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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)])

C4⋊Dic3 is a maximal subgroup of
C6.Q16  C12.Q8  C2.Dic12  C8⋊Dic3  C241C4  C2.D24  D4⋊Dic3  Q82Dic3  C4×Dic6  C122Q8  C12.6Q8  C4×D12  Dic3.D4  C23.8D6  C23.9D6  C23.21D6  C12⋊Q8  Dic3.Q8  C4.Dic6  S3×C4⋊C4  C4⋊C47S3  C4.D12  C4⋊C4⋊S3  C12.48D4  C23.26D6  C127D4  D4×Dic3  D63D4  Q8×Dic3  D63Q8  C4⋊Dic9  Dic3⋊Dic3  C12⋊Dic3  Q8.D12  SL2(𝔽3)⋊Q8  C24.4D6  (C2×C4).S4  C30.Q8  C605C4  C60⋊C4  C42.Q8  C84⋊C4  C6.S3≀C2  (C3×C6).9D12  C6.2PSU3(𝔽2)  C339(C4⋊C4)
C4⋊Dic3 is a maximal quotient of
C12⋊C8  C8⋊Dic3  C241C4  C24.C4  C6.C42  C4⋊Dic9  Dic3⋊Dic3  C12⋊Dic3  C24.4D6  C30.Q8  C605C4  C60⋊C4  C42.Q8  C84⋊C4  (C3×C6).9D12  C6.2PSU3(𝔽2)  C339(C4⋊C4)

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

Export

Subgroup lattice of C4⋊Dic3 in TeX
Character table of C4⋊Dic3 in TeX

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