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G = C427D6order 192 = 26·3

5th semidirect product of C42 and D6 acting via D6/C3=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C427D6, D12.13D4, Dic6.13D4, (C2×Q8)⋊5D6, C4.51(S3×D4), (C2×D4).51D6, C4.4D44S3, C12.28(C2×D4), (C6×Q8)⋊2C22, (C4×C12)⋊13C22, C6.51C22≀C2, D46D6.4C2, C12.D45C2, C33(D4.9D4), C424S311C2, (C22×C6).22D4, Q8.11D62C2, (C6×D4).67C22, C4.Dic36C22, C2.19(C232D6), (C2×C12).379C23, C4○D12.19C22, C23.10(C3⋊D4), (C2×C6).510(C2×D4), (C3×C4.4D4)⋊4C2, C22.31(C2×C3⋊D4), (C2×C4).116(C22×S3), SmallGroup(192,620)

Series: Derived Chief Lower central Upper central

C1C2×C12 — C427D6
C1C3C6C12C2×C12C4○D12D46D6 — C427D6
C3C6C2×C12 — C427D6
C1C2C2×C4C4.4D4

Generators and relations for C427D6
 G = < a,b,c,d | a4=b4=c6=d2=1, ab=ba, cac-1=a-1b2, dad=a-1b, cbc-1=b-1, bd=db, dcd=c-1 >

Subgroups: 496 in 152 conjugacy classes, 39 normal (19 characteristic)
C1, C2, C2 [×5], C3, C4 [×2], C4 [×4], C22, C22 [×8], S3 [×2], C6, C6 [×3], C8 [×2], C2×C4, C2×C4 [×6], D4 [×10], Q8 [×4], C23 [×2], C23 [×2], Dic3 [×2], C12 [×2], C12 [×2], D6 [×4], C2×C6, C2×C6 [×4], C42, C22⋊C4 [×2], M4(2) [×2], SD16 [×2], Q16 [×2], C2×D4, C2×D4 [×4], C2×Q8, C4○D4 [×4], C3⋊C8 [×2], Dic6 [×2], C4×S3 [×2], D12 [×2], C2×Dic3 [×2], C3⋊D4 [×6], C2×C12, C2×C12 [×2], C3×D4 [×2], C3×Q8 [×2], C22×S3 [×2], C22×C6 [×2], C4.D4, C4≀C2 [×2], C4.4D4, C8.C22 [×2], 2+ 1+4, C4.Dic3 [×2], Q82S3 [×2], C3⋊Q16 [×2], C4×C12, C3×C22⋊C4 [×2], C4○D12 [×2], S3×D4 [×2], D42S3 [×2], C2×C3⋊D4 [×2], C6×D4, C6×Q8, D4.9D4, C424S3 [×2], C12.D4, Q8.11D6 [×2], C3×C4.4D4, D46D6, C427D6
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, D4.9D4, C232D6, C427D6

Character table of C427D6

 class 12A2B2C2D2E2F34A4B4C4D4E4F4G6A6B6C6D6E8A8B12A12B12C12D12E12F12G12H
 size 112441212222448121222288242444444488
ρ1111111111111111111111111111111    trivial
ρ21111111111-1-1-11111111-1-1-1-111-1-1-1-1    linear of order 2
ρ3111-1-1-11111-1-11-11111-1-1-11-1-111-1-111    linear of order 2
ρ4111-1-1-1111111-1-11111-1-11-1111111-1-1    linear of order 2
ρ511111-1-1111-1-1-1-1-11111111-1-111-1-1-1-1    linear of order 2
ρ611111-1-1111111-1-111111-1-111111111    linear of order 2
ρ7111-1-11-111111-11-1111-1-1-11111111-1-1    linear of order 2
ρ8111-1-11-1111-1-111-1111-1-11-1-1-111-1-111    linear of order 2
ρ9222-2-200-12222-200-1-1-11100-1-1-1-1-1-111    orthogonal lifted from D6
ρ10222-2-200-122-2-2200-1-1-1110011-1-111-1-1    orthogonal lifted from D6
ρ1122-2002022-2000-202-2-2000000-220000    orthogonal lifted from D4
ρ1222-2000-22-22000022-2-20000002-20000    orthogonal lifted from D4
ρ132222-2002-2-2000002222-20000-2-20000    orthogonal lifted from D4
ρ1422-200-2022-2000202-2-2000000-220000    orthogonal lifted from D4
ρ152222200-12222200-1-1-1-1-100-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ1622-200022-220000-22-2-20000002-20000    orthogonal lifted from D4
ρ172222200-122-2-2-200-1-1-1-1-10011-1-11111    orthogonal lifted from D6
ρ18222-22002-2-200000222-220000-2-20000    orthogonal lifted from D4
ρ192222-200-1-2-200000-1-1-1-1100-3-311--3--3--3-3    complex lifted from C3⋊D4
ρ20222-2200-1-2-200000-1-1-11-100--3--311-3-3--3-3    complex lifted from C3⋊D4
ρ21222-2200-1-2-200000-1-1-11-100-3-311--3--3-3--3    complex lifted from C3⋊D4
ρ222222-200-1-2-200000-1-1-1-1100--3--311-3-3-3--3    complex lifted from C3⋊D4
ρ2344-40000-2-4400000-222000000-220000    orthogonal lifted from S3×D4
ρ2444-40000-24-400000-2220000002-20000    orthogonal lifted from S3×D4
ρ254-400000400-2i2i000-40000002i-2i002i-2i00    complex lifted from D4.9D4
ρ264-4000004002i-2i000-4000000-2i2i00-2i2i00    complex lifted from D4.9D4
ρ274-400000-200-2i2i0002-2-32-300004ζ3243ζ32004ζ343ζ300    complex faithful
ρ284-400000-2002i-2i00022-3-2-3000043ζ34ζ30043ζ324ζ3200    complex faithful
ρ294-400000-2002i-2i0002-2-32-3000043ζ324ζ320043ζ34ζ300    complex faithful
ρ304-400000-200-2i2i00022-3-2-300004ζ343ζ3004ζ3243ζ3200    complex faithful

Smallest permutation representation of C427D6
On 48 points
Generators in S48
(1 24 13 32)(2 47 14 38)(3 20 15 34)(4 43 16 40)(5 22 17 36)(6 45 18 42)(7 23 28 31)(8 46 29 37)(9 19 30 33)(10 48 25 39)(11 21 26 35)(12 44 27 41)
(1 16 29 11)(2 12 30 17)(3 18 25 7)(4 8 26 13)(5 14 27 9)(6 10 28 15)(19 22 38 41)(20 42 39 23)(21 24 40 37)(31 34 45 48)(32 43 46 35)(33 36 47 44)
(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 6)(2 5)(3 4)(7 13)(8 18)(9 17)(10 16)(11 15)(12 14)(19 38)(20 37)(21 42)(22 41)(23 40)(24 39)(25 26)(27 30)(28 29)(31 35)(32 34)(43 45)(46 48)

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

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

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

Matrix representation of C427D6 in GL4(𝔽73) generated by

17666366
710756
1071766
6617710
,
0010
0001
72000
07200
,
001330
004343
133000
434300
,
004360
003030
301300
434300
G:=sub<GL(4,GF(73))| [17,7,10,66,66,10,7,17,63,7,17,7,66,56,66,10],[0,0,72,0,0,0,0,72,1,0,0,0,0,1,0,0],[0,0,13,43,0,0,30,43,13,43,0,0,30,43,0,0],[0,0,30,43,0,0,13,43,43,30,0,0,60,30,0,0] >;

C427D6 in GAP, Magma, Sage, TeX

C_4^2\rtimes_7D_6
% in TeX

G:=Group("C4^2:7D6");
// GroupNames label

G:=SmallGroup(192,620);
// by ID

G=gap.SmallGroup(192,620);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,254,219,184,1123,570,297,136,1684,6278]);
// Polycyclic

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

Export

Character table of C427D6 in TeX

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