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G = C3×C22⋊C4  order 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, C22⋊2C12, 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

Derived series
C1 — C2 — C3×C22⋊C4
Chief series
C1 — C2 — C22 — C2×C6 — C2×C12 — C3×C22⋊C4
Lower central
C1 — C2 — C3×C22⋊C4
Upper central
C1 — C2×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 >

C1
C2
C2
C2
2C2
2C2
C3
C22
C22
C22
2C4
2C22
2C4
2C22
C6
C6
C6
2C6
2C6
C23
C2×C4
C2×C4
C2×C6
C2×C6
C2×C6
2C12
2C12
2C2×C6
2C2×C6
C22⋊C4
C2×C12
C22×C6
C2×C12
C3×C22⋊C4

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
ρ171-11-11-1ζ32ζ3ii-i-iζ32ζ6ζ6ζ65ζ3ζ65ζ32ζ65ζ3ζ6ζ4ζ3ζ43ζ3ζ4ζ32ζ43ζ32ζ43ζ32ζ4ζ32ζ43ζ3ζ4ζ3    linear of order 12
ρ181-11-1-11ζ3ζ32i-ii-iζ3ζ65ζ65ζ6ζ32ζ6ζ65ζ32ζ6ζ3ζ43ζ32ζ4ζ32ζ43ζ3ζ4ζ3ζ43ζ3ζ4ζ3ζ43ζ32ζ4ζ32    linear of order 12
ρ191-11-11-1ζ3ζ32-i-iiiζ3ζ65ζ65ζ6ζ32ζ6ζ3ζ6ζ32ζ65ζ43ζ32ζ4ζ32ζ43ζ3ζ4ζ3ζ4ζ3ζ43ζ3ζ4ζ32ζ43ζ32    linear of order 12
ρ201-11-11-1ζ3ζ32ii-i-iζ3ζ65ζ65ζ6ζ32ζ6ζ3ζ6ζ32ζ65ζ4ζ32ζ43ζ32ζ4ζ3ζ43ζ3ζ43ζ3ζ4ζ3ζ43ζ32ζ4ζ32    linear of order 12
ρ211-11-1-11ζ3ζ32-ii-iiζ3ζ65ζ65ζ6ζ32ζ6ζ65ζ32ζ6ζ3ζ4ζ32ζ43ζ32ζ4ζ3ζ43ζ3ζ4ζ3ζ43ζ3ζ4ζ32ζ43ζ32    linear of order 12
ρ221-11-1-11ζ32ζ3-ii-iiζ32ζ6ζ6ζ65ζ3ζ65ζ6ζ3ζ65ζ32ζ4ζ3ζ43ζ3ζ4ζ32ζ43ζ32ζ4ζ32ζ43ζ32ζ4ζ3ζ43ζ3    linear of order 12
ρ231-11-11-1ζ32ζ3-i-iiiζ32ζ6ζ6ζ65ζ3ζ65ζ32ζ65ζ3ζ6ζ43ζ3ζ4ζ3ζ43ζ32ζ4ζ32ζ4ζ32ζ43ζ32ζ4ζ3ζ43ζ3    linear of order 12
ρ241-11-1-11ζ32ζ3i-ii-iζ32ζ6ζ6ζ65ζ3ζ65ζ6ζ3ζ65ζ32ζ43ζ3ζ4ζ3ζ43ζ32ζ4ζ32ζ43ζ32ζ4ζ32ζ43ζ3ζ4ζ3    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 6 19)(2 7 20)(3 8 17)(4 5 18)(9 15 21)(10 16 22)(11 13 23)(12 14 24)

(2 12)(4 10)(5 16)(7 14)(18 22)(20 24)

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

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

C3×C22⋊C4 is a maximal subgroup of
 C23.6D6  C23.16D6  Dic3.D4  C23.8D6  Dic3⋊4D4  D6⋊D4  C23.9D6  Dic3⋊D4  C23.11D6  C23.21D6  D4×C12  (C2×Q8)⋊C12  SL2(𝔽3)⋊5D4  D14⋊C12  C23.2F7  C22⋊(He3⋊C4)
C3×C22⋊C4 is a maximal quotient of
 D14⋊C12  C23.2F7

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
 

Export

Subgroup lattice of C3×C22⋊C4 in TeX
Character table of C3×C22⋊C4 in TeX

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