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G = D1218D4order 192 = 26·3

6th semidirect product of D12 and D4 acting via D4/C22=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D1218D4, Dic618D4, M4(2)⋊5D6, (C3×D4)⋊7D4, (C2×D4)⋊4D6, (C3×Q8)⋊7D4, D4○D122C2, C8⋊C221S3, C123D47C2, D44(C3⋊D4), C34(D44D4), C4○D4.21D6, Q85(C3⋊D4), C4.104(S3×D4), (C6×D4)⋊4C22, D12⋊C45C2, C6.63C22≀C2, D126C225C2, C12.194(C2×D4), (C22×S3).5D4, C22.35(S3×D4), Q83Dic36C2, C12.46D45C2, (C2×C12).13C23, (C4×Dic3)⋊5C22, C4.Dic38C22, C2.31(C232D6), C4○D12.23C22, (C2×D12).128C22, (C3×M4(2))⋊12C22, (C3×C8⋊C22)⋊5C2, (C2×C6).34(C2×D4), C4.50(C2×C3⋊D4), (C2×C4).13(C22×S3), (C3×C4○D4).11C22, SmallGroup(192,757)

Series: Derived Chief Lower central Upper central

C1C2×C12 — D1218D4
C1C3C6C2×C6C2×C12C2×D12D4○D12 — D1218D4
C3C6C2×C12 — D1218D4
C1C2C2×C4C8⋊C22

Generators and relations for D1218D4
 G = < a,b,c,d | a12=b2=c4=d2=1, bab=dad=a-1, cac-1=a5, cbc-1=ab, dbd=a10b, dcd=c-1 >

Subgroups: 624 in 168 conjugacy classes, 39 normal (37 characteristic)
C1, C2, C2 [×6], C3, C4 [×2], C4 [×4], C22, C22 [×11], S3 [×3], C6, C6 [×3], C8 [×2], C2×C4, C2×C4 [×5], D4, D4 [×15], Q8, Q8, C23 [×5], Dic3 [×3], C12 [×2], C12, D6 [×7], C2×C6, C2×C6 [×4], C42, M4(2), M4(2), D8 [×2], SD16 [×2], C2×D4, C2×D4 [×7], C4○D4, C4○D4 [×3], C3⋊C8, C24, Dic6, C4×S3 [×3], D12, D12 [×4], C2×Dic3, C3⋊D4 [×7], C2×C12, C2×C12, C3×D4, C3×D4 [×3], C3×Q8, C22×S3 [×2], C22×S3 [×2], C22×C6, C4.D4, C4≀C2 [×2], C41D4, C8⋊C22, C8⋊C22, 2+ 1+4, C4.Dic3, C4×Dic3, D4⋊S3, D4.S3, C3×M4(2), C3×D8, C3×SD16, C2×D12, C2×D12, C4○D12, C4○D12, S3×D4 [×3], Q83S3, C2×C3⋊D4 [×2], C6×D4, C3×C4○D4, D44D4, C12.46D4, D12⋊C4, Q83Dic3, D126C22, C123D4, C3×C8⋊C22, D4○D12, D1218D4
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×6], C23, D6 [×3], C2×D4 [×3], C3⋊D4 [×2], C22×S3, C22≀C2, S3×D4 [×2], C2×C3⋊D4, D44D4, C232D6, D1218D4

Character table of D1218D4

 class 12A2B2C2D2E2F2G34A4B4C4D4E4F6A6B6C6D6E8A8B12A12B12C24A24B
 size 1124812121222241212122488882444888
ρ1111111111111111111111111111    trivial
ρ2111-1-1-111111-1-1-1-111-1-1-11111-111    linear of order 2
ρ3111-111-1-1111-11-1-111-111-1111-1-1-1    linear of order 2
ρ41111-1-1-1-11111-111111-1-1-11111-1-1    linear of order 2
ρ5111-11-111111-1-11111-111-1-111-1-1-1    linear of order 2
ρ61111-111111111-1-1111-1-1-1-1111-1-1    linear of order 2
ρ711111-1-1-11111-1-1-1111111-111111    linear of order 2
ρ8111-1-11-1-1111-111111-1-1-11-111-111    linear of order 2
ρ922222000-1222000-1-1-1-1-120-1-1-1-1-1    orthogonal lifted from S3
ρ1022-2-200002-2220002-2-200002-2200    orthogonal lifted from D4
ρ1122-200-20022-202002-200000-22000    orthogonal lifted from D4
ρ12222-2-2000-122-2000-1-111120-1-11-1-1    orthogonal lifted from D6
ρ132220002-22-2-200002200000-2-2000    orthogonal lifted from D4
ρ142222-2000-1222000-1-1-111-20-1-1-111    orthogonal lifted from D6
ρ1522-2200002-22-20002-2200002-2-200    orthogonal lifted from D4
ρ16222-22000-122-2000-1-11-1-1-20-1-1111    orthogonal lifted from D6
ρ17222000-222-2-200002200000-2-2000    orthogonal lifted from D4
ρ1822-20020022-20-2002-200000-22000    orthogonal lifted from D4
ρ1922-2-20000-1-222000-111--3-300-11-1-3--3    complex lifted from C3⋊D4
ρ2022-220000-1-22-2000-11-1--3-300-111--3-3    complex lifted from C3⋊D4
ρ2122-2-20000-1-222000-111-3--300-11-1--3-3    complex lifted from C3⋊D4
ρ2222-220000-1-22-2000-11-1-3--300-111-3--3    complex lifted from C3⋊D4
ρ234-400000040000-22-400000000000    orthogonal lifted from D44D4
ρ2444400000-2-4-40000-2-20000022000    orthogonal lifted from S3×D4
ρ2544-400000-24-40000-22000002-2000    orthogonal lifted from S3×D4
ρ264-4000000400002-2-400000000000    orthogonal lifted from D44D4
ρ278-8000000-4000000400000000000    orthogonal faithful

Permutation representations of D1218D4
On 24 points - transitive group 24T366
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 21)(2 20)(3 19)(4 18)(5 17)(6 16)(7 15)(8 14)(9 13)(10 24)(11 23)(12 22)
(2 6)(3 11)(5 9)(8 12)(13 22 19 16)(14 15 20 21)(17 18 23 24)
(2 12)(3 11)(4 10)(5 9)(6 8)(13 19)(14 18)(15 17)(20 24)(21 23)

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

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

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

G:=TransitiveGroup(24,366);

Matrix representation of D1218D4 in GL8(ℤ)

00010000
00-110000
0-1000000
1-1000000
00000001
000000-11
00000-100
00001-100
,
0000-1100
00000100
0000001-1
0000000-1
-11000000
01000000
001-10000
000-10000
,
000-10000
00-100000
01000000
10000000
00000100
00001000
00000001
00000010
,
000-10000
00-100000
0-1000000
-10000000
0000000-1
000000-10
00000-100
0000-1000

G:=sub<GL(8,Integers())| [0,0,0,1,0,0,0,0,0,0,-1,-1,0,0,0,0,0,-1,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,-1,0,0,0,0,0,-1,0,0,0,0,0,0,1,1,0,0],[0,0,0,0,-1,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,-1,-1,-1,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,-1,-1,0,0,0,0],[0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0] >;

D1218D4 in GAP, Magma, Sage, TeX

D_{12}\rtimes_{18}D_4
% in TeX

G:=Group("D12:18D4");
// GroupNames label

G:=SmallGroup(192,757);
// by ID

G=gap.SmallGroup(192,757);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,254,219,570,1684,851,438,102,6278]);
// Polycyclic

G:=Group<a,b,c,d|a^12=b^2=c^4=d^2=1,b*a*b=d*a*d=a^-1,c*a*c^-1=a^5,c*b*c^-1=a*b,d*b*d=a^10*b,d*c*d=c^-1>;
// generators/relations

Export

Character table of D1218D4 in TeX

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