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G = D126C22order 96 = 25·3

4th semidirect product of D12 and C22 acting via C22/C2=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D4.6D6, C12.15D4, D126C22, C12.12C23, Dic65C22, D4⋊S35C2, (C2×D4)⋊2S3, (C6×D4)⋊2C2, C3⋊C83C22, C4○D123C2, C34(C8⋊C22), D4.S35C2, (C2×C4).17D6, (C2×C6).39D4, C6.45(C2×D4), C4.Dic36C2, C4.16(C3⋊D4), C4.12(C22×S3), (C3×D4).6C22, (C2×C12).30C22, C22.10(C3⋊D4), C2.9(C2×C3⋊D4), SmallGroup(96,139)

Series: Derived Chief Lower central Upper central

C1C12 — D126C22
C1C3C6C12D12C4○D12 — D126C22
C3C6C12 — D126C22
C1C2C2×C4C2×D4

Generators and relations for D126C22
 G = < a,b,c,d | a12=b2=c2=d2=1, bab=a-1, ac=ca, dad=a7, cbc=a6b, dbd=a3b, cd=dc >

Subgroups: 162 in 68 conjugacy classes, 29 normal (19 characteristic)
C1, C2, C2 [×4], C3, C4 [×2], C4, C22, C22 [×5], S3, C6, C6 [×3], C8 [×2], C2×C4, C2×C4, D4 [×2], D4 [×3], Q8, C23, Dic3, C12 [×2], D6, C2×C6, C2×C6 [×4], M4(2), D8 [×2], SD16 [×2], C2×D4, C4○D4, C3⋊C8 [×2], Dic6, C4×S3, D12, C3⋊D4, C2×C12, C3×D4 [×2], C3×D4, C22×C6, C8⋊C22, C4.Dic3, D4⋊S3 [×2], D4.S3 [×2], C4○D12, C6×D4, D126C22
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D6 [×3], C2×D4, C3⋊D4 [×2], C22×S3, C8⋊C22, C2×C3⋊D4, D126C22

Character table of D126C22

 class 12A2B2C2D2E34A4B4C6A6B6C6D6E6F6G8A8B12A12B
 size 1124412222122224444121244
ρ1111111111111111111111    trivial
ρ211-1-11-11-111-1-11-1-1111-11-1    linear of order 2
ρ3111-1-1-1111-1111-1-1-1-11111    linear of order 2
ρ411111-1111-11111111-1-111    linear of order 2
ρ511-11-1-11-111-1-1111-1-1-111-1    linear of order 2
ρ611-11-111-11-1-1-1111-1-11-11-1    linear of order 2
ρ711-1-1111-11-1-1-11-1-111-111-1    linear of order 2
ρ8111-1-111111111-1-1-1-1-1-111    linear of order 2
ρ9222220-1220-1-1-1-1-1-1-100-1-1    orthogonal lifted from S3
ρ10222-2-20-1220-1-1-1111100-1-1    orthogonal lifted from D6
ρ1122-22-20-1-22011-1-1-11100-11    orthogonal lifted from D6
ρ1222-200022-20-2-22000000-22    orthogonal lifted from D4
ρ132220002-2-20222000000-2-2    orthogonal lifted from D4
ρ1422-2-220-1-22011-111-1-100-11    orthogonal lifted from D6
ρ15222000-1-2-20-1-1-1--3-3--3-30011    complex lifted from C3⋊D4
ρ16222000-1-2-20-1-1-1-3--3-3--30011    complex lifted from C3⋊D4
ρ1722-2000-12-2011-1-3--3--3-3001-1    complex lifted from C3⋊D4
ρ1822-2000-12-2011-1--3-3-3--3001-1    complex lifted from C3⋊D4
ρ194-40000400000-400000000    orthogonal lifted from C8⋊C22
ρ204-40000-2000-2-32-3200000000    complex faithful
ρ214-40000-20002-3-2-3200000000    complex faithful

Permutation representations of D126C22
On 24 points - transitive group 24T118
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 14)(2 13)(3 24)(4 23)(5 22)(6 21)(7 20)(8 19)(9 18)(10 17)(11 16)(12 15)
(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)
(2 8)(4 10)(6 12)(13 16)(14 23)(15 18)(17 20)(19 22)(21 24)

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

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

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

G:=TransitiveGroup(24,118);

D126C22 is a maximal subgroup of
D12.2D4  D12.3D4  D12.14D4  C428D6  C24.23D4  C24.44D4  D1218D4  D12.38D4  D813D6  SD1613D6  S3×C8⋊C22  D84D6  C12.C24  D12.32C23  D12.33C23  D366C22  D1220D6  D12.28D6  D129D6  D12.7D6  C62.131D4  C60.36D4  D6030C22  D1210D10  D20.9D6  D4.D30
D126C22 is a maximal quotient of
C4⋊C4.225D6  C4○D12⋊C4  C4⋊C4.228D6  C4⋊C4.230D6  D4.3Dic6  C42.48D6  D4.1D12  C42.51D6  C6.Q16⋊C2  D1217D4  C4⋊D4⋊S3  C3⋊C85D4  C42.72D6  C122D8  C42.74D6  Dic69D4  C42.76D6  D125Q8  C42.82D6  Dic65Q8  (C6×D4)⋊6C4  (C2×C6)⋊8D8  (C3×D4).31D4  D366C22  D1220D6  D12.28D6  D129D6  D12.7D6  C62.131D4  C60.36D4  D6030C22  D1210D10  D20.9D6  D4.D30

Matrix representation of D126C22 in GL4(𝔽7) generated by

6463
2026
1115
1630
,
6000
0532
4355
1165
,
0145
1035
0010
0006
,
0660
6060
0010
0006
G:=sub<GL(4,GF(7))| [6,2,1,1,4,0,1,6,6,2,1,3,3,6,5,0],[6,0,4,1,0,5,3,1,0,3,5,6,0,2,5,5],[0,1,0,0,1,0,0,0,4,3,1,0,5,5,0,6],[0,6,0,0,6,0,0,0,6,6,1,0,0,0,0,6] >;

D126C22 in GAP, Magma, Sage, TeX

D_{12}\rtimes_6C_2^2
% in TeX

G:=Group("D12:6C2^2");
// GroupNames label

G:=SmallGroup(96,139);
// by ID

G=gap.SmallGroup(96,139);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-3,103,218,188,579,159,69,2309]);
// Polycyclic

G:=Group<a,b,c,d|a^12=b^2=c^2=d^2=1,b*a*b=a^-1,a*c=c*a,d*a*d=a^7,c*b*c=a^6*b,d*b*d=a^3*b,c*d=d*c>;
// generators/relations

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Character table of D126C22 in TeX

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