,29,43)(27,45,30,48) );

G=PermutationGroup([(1,27,16,24),(2,28,17,19),(3,29,18,20),(4,30,13,21),(5,25,14,22),(6,26,15,23),(7,38,45,33),(8,39,46,34),(9,40,47,35),(10,41,48,36),(11,42,43,31),(12,37,44,32)], [(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)], [(1,38,4,41),(2,37,5,40),(3,42,6,39),(7,21,10,24),(8,20,11,23),(9,19,12,22),(13,36,16,33),(14,35,17,32),(15,34,18,31),(25,47,28,44),(26,46,29,43),(27,45,30,48)])

Matrix representation of C4×Dic3 in GL3(𝔽13) generated by

800
0120
0012
,
100
0012
011
,
1200
008
080
G:=sub<GL(3,GF(13))| [8,0,0,0,12,0,0,0,12],[1,0,0,0,0,1,0,12,1],[12,0,0,0,0,8,0,8,0] >;

C4×Dic3 in GAP, Magma, Sage, TeX

C_4\times {\rm Dic}_3
% in TeX

G:=Group("C4xDic3");
// GroupNames label

G:=SmallGroup(48,11);
// by ID

G=gap.SmallGroup(48,11);
# by ID

G:=PCGroup([5,-2,-2,-2,-2,-3,20,46,804]);
// Polycyclic

G:=Group<a,b,c|a^4=b^6=1,c^2=b^3,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

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