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G = C23.7D6order 96 = 25·3

2nd non-split extension by C23 of D6 acting via D6/C3=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C23.7D6, C232Dic3, (C2×C4)⋊Dic3, (C2×C12)⋊1C4, (C2×C6).2D4, C32(C23⋊C4), (C22×C6)⋊2C4, (C6×D4).6C2, (C2×D4).3S3, C6.D42C2, C6.15(C22⋊C4), C22.2(C3⋊D4), (C22×C6).6C22, C22.3(C2×Dic3), C2.5(C6.D4), (C2×C6).29(C2×C4), SmallGroup(96,41)

Series: Derived Chief Lower central Upper central

C1C2×C6 — C23.7D6
C1C3C6C2×C6C22×C6C6.D4 — C23.7D6
C3C6C2×C6 — C23.7D6
C1C2C23C2×D4

Generators and relations for C23.7D6
 G = < a,b,c,d,e | a2=b2=c2=d6=1, e2=ba=ab, dad-1=eae-1=ac=ca, ebe-1=bc=cb, bd=db, cd=dc, ce=ec, ede-1=bcd-1 >

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

Character table of C23.7D6

 class 12A2B2C2D2E34A4B4C4D4E6A6B6C6D6E6F6G12A12B
 size 1122242412121212222444444
ρ1111111111111111111111    trivial
ρ211111-11-11-11-1111-1-111-1-1    linear of order 2
ρ311111111-1-1-1-1111111111    linear of order 2
ρ411111-11-1-11-11111-1-111-1-1    linear of order 2
ρ511-11-111-1ii-i-i11111-1-1-1-1    linear of order 4
ρ611-11-1-111-iii-i111-1-1-1-111    linear of order 4
ρ711-11-1-111i-i-ii111-1-1-1-111    linear of order 4
ρ811-11-111-1-i-iii11111-1-1-1-1    linear of order 4
ρ922-2-220200000-22-2002-200    orthogonal lifted from D4
ρ10222222-120000-1-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ1122222-2-1-20000-1-1-111-1-111    orthogonal lifted from D6
ρ12222-2-20200000-22-200-2200    orthogonal lifted from D4
ρ1322-22-22-1-20000-1-1-1-1-11111    symplectic lifted from Dic3, Schur index 2
ρ1422-22-2-2-120000-1-1-11111-1-1    symplectic lifted from Dic3, Schur index 2
ρ1522-2-220-1000001-11-3--3-11--3-3    complex lifted from C3⋊D4
ρ16222-2-20-1000001-11--3-31-1--3-3    complex lifted from C3⋊D4
ρ17222-2-20-1000001-11-3--31-1-3--3    complex lifted from C3⋊D4
ρ1822-2-220-1000001-11--3-3-11-3--3    complex lifted from C3⋊D4
ρ194-400004000000-40000000    orthogonal lifted from C23⋊C4
ρ204-40000-200000-2-322-3000000    complex faithful
ρ214-40000-2000002-32-2-3000000    complex faithful

Permutation representations of C23.7D6
On 24 points - transitive group 24T96
Generators in S24
(2 13)(4 15)(6 17)(8 21)(10 23)(12 19)
(7 20)(8 21)(9 22)(10 23)(11 24)(12 19)
(1 18)(2 13)(3 14)(4 15)(5 16)(6 17)(7 20)(8 21)(9 22)(10 23)(11 24)(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 19)(2 11 13 24)(3 23)(4 9 15 22)(5 21)(6 7 17 20)(8 16)(10 14)(12 18)

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

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

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

G:=TransitiveGroup(24,96);

On 24 points - transitive group 24T98
Generators in S24
(1 20)(2 24)(3 22)(4 23)(5 21)(6 19)(7 14)(8 18)(9 16)(10 13)(11 17)(12 15)
(7 11)(8 12)(9 10)(13 16)(14 17)(15 18)
(1 4)(2 5)(3 6)(7 11)(8 12)(9 10)(13 16)(14 17)(15 18)(19 22)(20 23)(21 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 14 20 11)(2 16 24 10)(3 18 22 12)(4 17 23 7)(5 13 21 9)(6 15 19 8)

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

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

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

G:=TransitiveGroup(24,98);

On 24 points - transitive group 24T111
Generators in S24
(2 20)(4 22)(6 24)(7 16)(9 18)(11 14)
(1 12)(2 7)(3 8)(4 9)(5 10)(6 11)(13 23)(14 24)(15 19)(16 20)(17 21)(18 22)
(1 19)(2 20)(3 21)(4 22)(5 23)(6 24)(7 16)(8 17)(9 18)(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 12 14)(2 10 16 5)(3 4 8 18)(7 23 20 13)(9 21 22 17)(11 19 24 15)

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

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

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

G:=TransitiveGroup(24,111);

C23.7D6 is a maximal subgroup of
C23.D12  C23.2D12  C23.3D12  C23.4D12  C245Dic3  (C22×C12)⋊C4  C424Dic3  C425Dic3  C23⋊C45S3  S3×C23⋊C4  C246D6  C22⋊C4⋊D6  (C6×D4)⋊10C4  2+ 1+4.5S3  2+ 1+47S3  C232Dic9  C62.31D4  C62.38D4  (C2×C6).D20  C23.7D30  (C2×C60)⋊C4  C3⋊(C23⋊F5)
C23.7D6 is a maximal quotient of
C24.3Dic3  C24.12D6  (C2×C12)⋊C8  C245Dic3  (C6×D4)⋊C4  (C6×Q8)⋊C4  (C22×C12)⋊C4  C424Dic3  C42.Dic3  C425Dic3  C42.3Dic3  C232Dic9  C62.31D4  C62.38D4  (C2×C6).D20  C23.7D30  (C2×C60)⋊C4  C3⋊(C23⋊F5)

Matrix representation of C23.7D6 in GL4(𝔽7) generated by

2604
2641
0060
5210
,
0632
6042
0060
0001
,
6000
0600
0060
0006
,
3541
6446
5554
0002
,
2141
3300
1633
4436
G:=sub<GL(4,GF(7))| [2,2,0,5,6,6,0,2,0,4,6,1,4,1,0,0],[0,6,0,0,6,0,0,0,3,4,6,0,2,2,0,1],[6,0,0,0,0,6,0,0,0,0,6,0,0,0,0,6],[3,6,5,0,5,4,5,0,4,4,5,0,1,6,4,2],[2,3,1,4,1,3,6,4,4,0,3,3,1,0,3,6] >;

C23.7D6 in GAP, Magma, Sage, TeX

C_2^3._7D_6
% in TeX

G:=Group("C2^3.7D6");
// GroupNames label

G:=SmallGroup(96,41);
// by ID

G=gap.SmallGroup(96,41);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-3,24,121,188,579,2309]);
// Polycyclic

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

Export

Subgroup lattice of C23.7D6 in TeX
Character table of C23.7D6 in TeX

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