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G = C4⋊Dic3order 48 = 24·3

The semidirect product of C4 and Dic3 acting via Dic3/C6=C2

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C4⋊Dic3, C121C4, C6.4D4, C6.2Q8, C2.1D12, C2.2Dic6, C22.5D6, C32(C4⋊C4), (C2×C4).3S3, C6.8(C2×C4), (C2×C12).2C2, (C2×C6).5C22, C2.4(C2×Dic3), (C2×Dic3).2C2, SmallGroup(48,13)

Series: Derived Chief Lower central Upper central

C1C6 — C4⋊Dic3
C1C3C6C2×C6C2×Dic3 — C4⋊Dic3
C3C6 — C4⋊Dic3
C1C22C2×C4

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

6C4
6C4
3C2×C4
3C2×C4
2Dic3
2Dic3
3C4⋊C4

Character table of C4⋊Dic3

 class 12A2B2C34A4B4C4D4E4F6A6B6C12A12B12C12D
 size 111122266662222222
ρ1111111111111111111    trivial
ρ211111-1-11-11-1111-1-1-1-1    linear of order 2
ρ31111111-1-1-1-11111111    linear of order 2
ρ411111-1-1-11-11111-1-1-1-1    linear of order 2
ρ51-11-111-1-i-iii1-1-1-1-111    linear of order 4
ρ61-11-111-1ii-i-i1-1-1-1-111    linear of order 4
ρ71-11-11-11-iii-i1-1-111-1-1    linear of order 4
ρ81-11-11-11i-i-ii1-1-111-1-1    linear of order 4
ρ922-2-22000000-2-220000    orthogonal lifted from D4
ρ102222-1220000-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ1122-2-2-100000011-13-33-3    orthogonal lifted from D12
ρ1222-2-2-100000011-1-33-33    orthogonal lifted from D12
ρ132222-1-2-20000-1-1-11111    orthogonal lifted from D6
ρ142-22-2-1-220000-111-1-111    symplectic lifted from Dic3, Schur index 2
ρ152-2-222000000-22-20000    symplectic lifted from Q8, Schur index 2
ρ162-2-22-10000001-113-3-33    symplectic lifted from Dic6, Schur index 2
ρ172-22-2-12-20000-11111-1-1    symplectic lifted from Dic3, Schur index 2
ρ182-2-22-10000001-11-333-3    symplectic lifted from Dic6, Schur index 2

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

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

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

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

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

100
036
0710
,
1200
0012
011
,
800
050
088
G:=sub<GL(3,GF(13))| [1,0,0,0,3,7,0,6,10],[12,0,0,0,0,1,0,12,1],[8,0,0,0,5,8,0,0,8] >;

C4⋊Dic3 in GAP, Magma, Sage, TeX

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

G:=Group("C4:Dic3");
// GroupNames label

G:=SmallGroup(48,13);
// by ID

G=gap.SmallGroup(48,13);
# by ID

G:=PCGroup([5,-2,-2,-2,-2,-3,20,101,46,804]);
// Polycyclic

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

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