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G = C12.D4order 96 = 25·3

8th non-split extension by C12 of D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C12.8D4, C23.2Dic3, (C2×C4).3D6, (C6×D4).2C2, (C2×D4).2S3, C32(C4.D4), C4.Dic33C2, (C22×C6).2C4, C4.13(C3⋊D4), C6.14(C22⋊C4), (C2×C12).17C22, C22.2(C2×Dic3), C2.4(C6.D4), (C2×C6).28(C2×C4), SmallGroup(96,40)

Series: Derived Chief Lower central Upper central

C1C2×C6 — C12.D4
C1C3C6C12C2×C12C4.Dic3 — C12.D4
C3C6C2×C6 — C12.D4
C1C2C2×C4C2×D4

Generators and relations for C12.D4
 G = < a,b,c | a12=1, b4=a6, c2=a9, bab-1=a-1, cac-1=a5, cbc-1=a3b3 >

2C2
4C2
4C2
2C22
2C22
4C22
4C22
2C6
4C6
4C6
2D4
2D4
6C8
6C8
2C2×C6
2C2×C6
4C2×C6
4C2×C6
3M4(2)
3M4(2)
2C3×D4
2C3⋊C8
2C3⋊C8
2C3×D4
3C4.D4

Character table of C12.D4

 class 12A2B2C2D34A4B6A6B6C6D6E6F6G8A8B8C8D12A12B
 size 1124422222244441212121244
ρ1111111111111111111111    trivial
ρ2111111111111111-1-1-1-111    linear of order 2
ρ3111-1-1111111-1-1-1-1-11-1111    linear of order 2
ρ4111-1-1111111-1-1-1-11-11-111    linear of order 2
ρ51111-11-1-11111-11-1i-i-ii-1-1    linear of order 4
ρ61111-11-1-11111-11-1-iii-i-1-1    linear of order 4
ρ7111-111-1-1111-11-11-i-iii-1-1    linear of order 4
ρ8111-111-1-1111-11-11ii-i-i-1-1    linear of order 4
ρ9222-2-2-122-1-1-111110000-1-1    orthogonal lifted from D6
ρ1022222-122-1-1-1-1-1-1-10000-1-1    orthogonal lifted from S3
ρ1122-20022-2-2-22000000002-2    orthogonal lifted from D4
ρ1222-2002-22-2-2200000000-22    orthogonal lifted from D4
ρ13222-22-1-2-2-1-1-11-11-1000011    symplectic lifted from Dic3, Schur index 2
ρ142222-2-1-2-2-1-1-1-11-11000011    symplectic lifted from Dic3, Schur index 2
ρ1522-200-12-211-1--3-3-3--30000-11    complex lifted from C3⋊D4
ρ1622-200-12-211-1-3--3--3-30000-11    complex lifted from C3⋊D4
ρ1722-200-1-2211-1-3-3--3--300001-1    complex lifted from C3⋊D4
ρ1822-200-1-2211-1--3--3-3-300001-1    complex lifted from C3⋊D4
ρ194-400040000-40000000000    orthogonal lifted from C4.D4
ρ204-4000-200-2-32-320000000000    complex faithful
ρ214-4000-2002-3-2-320000000000    complex faithful

Permutation representations of C12.D4
On 24 points - transitive group 24T99
Generators in S24
(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 16 10 19 7 22 4 13)(2 15 11 18 8 21 5 24)(3 14 12 17 9 20 6 23)
(1 19 10 16 7 13 4 22)(2 24 11 21 8 18 5 15)(3 17 12 14 9 23 6 20)

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

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

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

G:=TransitiveGroup(24,99);

C12.D4 is a maximal subgroup of
C3⋊C2≀C4  (C2×D4).D6  C245Dic3  (C22×C12)⋊C4  S3×C4.D4  M4(2).19D6  C427D6  C428D6  C24.23D4  C24.44D4  M4(2).D6  M4(2).13D6  (C6×D4).16C4  2+ 1+46S3  2+ 1+4.4S3  C36.D4  C12.D12  (C6×D4).S3  C60.28D4  C60.8D4  C5⋊(C12.D4)
C12.D4 is a maximal quotient of
C24.3Dic3  C12.(C4⋊C4)  C42.7D6  C12.9D8  C12.5Q16  C36.D4  C12.D12  (C6×D4).S3  C60.28D4  C60.8D4  C5⋊(C12.D4)

Matrix representation of C12.D4 in GL4(𝔽7) generated by

1062
2101
5554
4350
,
6032
0512
4352
1135
,
2301
4241
4355
1135
G:=sub<GL(4,GF(7))| [1,2,5,4,0,1,5,3,6,0,5,5,2,1,4,0],[6,0,4,1,0,5,3,1,3,1,5,3,2,2,2,5],[2,4,4,1,3,2,3,1,0,4,5,3,1,1,5,5] >;

C12.D4 in GAP, Magma, Sage, TeX

C_{12}.D_4
% in TeX

G:=Group("C12.D4");
// GroupNames label

G:=SmallGroup(96,40);
// by ID

G=gap.SmallGroup(96,40);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-3,24,121,188,86,579,2309]);
// Polycyclic

G:=Group<a,b,c|a^12=1,b^4=a^6,c^2=a^9,b*a*b^-1=a^-1,c*a*c^-1=a^5,c*b*c^-1=a^3*b^3>;
// generators/relations

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Subgroup lattice of C12.D4 in TeX
Character table of C12.D4 in TeX

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