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G = C3×C22⋊C4order 48 = 24·3

Direct product of C3 and C22⋊C4

direct product, metabelian, nilpotent (class 2), monomial, 2-elementary

Aliases: C3×C22⋊C4, C6.12D4, C222C12, C23.2C6, (C2×C4)⋊1C6, (C2×C6)⋊1C4, (C2×C12)⋊2C2, C2.1(C3×D4), C6.10(C2×C4), C2.1(C2×C12), C22.2(C2×C6), (C22×C6).1C2, (C2×C6).13C22, SmallGroup(48,21)

Series: Derived Chief Lower central Upper central

C1C2 — C3×C22⋊C4
C1C2C22C2×C6C2×C12 — C3×C22⋊C4
C1C2 — C3×C22⋊C4
C1C2×C6 — C3×C22⋊C4

Generators and relations for C3×C22⋊C4
 G = < a,b,c,d | a3=b2=c2=d4=1, ab=ba, ac=ca, ad=da, dbd-1=bc=cb, cd=dc >

2C2
2C2
2C4
2C22
2C4
2C22
2C6
2C6
2C12
2C12
2C2×C6
2C2×C6

Character table of C3×C22⋊C4

 class 12A2B2C2D2E3A3B4A4B4C4D6A6B6C6D6E6F6G6H6I6J12A12B12C12D12E12F12G12H
 size 111122112222111111222222222222
ρ1111111111111111111111111111111    trivial
ρ21111-1-111-111-1111111-1-1-1-11111-1-1-1-1    linear of order 2
ρ311111111-1-1-1-11111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ41111-1-1111-1-11111111-1-1-1-1-1-1-1-11111    linear of order 2
ρ5111111ζ32ζ3-1-1-1-1ζ32ζ32ζ32ζ3ζ3ζ3ζ32ζ3ζ3ζ32ζ65ζ65ζ6ζ6ζ6ζ6ζ65ζ65    linear of order 6
ρ61111-1-1ζ32ζ31-1-11ζ32ζ32ζ32ζ3ζ3ζ3ζ6ζ65ζ65ζ6ζ65ζ65ζ6ζ6ζ32ζ32ζ3ζ3    linear of order 6
ρ71111-1-1ζ3ζ321-1-11ζ3ζ3ζ3ζ32ζ32ζ32ζ65ζ6ζ6ζ65ζ6ζ6ζ65ζ65ζ3ζ3ζ32ζ32    linear of order 6
ρ81111-1-1ζ3ζ32-111-1ζ3ζ3ζ3ζ32ζ32ζ32ζ65ζ6ζ6ζ65ζ32ζ32ζ3ζ3ζ65ζ65ζ6ζ6    linear of order 6
ρ91111-1-1ζ32ζ3-111-1ζ32ζ32ζ32ζ3ζ3ζ3ζ6ζ65ζ65ζ6ζ3ζ3ζ32ζ32ζ6ζ6ζ65ζ65    linear of order 6
ρ10111111ζ32ζ31111ζ32ζ32ζ32ζ3ζ3ζ3ζ32ζ3ζ3ζ32ζ3ζ3ζ32ζ32ζ32ζ32ζ3ζ3    linear of order 3
ρ11111111ζ3ζ32-1-1-1-1ζ3ζ3ζ3ζ32ζ32ζ32ζ3ζ32ζ32ζ3ζ6ζ6ζ65ζ65ζ65ζ65ζ6ζ6    linear of order 6
ρ12111111ζ3ζ321111ζ3ζ3ζ3ζ32ζ32ζ32ζ3ζ32ζ32ζ3ζ32ζ32ζ3ζ3ζ3ζ3ζ32ζ32    linear of order 3
ρ131-11-11-111ii-i-i1-1-1-11-11-11-1i-ii-i-ii-ii    linear of order 4
ρ141-11-1-1111-ii-ii1-1-1-11-1-11-11i-ii-ii-ii-i    linear of order 4
ρ151-11-11-111-i-iii1-1-1-11-11-11-1-ii-iii-ii-i    linear of order 4
ρ161-11-1-1111i-ii-i1-1-1-11-1-11-11-ii-ii-ii-ii    linear of order 4
ρ171-11-11-1ζ32ζ62ζ2ζ2ζ2ζ2ζ32ζ6ζ6ζ65ζ62ζ65ζ32ζ65ζ62ζ6ζ4ζ3ζ43ζ3ζ4ζ32ζ43ζ32ζ43ζ32ζ4ζ32ζ43ζ3ζ4ζ3    linear of order 12
ρ181-11-1-11ζ62ζ32ζ2ζ2ζ2ζ2ζ62ζ65ζ65ζ6ζ32ζ6ζ65ζ32ζ6ζ62ζ43ζ32ζ4ζ32ζ43ζ3ζ4ζ3ζ43ζ3ζ4ζ3ζ43ζ32ζ4ζ32    linear of order 12
ρ191-11-11-1ζ62ζ32ζ2ζ2ζ2ζ2ζ62ζ65ζ65ζ6ζ32ζ6ζ62ζ6ζ32ζ65ζ43ζ32ζ4ζ32ζ43ζ3ζ4ζ3ζ4ζ3ζ43ζ3ζ4ζ32ζ43ζ32    linear of order 12
ρ201-11-11-1ζ62ζ32ζ2ζ2ζ2ζ2ζ62ζ65ζ65ζ6ζ32ζ6ζ62ζ6ζ32ζ65ζ4ζ32ζ43ζ32ζ4ζ3ζ43ζ3ζ43ζ3ζ4ζ3ζ43ζ32ζ4ζ32    linear of order 12
ρ211-11-1-11ζ62ζ32ζ2ζ2ζ2ζ2ζ62ζ65ζ65ζ6ζ32ζ6ζ65ζ32ζ6ζ62ζ4ζ32ζ43ζ32ζ4ζ3ζ43ζ3ζ4ζ3ζ43ζ3ζ4ζ32ζ43ζ32    linear of order 12
ρ221-11-1-11ζ32ζ62ζ2ζ2ζ2ζ2ζ32ζ6ζ6ζ65ζ62ζ65ζ6ζ62ζ65ζ32ζ4ζ3ζ43ζ3ζ4ζ32ζ43ζ32ζ4ζ32ζ43ζ32ζ4ζ3ζ43ζ3    linear of order 12
ρ231-11-11-1ζ32ζ62ζ2ζ2ζ2ζ2ζ32ζ6ζ6ζ65ζ62ζ65ζ32ζ65ζ62ζ6ζ43ζ3ζ4ζ3ζ43ζ32ζ4ζ32ζ4ζ32ζ43ζ32ζ4ζ3ζ43ζ3    linear of order 12
ρ241-11-1-11ζ32ζ62ζ2ζ2ζ2ζ2ζ32ζ6ζ6ζ65ζ62ζ65ζ6ζ62ζ65ζ32ζ43ζ3ζ4ζ3ζ43ζ32ζ4ζ32ζ43ζ32ζ4ζ32ζ43ζ3ζ4ζ3    linear of order 12
ρ252-2-2200220000-22-2-2-22000000000000    orthogonal lifted from D4
ρ2622-2-200220000-2-222-2-2000000000000    orthogonal lifted from D4
ρ2722-2-200-1--3-1+-300001+-31+-3-1--3-1+-31--31--3000000000000    complex lifted from C3×D4
ρ282-2-2200-1--3-1+-300001+-3-1--31+-31--31--3-1+-3000000000000    complex lifted from C3×D4
ρ292-2-2200-1+-3-1--300001--3-1+-31--31+-31+-3-1--3000000000000    complex lifted from C3×D4
ρ3022-2-200-1+-3-1--300001--31--3-1+-3-1--31+-31+-3000000000000    complex lifted from C3×D4

Permutation representations of C3×C22⋊C4
On 24 points - transitive group 24T39
Generators in S24
(1 19 6)(2 20 7)(3 17 8)(4 18 5)(9 15 21)(10 16 22)(11 13 23)(12 14 24)
(2 12)(4 10)(5 22)(7 24)(14 20)(16 18)
(1 11)(2 12)(3 9)(4 10)(5 22)(6 23)(7 24)(8 21)(13 19)(14 20)(15 17)(16 18)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)

G:=sub<Sym(24)| (1,19,6)(2,20,7)(3,17,8)(4,18,5)(9,15,21)(10,16,22)(11,13,23)(12,14,24), (2,12)(4,10)(5,22)(7,24)(14,20)(16,18), (1,11)(2,12)(3,9)(4,10)(5,22)(6,23)(7,24)(8,21)(13,19)(14,20)(15,17)(16,18), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24)>;

G:=Group( (1,19,6)(2,20,7)(3,17,8)(4,18,5)(9,15,21)(10,16,22)(11,13,23)(12,14,24), (2,12)(4,10)(5,22)(7,24)(14,20)(16,18), (1,11)(2,12)(3,9)(4,10)(5,22)(6,23)(7,24)(8,21)(13,19)(14,20)(15,17)(16,18), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24) );

G=PermutationGroup([(1,19,6),(2,20,7),(3,17,8),(4,18,5),(9,15,21),(10,16,22),(11,13,23),(12,14,24)], [(2,12),(4,10),(5,22),(7,24),(14,20),(16,18)], [(1,11),(2,12),(3,9),(4,10),(5,22),(6,23),(7,24),(8,21),(13,19),(14,20),(15,17),(16,18)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24)])

G:=TransitiveGroup(24,39);

Matrix representation of C3×C22⋊C4 in GL3(𝔽13) generated by

300
090
009
,
100
0120
031
,
100
0120
0012
,
500
081
005
G:=sub<GL(3,GF(13))| [3,0,0,0,9,0,0,0,9],[1,0,0,0,12,3,0,0,1],[1,0,0,0,12,0,0,0,12],[5,0,0,0,8,0,0,1,5] >;

C3×C22⋊C4 in GAP, Magma, Sage, TeX

C_3\times C_2^2\rtimes C_4
% in TeX

G:=Group("C3xC2^2:C4");
// GroupNames label

G:=SmallGroup(48,21);
// by ID

G=gap.SmallGroup(48,21);
# by ID

G:=PCGroup([5,-2,-2,-3,-2,-2,120,141]);
// Polycyclic

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

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