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G = D121D4order 192 = 26·3

1st semidirect product of D12 and D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D121D4, Dic61D4, M4(2)⋊2D6, C23.7D12, C8⋊D65C2, C4.79(S3×D4), D46D62C2, C123D41C2, C31(D44D4), (C2×D4).14D6, C4.D42S3, C12.92(C2×D4), D12⋊C42C2, C6.16C22≀C2, (C2×C12).4C23, (C22×C6).20D4, (C2×D12)⋊11C22, C4○D12.2C22, (C4×Dic3)⋊2C22, (C6×D4).14C22, C22.11(C2×D12), C2.19(D6⋊D4), (C3×M4(2))⋊9C22, (C2×C6).21(C2×D4), (C3×C4.D4)⋊4C2, (C2×C4).4(C22×S3), SmallGroup(192,306)

Series: Derived Chief Lower central Upper central

C1C2×C12 — D121D4
C1C3C6C12C2×C12C4○D12D46D6 — D121D4
C3C6C2×C12 — D121D4
C1C2C2×C4C4.D4

Generators and relations for D121D4
 G = < a,b,c,d | a12=b2=c4=d2=1, bab=dad=a-1, cac-1=a5, cbc-1=ab, dbd=a7b, dcd=c-1 >

Subgroups: 656 in 168 conjugacy classes, 39 normal (17 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 [×16], Q8 [×2], C23 [×2], C23 [×3], Dic3 [×4], C12 [×2], D6 [×7], C2×C6, C2×C6 [×4], C42, M4(2) [×2], D8 [×2], SD16 [×2], C2×D4, C2×D4 [×7], C4○D4 [×4], C24 [×2], Dic6 [×2], C4×S3 [×2], D12 [×2], D12 [×2], C2×Dic3 [×3], C3⋊D4 [×10], C2×C12, C3×D4 [×2], C22×S3 [×3], C22×C6 [×2], C4.D4, C4≀C2 [×2], C41D4, C8⋊C22 [×2], 2+ 1+4, C24⋊C2 [×2], D24 [×2], C4×Dic3, C3×M4(2) [×2], C2×D12, C4○D12 [×2], S3×D4 [×2], D42S3 [×2], C2×C3⋊D4 [×4], C6×D4, D44D4, D12⋊C4 [×2], C3×C4.D4, C8⋊D6 [×2], C123D4, D46D6, D121D4
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×6], C23, D6 [×3], C2×D4 [×3], D12 [×2], C22×S3, C22≀C2, C2×D12, S3×D4 [×2], D44D4, D6⋊D4, D121D4

Character table of D121D4

 class 12A2B2C2D2E2F2G34A4B4C4D4E4F6A6B6C6D8A8B12A12B24A24B24C24D
 size 1124412122422212121212248888448888
ρ1111111111111111111111111111    trivial
ρ21111111-1111-111-11111-1-111-1-1-1-1    linear of order 2
ρ3111-1-1-11-111111-1111-1-11-111-11-11    linear of order 2
ρ4111-1-1-111111-11-1-111-1-1-11111-11-1    linear of order 2
ρ511111-1-111111-1-111111-1-111-1-1-1-1    linear of order 2
ρ611111-1-1-1111-1-1-1-1111111111111    linear of order 2
ρ7111-1-11-1-11111-11111-1-1-11111-11-1    linear of order 2
ρ8111-1-11-11111-1-11-111-1-11-111-11-11    linear of order 2
ρ922-2002002-2200-202-200002-20000    orthogonal lifted from D4
ρ10222-2-2000-1220000-1-1112-2-1-11-11-1    orthogonal lifted from D6
ρ1122222000-1220000-1-1-1-122-1-1-1-1-1-1    orthogonal lifted from S3
ρ12222-2-2000-1220000-1-111-22-1-1-11-11    orthogonal lifted from D6
ρ13222-220002-2-20000222-200-2-20000    orthogonal lifted from D4
ρ1422222000-1220000-1-1-1-1-2-2-1-11111    orthogonal lifted from D6
ρ152222-20002-2-2000022-2200-2-20000    orthogonal lifted from D4
ρ1622-200-2002-2200202-200002-20000    orthogonal lifted from D4
ρ1722-20002022-20-2002-20000-220000    orthogonal lifted from D4
ρ1822-2000-2022-202002-20000-220000    orthogonal lifted from D4
ρ192222-2000-1-2-20000-1-11-100113-3-33    orthogonal lifted from D12
ρ20222-22000-1-2-20000-1-1-11001133-3-3    orthogonal lifted from D12
ρ21222-22000-1-2-20000-1-1-110011-3-333    orthogonal lifted from D12
ρ222222-2000-1-2-20000-1-11-10011-333-3    orthogonal lifted from D12
ρ234-4000000400-2002-400000000000    orthogonal lifted from D44D4
ρ2444-400000-24-40000-2200002-20000    orthogonal lifted from S3×D4
ρ2544-400000-2-440000-220000-220000    orthogonal lifted from S3×D4
ρ264-4000000400200-2-400000000000    orthogonal lifted from D44D4
ρ278-8000000-4000000400000000000    orthogonal faithful

Permutation representations of D121D4
On 24 points - transitive group 24T343
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 22)(14 21)(15 20)(16 19)(17 18)(23 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), (1,21)(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)(13,22)(14,21)(15,20)(16,19)(17,18)(23,24)>;

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), (1,21)(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)(13,22)(14,21)(15,20)(16,19)(17,18)(23,24) );

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)], [(1,21),(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),(13,22),(14,21),(15,20),(16,19),(17,18),(23,24)])

G:=TransitiveGroup(24,343);

Matrix representation of D121D4 in GL6(𝔽73)

110000
7200000
00722500
0070100
00111701
00620720
,
6670000
1470000
007242710
00052703
000491163
000246311
,
7200000
110000
0014800
0037200
006249720
001049072
,
7200000
110000
0072000
0070100
00116601
0063710

G:=sub<GL(6,GF(73))| [1,72,0,0,0,0,1,0,0,0,0,0,0,0,72,70,11,62,0,0,25,1,17,0,0,0,0,0,0,72,0,0,0,0,1,0],[66,14,0,0,0,0,7,7,0,0,0,0,0,0,72,0,0,0,0,0,42,52,49,24,0,0,71,70,11,63,0,0,0,3,63,11],[72,1,0,0,0,0,0,1,0,0,0,0,0,0,1,3,62,10,0,0,48,72,49,49,0,0,0,0,72,0,0,0,0,0,0,72],[72,1,0,0,0,0,0,1,0,0,0,0,0,0,72,70,11,63,0,0,0,1,66,7,0,0,0,0,0,1,0,0,0,0,1,0] >;

D121D4 in GAP, Magma, Sage, TeX

D_{12}\rtimes_1D_4
% in TeX

G:=Group("D12:1D4");
// GroupNames label

G:=SmallGroup(192,306);
// by ID

G=gap.SmallGroup(192,306);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,254,219,226,1123,570,136,1684,438,6278]);
// 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^7*b,d*c*d=c^-1>;
// generators/relations

Export

Character table of D121D4 in TeX

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