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G = C23.6D6order 96 = 25·3

1st non-split extension by C23 of D6 acting via D6/C3=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C23.6D6, C22.2D12, (C22×S3)⋊C4, (C2×Dic3)⋊C4, C22⋊C41S3, C31(C23⋊C4), (C2×C6).27D4, C2.4(D6⋊C4), C22.3(C4×S3), C6.D41C2, C6.2(C22⋊C4), C22.8(C3⋊D4), (C22×C6).5C22, (C2×C6).1(C2×C4), (C3×C22⋊C4)⋊1C2, (C2×C3⋊D4).1C2, SmallGroup(96,13)

Series: Derived Chief Lower central Upper central

C1C2×C6 — C23.6D6
C1C3C6C2×C6C22×C6C2×C3⋊D4 — C23.6D6
C3C6C2×C6 — C23.6D6
C1C2C23C22⋊C4

Generators and relations for C23.6D6
 G = < a,b,c,d,e | a2=b2=c2=1, d6=a, e2=abc, ab=ba, eae-1=ac=ca, ad=da, dbd-1=ebe-1=bc=cb, cd=dc, ce=ec, ede-1=bcd5 >

2C2
2C2
2C2
12C2
4C4
4C22
6C22
6C4
12C4
12C22
2C6
2C6
2C6
4S3
2C2×C4
3C23
3C2×C4
6C2×C4
6D4
6D4
2Dic3
2D6
4Dic3
4D6
4C2×C6
4C12
3C22⋊C4
3C2×D4
2C2×C12
2C3⋊D4
2C3⋊D4
2C2×Dic3
3C23⋊C4

Character table of C23.6D6

 class 12A2B2C2D2E34A4B4C4D4E6A6B6C6D6E12A12B12C12D
 size 1122212244121212222444444
ρ1111111111111111111111    trivial
ρ21111111-1-1-1-1111111-1-1-1-1    linear of order 2
ρ311111-11-1-111-111111-1-1-1-1    linear of order 2
ρ411111-1111-1-1-1111111111    linear of order 2
ρ511-11-1-11-iii-i11-1-11-1-i-iii    linear of order 4
ρ611-11-111-ii-ii-11-1-11-1-i-iii    linear of order 4
ρ711-11-1-11i-i-ii11-1-11-1ii-i-i    linear of order 4
ρ811-11-111i-ii-i-11-1-11-1ii-i-i    linear of order 4
ρ9222-2-20200000222-2-20000    orthogonal lifted from D4
ρ1022-2-2202000002-2-2-220000    orthogonal lifted from D4
ρ11222220-1-2-2000-1-1-1-1-11111    orthogonal lifted from D6
ρ12222220-122000-1-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ1322-2-220-100000-1111-1-33-33    orthogonal lifted from D12
ρ1422-2-220-100000-1111-13-33-3    orthogonal lifted from D12
ρ1522-22-20-1-2i2i000-111-11ii-i-i    complex lifted from C4×S3
ρ1622-22-20-12i-2i000-111-11-i-iii    complex lifted from C4×S3
ρ17222-2-20-100000-1-1-111--3-3-3--3    complex lifted from C3⋊D4
ρ18222-2-20-100000-1-1-111-3--3--3-3    complex lifted from C3⋊D4
ρ194-40000400000-400000000    orthogonal lifted from C23⋊C4
ρ204-40000-2000002-2-32-3000000    complex faithful
ρ214-40000-20000022-3-2-3000000    complex faithful

Permutation representations of C23.6D6
On 24 points - transitive group 24T94
Generators in S24
(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)
(1 10)(3 12)(5 8)(14 20)(16 22)(18 24)
(1 10)(2 11)(3 12)(4 7)(5 8)(6 9)(13 19)(14 20)(15 21)(16 22)(17 23)(18 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)
(1 21)(2 14 11 20)(3 13)(4 18 7 24)(5 17)(6 22 9 16)(8 23)(10 15)(12 19)

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

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

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

G:=TransitiveGroup(24,94);

On 24 points - transitive group 24T103
Generators in S24
(1 7)(2 8)(3 9)(4 10)(5 11)(6 12)(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)
(2 17)(4 19)(6 21)(8 23)(10 13)(12 15)
(1 16)(2 17)(3 18)(4 19)(5 20)(6 21)(7 22)(8 23)(9 24)(10 13)(11 14)(12 15)
(1 2 3 4 5 6 7 8 9 10 11 12)(13 14 15 16 17 18 19 20 21 22 23 24)
(1 6 22 12)(2 14 8 5)(3 19 24 13)(4 9 10 18)(7 15 16 21)(11 23 20 17)

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

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

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

G:=TransitiveGroup(24,103);

On 24 points - transitive group 24T108
Generators in S24
(1 7)(2 8)(3 9)(4 10)(5 11)(6 12)(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)
(1 14)(2 8)(3 16)(4 10)(5 18)(6 12)(7 20)(9 22)(11 24)(13 19)(15 21)(17 23)
(1 20)(2 21)(3 22)(4 23)(5 24)(6 13)(7 14)(8 15)(9 16)(10 17)(11 18)(12 19)
(1 2 3 4 5 6 7 8 9 10 11 12)(13 14 15 16 17 18 19 20 21 22 23 24)
(1 9)(2 15 21 8)(3 14)(4 6 23 13)(7 22)(10 19 17 12)(11 18)(16 20)

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

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

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

G:=TransitiveGroup(24,108);

On 24 points - transitive group 24T119
Generators in S24
(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)
(1 7)(2 11)(3 9)(4 10)(5 8)(6 12)(13 16)(14 23)(15 18)(17 20)(19 22)(21 24)
(1 4)(2 5)(3 6)(7 10)(8 11)(9 12)(13 19)(14 20)(15 21)(16 22)(17 23)(18 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)
(1 13 10 16)(2 21 8 24)(3 17 12 20)(4 19 7 22)(5 15 11 18)(6 23 9 14)

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

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

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

G:=TransitiveGroup(24,119);

C23.6D6 is a maximal subgroup of
C23⋊C45S3  C23⋊D12  C23.5D12  S3×C23⋊C4  (C2×D12)⋊13C4  C246D6  C22⋊C4⋊D6  C22.D36  C62.31D4  C62.32D4  C62.110D4  C158(C23⋊C4)  C159(C23⋊C4)  C23.6D30  D10.D12  D10.4D12
C23.6D6 is a maximal quotient of
C6.C4≀C2  C4⋊Dic3⋊C4  C23.35D12  (C22×S3)⋊C8  (C2×Dic3)⋊C8  C22.2D24  C3⋊C2≀C4  (C2×D4).D6  C23.D12  C23.2D12  C23.3D12  C23.4D12  (C2×C4).D12  (C2×C12).D4  C24.13D6  C22.D36  C62.31D4  C62.32D4  C62.110D4  C158(C23⋊C4)  C159(C23⋊C4)  C23.6D30  D10.D12  D10.4D12

Matrix representation of C23.6D6 in GL4(𝔽7) generated by

0145
1035
0010
0006
,
1526
1553
0010
5210
,
6000
0600
0060
0006
,
1052
2161
5554
4350
,
4226
3054
6143
4456
G:=sub<GL(4,GF(7))| [0,1,0,0,1,0,0,0,4,3,1,0,5,5,0,6],[1,1,0,5,5,5,0,2,2,5,1,1,6,3,0,0],[6,0,0,0,0,6,0,0,0,0,6,0,0,0,0,6],[1,2,5,4,0,1,5,3,5,6,5,5,2,1,4,0],[4,3,6,4,2,0,1,4,2,5,4,5,6,4,3,6] >;

C23.6D6 in GAP, Magma, Sage, TeX

C_2^3._6D_6
% in TeX

G:=Group("C2^3.6D6");
// GroupNames label

G:=SmallGroup(96,13);
// by ID

G=gap.SmallGroup(96,13);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-3,121,31,362,297,2309]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^2=1,d^6=a,e^2=a*b*c,a*b=b*a,e*a*e^-1=a*c=c*a,a*d=d*a,d*b*d^-1=e*b*e^-1=b*c=c*b,c*d=d*c,c*e=e*c,e*d*e^-1=b*c*d^5>;
// generators/relations

Export

Subgroup lattice of C23.6D6 in TeX
Character table of C23.6D6 in TeX

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