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G = C123D4order 96 = 25·3

3rd semidirect product of C12 and D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C123D4, Dic31D4, C23.15D6, (C2×D4)⋊6S3, (C6×D4)⋊4C2, (C2×D12)⋊9C2, C32(C41D4), C41(C3⋊D4), (C2×C4).52D6, C6.52(C2×D4), C2.28(S3×D4), (C4×Dic3)⋊6C2, (C2×C6).55C23, (C2×C12).35C22, C22.62(C22×S3), (C22×C6).22C22, (C22×S3).12C22, (C2×Dic3).39C22, (C2×C3⋊D4)⋊7C2, C2.16(C2×C3⋊D4), SmallGroup(96,147)

Series: Derived Chief Lower central Upper central

C1C2×C6 — C123D4
C1C3C6C2×C6C22×S3C2×D12 — C123D4
C3C2×C6 — C123D4
C1C22C2×D4

Generators and relations for C123D4
 G = < a,b,c | a12=b4=c2=1, bab-1=a5, cac=a-1, cbc=b-1 >

Subgroups: 290 in 108 conjugacy classes, 37 normal (13 characteristic)
C1, C2, C2 [×2], C2 [×4], C3, C4 [×2], C4 [×4], C22, C22 [×12], S3 [×2], C6, C6 [×2], C6 [×2], C2×C4, C2×C4 [×2], D4 [×12], C23 [×2], C23 [×2], Dic3 [×4], C12 [×2], D6 [×6], C2×C6, C2×C6 [×6], C42, C2×D4, C2×D4 [×5], D12 [×2], C2×Dic3 [×2], C3⋊D4 [×8], C2×C12, C3×D4 [×2], C22×S3 [×2], C22×C6 [×2], C41D4, C4×Dic3, C2×D12, C2×C3⋊D4 [×4], C6×D4, C123D4
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×6], C23, D6 [×3], C2×D4 [×3], C3⋊D4 [×2], C22×S3, C41D4, S3×D4 [×2], C2×C3⋊D4, C123D4

Character table of C123D4

 class 12A2B2C2D2E2F2G34A4B4C4D4E4F6A6B6C6D6E6F6G12A12B
 size 11114412122226666222444444
ρ1111111111111111111111111    trivial
ρ211111-1-111-1-11-11-11111-1-11-1-1    linear of order 2
ρ311111-11-11-1-1-11-111111-1-11-1-1    linear of order 2
ρ4111111-1-1111-1-1-1-1111111111    linear of order 2
ρ51111-11-111-1-1-11-11111-111-1-1-1    linear of order 2
ρ61111-1-111111-1-1-1-1111-1-1-1-111    linear of order 2
ρ71111-1-1-1-11111111111-1-1-1-111    linear of order 2
ρ81111-111-11-1-11-11-1111-111-1-1-1    linear of order 2
ρ92-22-20000200020-2-22-2000000    orthogonal lifted from D4
ρ102-2-220000200-20202-2-2000000    orthogonal lifted from D4
ρ112-22-200002000-202-22-2000000    orthogonal lifted from D4
ρ1222-2-2000022-20000-2-220000-22    orthogonal lifted from D4
ρ1322222-200-1-2-20000-1-1-1-111-111    orthogonal lifted from D6
ρ1422-2-200002-220000-2-2200002-2    orthogonal lifted from D4
ρ1522222200-1220000-1-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ162-2-22000020020-202-2-2000000    orthogonal lifted from D4
ρ172222-2-200-1220000-1-1-11111-1-1    orthogonal lifted from D6
ρ182222-2200-1-2-20000-1-1-11-1-1111    orthogonal lifted from D6
ρ1922-2-20000-12-2000011-1--3--3-3-31-1    complex lifted from C3⋊D4
ρ2022-2-20000-12-2000011-1-3-3--3--31-1    complex lifted from C3⋊D4
ρ2122-2-20000-1-22000011-1-3--3-3--3-11    complex lifted from C3⋊D4
ρ2222-2-20000-1-22000011-1--3-3--3-3-11    complex lifted from C3⋊D4
ρ234-4-440000-2000000-222000000    orthogonal lifted from S3×D4
ρ244-44-40000-20000002-22000000    orthogonal lifted from S3×D4

Smallest permutation representation of C123D4
On 48 points
Generators in S48
(1 2 3 4 5 6 7 8 9 10 11 12)(13 14 15 16 17 18 19 20 21 22 23 24)(25 26 27 28 29 30 31 32 33 34 35 36)(37 38 39 40 41 42 43 44 45 46 47 48)
(1 48 29 21)(2 41 30 14)(3 46 31 19)(4 39 32 24)(5 44 33 17)(6 37 34 22)(7 42 35 15)(8 47 36 20)(9 40 25 13)(10 45 26 18)(11 38 27 23)(12 43 28 16)
(2 12)(3 11)(4 10)(5 9)(6 8)(13 44)(14 43)(15 42)(16 41)(17 40)(18 39)(19 38)(20 37)(21 48)(22 47)(23 46)(24 45)(25 33)(26 32)(27 31)(28 30)(34 36)

G:=sub<Sym(48)| (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45,46,47,48), (1,48,29,21)(2,41,30,14)(3,46,31,19)(4,39,32,24)(5,44,33,17)(6,37,34,22)(7,42,35,15)(8,47,36,20)(9,40,25,13)(10,45,26,18)(11,38,27,23)(12,43,28,16), (2,12)(3,11)(4,10)(5,9)(6,8)(13,44)(14,43)(15,42)(16,41)(17,40)(18,39)(19,38)(20,37)(21,48)(22,47)(23,46)(24,45)(25,33)(26,32)(27,31)(28,30)(34,36)>;

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)(25,26,27,28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45,46,47,48), (1,48,29,21)(2,41,30,14)(3,46,31,19)(4,39,32,24)(5,44,33,17)(6,37,34,22)(7,42,35,15)(8,47,36,20)(9,40,25,13)(10,45,26,18)(11,38,27,23)(12,43,28,16), (2,12)(3,11)(4,10)(5,9)(6,8)(13,44)(14,43)(15,42)(16,41)(17,40)(18,39)(19,38)(20,37)(21,48)(22,47)(23,46)(24,45)(25,33)(26,32)(27,31)(28,30)(34,36) );

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),(25,26,27,28,29,30,31,32,33,34,35,36),(37,38,39,40,41,42,43,44,45,46,47,48)], [(1,48,29,21),(2,41,30,14),(3,46,31,19),(4,39,32,24),(5,44,33,17),(6,37,34,22),(7,42,35,15),(8,47,36,20),(9,40,25,13),(10,45,26,18),(11,38,27,23),(12,43,28,16)], [(2,12),(3,11),(4,10),(5,9),(6,8),(13,44),(14,43),(15,42),(16,41),(17,40),(18,39),(19,38),(20,37),(21,48),(22,47),(23,46),(24,45),(25,33),(26,32),(27,31),(28,30),(34,36)])

C123D4 is a maximal subgroup of
C23.2D12  D121D4  Dic3.SD16  Dic62D4  C4⋊C4.D6  D123D4  Dic3⋊D8  C245D4  C2411D4  Dic35SD16  C2415D4  C249D4  D1218D4  2+ 1+46S3  C42.228D6  Dic624D4  C42.114D6  C42.116D6  C248D6  C24.45D6  C24.47D6  C12⋊(C4○D4)  Dic620D4  C6.382+ 1+4  D1219D4  C6.442+ 1+4  C6.472+ 1+4  C6.482+ 1+4  C6.662+ 1+4  C6.672+ 1+4  C6.682+ 1+4  C42.233D6  C4220D6  C42.143D6  S3×C41D4  C4228D6  Dic611D4  D4×C3⋊D4  C24.52D6  C6.1462+ 1+4  (C2×C12)⋊17D4  C6.1482+ 1+4  C36⋊D4  C12⋊D12  C62.84C23  C62.121C23  C62.258C23  C12⋊D20  C6010D4  Dic155D4  C603D4
C123D4 is a maximal quotient of
C24.18D6  C24.24D6  C233D12  (C4×Dic3)⋊8C4  (C2×D12)⋊10C4  (C2×C12).290D4  C42.64D6  C42.214D6  C42.65D6  C12⋊D8  C42.74D6  C124SD16  C126SD16  C42.80D6  C123Q16  C245D4  C2411D4  C24.22D4  C24.31D4  C24.43D4  C2415D4  C249D4  C24.26D4  C24.37D4  C24.28D4  C24.30D6  C24.32D6  C36⋊D4  C12⋊D12  C62.84C23  C62.121C23  C62.258C23  C12⋊D20  C6010D4  Dic155D4  C603D4

Matrix representation of C123D4 in GL4(𝔽13) generated by

01200
1000
00121
00120
,
01200
1000
00411
0029
,
1000
01200
0001
0010
G:=sub<GL(4,GF(13))| [0,1,0,0,12,0,0,0,0,0,12,12,0,0,1,0],[0,1,0,0,12,0,0,0,0,0,4,2,0,0,11,9],[1,0,0,0,0,12,0,0,0,0,0,1,0,0,1,0] >;

C123D4 in GAP, Magma, Sage, TeX

C_{12}\rtimes_3D_4
% in TeX

G:=Group("C12:3D4");
// GroupNames label

G:=SmallGroup(96,147);
// by ID

G=gap.SmallGroup(96,147);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-3,217,103,218,188,2309]);
// Polycyclic

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

Export

Character table of C123D4 in TeX

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