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G = C424S3order 96 = 25·3

3rd semidirect product of C42 and S3 acting via S3/C3=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D121C4, C424S3, Dic61C4, C4.17D12, C12.33D4, C31C4≀C2, (C4×C12)⋊6C2, C4.6(C4×S3), (C2×C6).26D4, (C2×C4).66D6, C12.16(C2×C4), C2.3(D6⋊C4), C4○D12.1C2, C4.Dic31C2, C6.1(C22⋊C4), (C2×C12).96C22, C22.7(C3⋊D4), SmallGroup(96,12)

Series: Derived Chief Lower central Upper central

C1C12 — C424S3
C1C3C6C2×C6C2×C12C4○D12 — C424S3
C3C6C12 — C424S3
C1C4C2×C4C42

Generators and relations for C424S3
 G = < a,b,c,d | a4=b4=c3=d2=1, dad=ab=ba, ac=ca, bc=cb, dbd=b-1, dcd=c-1 >

2C2
12C2
2C4
2C4
6C22
6C4
2C6
4S3
2C2×C4
3D4
3Q8
6C8
6D4
6C2×C4
2C12
2C12
2D6
2Dic3
3M4(2)
3C4○D4
2C3⋊D4
2C4×S3
2C3⋊C8
2C2×C12
3C4≀C2

Character table of C424S3

 class 12A2B2C34A4B4C4D4E4F4G4H6A6B6C8A8B12A12B12C12D12E12F12G12H12I12J12K12L
 size 1121221122222122221212222222222222
ρ1111111111111111111111111111111    trivial
ρ2111-1111-11-1-1-1-111111-1-11-1-11-1-11-1-11    linear of order 2
ρ3111-111111111-1111-1-1111111111111    linear of order 2
ρ41111111-11-1-1-11111-1-1-1-11-1-11-1-11-1-11    linear of order 2
ρ511-111-1-1-i1ii-i-11-1-1-iiii1-i-i-1-i-i1ii-1    linear of order 4
ρ611-111-1-1i1-i-ii-11-1-1i-i-i-i1ii-1ii1-i-i-1    linear of order 4
ρ711-1-11-1-1-i1ii-i11-1-1i-iii1-i-i-1-i-i1ii-1    linear of order 4
ρ811-1-11-1-1i1-i-ii11-1-1-ii-i-i1ii-1ii1-i-i-1    linear of order 4
ρ922202-2-20-200002220000-200-200-200-2    orthogonal lifted from D4
ρ102220-122-22-2-2-20-1-1-10011-111-111-111-1    orthogonal lifted from D6
ρ112220-122222220-1-1-100-1-1-1-1-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ1222-202220-200002-2-20000-200200-2002    orthogonal lifted from D4
ρ1322-20-1220-20000-11100-3313-3-13-31-33-1    orthogonal lifted from D12
ρ1422-20-1220-20000-111003-31-33-1-3313-3-1    orthogonal lifted from D12
ρ152220-1-2-20-20000-1-1-100--3-31--3-31--3-31--3-31    complex lifted from C3⋊D4
ρ162-20022i-2i1-i0-1-i1+i-1+i0-200001+i1+i0-1+i-1+i-2i1-i1-i0-1-i-1-i2i    complex lifted from C4≀C2
ρ172220-1-2-20-20000-1-1-100-3--31-3--31-3--31-3--31    complex lifted from C3⋊D4
ρ1822-20-1-2-22i2-2i-2i2i0-11100ii-1-i-i1-i-i-1ii1    complex lifted from C4×S3
ρ192-2002-2i2i1+i0-1+i1-i-1-i0-200001-i1-i0-1-i-1-i2i1+i1+i0-1+i-1+i-2i    complex lifted from C4≀C2
ρ2022-20-1-2-2-2i22i2i-2i0-11100-i-i-1ii1ii-1-i-i1    complex lifted from C4×S3
ρ212-2002-2i2i-1-i01-i-1+i1+i0-20000-1+i-1+i01+i1+i2i-1-i-1-i01-i1-i-2i    complex lifted from C4≀C2
ρ222-200-12i-2i-1+i01+i-1-i1-i01--3-300ζ43ζ33+1ζ43ζ3232+13ζ4ζ32432ζ4ζ343iζ4ζ33+1ζ4ζ3232+1-3ζ43ζ324332ζ43ζ3433-i    complex faithful
ρ232-20022i-2i-1+i01+i-1-i1-i0-20000-1-i-1-i01-i1-i-2i-1+i-1+i01+i1+i2i    complex lifted from C4≀C2
ρ242-200-1-2i2i1+i0-1+i1-i-1-i01-3--300ζ4ζ343ζ4ζ324323ζ43ζ3232+1ζ43ζ33+1-iζ43ζ3433ζ43ζ324332-3ζ4ζ3232+1ζ4ζ33+1i    complex faithful
ρ252-200-12i-2i1-i0-1-i1+i-1+i01-3--300ζ43ζ3433ζ43ζ324332-3ζ4ζ3232+1ζ4ζ33+1iζ4ζ343ζ4ζ324323ζ43ζ3232+1ζ43ζ33+1-i    complex faithful
ρ262-200-1-2i2i-1-i01-i-1+i1+i01-3--300ζ4ζ3232+1ζ4ζ33+13ζ43ζ3433ζ43ζ324332-iζ43ζ3232+1ζ43ζ33+1-3ζ4ζ343ζ4ζ32432i    complex faithful
ρ272-200-1-2i2i1+i0-1+i1-i-1-i01--3-300ζ4ζ32432ζ4ζ343-3ζ43ζ33+1ζ43ζ3232+1-iζ43ζ324332ζ43ζ34333ζ4ζ33+1ζ4ζ3232+1i    complex faithful
ρ282-200-12i-2i1-i0-1-i1+i-1+i01--3-300ζ43ζ324332ζ43ζ34333ζ4ζ33+1ζ4ζ3232+1iζ4ζ32432ζ4ζ343-3ζ43ζ33+1ζ43ζ3232+1-i    complex faithful
ρ292-200-1-2i2i-1-i01-i-1+i1+i01--3-300ζ4ζ33+1ζ4ζ3232+1-3ζ43ζ324332ζ43ζ3433-iζ43ζ33+1ζ43ζ3232+13ζ4ζ32432ζ4ζ343i    complex faithful
ρ302-200-12i-2i-1+i01+i-1-i1-i01-3--300ζ43ζ3232+1ζ43ζ33+1-3ζ4ζ343ζ4ζ32432iζ4ζ3232+1ζ4ζ33+13ζ43ζ3433ζ43ζ324332-i    complex faithful

Permutation representations of C424S3
On 24 points - transitive group 24T117
Generators in S24
(13 14 15 16)(17 18 19 20)(21 22 23 24)
(1 10 4 7)(2 11 5 8)(3 12 6 9)(13 16 15 14)(17 20 19 18)(21 24 23 22)
(1 2 3)(4 5 6)(7 8 9)(10 11 12)(13 17 21)(14 18 22)(15 19 23)(16 20 24)
(1 18)(2 14)(3 22)(4 20)(5 16)(6 24)(7 17)(8 13)(9 21)(10 19)(11 15)(12 23)

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

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

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

G:=TransitiveGroup(24,117);

C424S3 is a maximal subgroup of
D2411C4  D244C4  S3×C4≀C2  C423D6  Q85D12  M4(2).22D6  C42.196D6  C425D6  Q8.14D12  D4.10D12  C426D6  C427D6  D12.14D4  C428D6  D12.15D4  C424D9  C42⋊D9  D122Dic3  C12.80D12  C122⋊C2  (C4×C12)⋊S3  C60.99D4  D6016C4  D607C4  D122F5  D605C4
C424S3 is a maximal quotient of
C6.C4≀C2  C4⋊Dic3⋊C4  C4.8Dic12  C4.17D24  C42.D6  C42.2D6  C12.8C42  C424D9  D122Dic3  C12.80D12  C122⋊C2  C60.99D4  D6016C4  D607C4  D122F5  D605C4

Matrix representation of C424S3 in GL2(𝔽13) generated by

80
01
,
50
08
,
90
03
,
01
10
G:=sub<GL(2,GF(13))| [8,0,0,1],[5,0,0,8],[9,0,0,3],[0,1,1,0] >;

C424S3 in GAP, Magma, Sage, TeX

C_4^2\rtimes_4S_3
% in TeX

G:=Group("C4^2:4S3");
// GroupNames label

G:=SmallGroup(96,12);
// by ID

G=gap.SmallGroup(96,12);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-3,121,31,362,579,69,2309]);
// Polycyclic

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

Export

Subgroup lattice of C424S3 in TeX
Character table of C424S3 in TeX

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