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G = C42.62D6order 192 = 26·3

62nd non-split extension by C42 of D6 acting via D6/C3=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C42.62D6, (C2×D4).45D6, (C2×C12).81D4, (C2×Q8).59D6, C122Q817C2, C4.4D4.5S3, C12.66(C4○D4), Q82Dic320C2, C42.S38C2, (C6×D4).61C22, (C6×Q8).53C22, C4.20(D42S3), C2.17(D4⋊D6), C6.118(C8⋊C22), (C2×C12).373C23, (C4×C12).104C22, D4⋊Dic3.12C2, C6.41(C4.4D4), C2.18(Q8.14D6), C2.8(C23.12D6), C6.119(C8.C22), C4⋊Dic3.150C22, C34(C42.28C22), (C2×C6).504(C2×D4), (C2×C4).60(C3⋊D4), (C2×C3⋊C8).120C22, (C3×C4.4D4).3C2, (C2×C4).473(C22×S3), C22.179(C2×C3⋊D4), SmallGroup(192,614)

Series: Derived Chief Lower central Upper central

C1C2×C12 — C42.62D6
C1C3C6C2×C6C2×C12C2×C3⋊C8C42.S3 — C42.62D6
C3C6C2×C12 — C42.62D6
C1C22C42C4.4D4

Generators and relations for C42.62D6
 G = < a,b,c,d | a4=b4=c6=1, d2=a2, ab=ba, cac-1=a-1b2, dad-1=a-1, cbc-1=dbd-1=b-1, dcd-1=a2bc-1 >

Subgroups: 272 in 100 conjugacy classes, 39 normal (27 characteristic)
C1, C2 [×3], C2, C3, C4 [×2], C4 [×5], C22, C22 [×3], C6 [×3], C6, C8 [×2], C2×C4 [×3], C2×C4 [×3], D4 [×2], Q8 [×4], C23, Dic3 [×2], C12 [×2], C12 [×3], C2×C6, C2×C6 [×3], C42, C22⋊C4 [×2], C4⋊C4 [×3], C2×C8 [×2], C2×D4, C2×Q8, C2×Q8, C3⋊C8 [×2], Dic6 [×2], C2×Dic3 [×2], C2×C12 [×3], C2×C12, C3×D4 [×2], C3×Q8 [×2], C22×C6, C8⋊C4, D4⋊C4 [×2], Q8⋊C4 [×2], C4.4D4, C4⋊Q8, C2×C3⋊C8 [×2], C4⋊Dic3 [×2], C4⋊Dic3, C4×C12, C3×C22⋊C4 [×2], C2×Dic6, C6×D4, C6×Q8, C42.28C22, C42.S3, D4⋊Dic3 [×2], Q82Dic3 [×2], C122Q8, C3×C4.4D4, C42.62D6
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D6 [×3], C2×D4, C4○D4 [×2], C3⋊D4 [×2], C22×S3, C4.4D4, C8⋊C22, C8.C22, D42S3 [×2], C2×C3⋊D4, C42.28C22, C23.12D6, D4⋊D6, Q8.14D6, C42.62D6

Character table of C42.62D6

 class 12A2B2C2D34A4B4C4D4E4F4G6A6B6C6D6E8A8B8C8D12A12B12C12D12E12F12G12H
 size 111182224482424222881212121244444488
ρ1111111111111111111111111111111    trivial
ρ211111111111-1-111111-1-1-1-111111111    linear of order 2
ρ31111-1111-1-111-1111-1-1-111-1-1-1-1-11111    linear of order 2
ρ41111-1111-1-11-11111-1-11-1-11-1-1-1-11111    linear of order 2
ρ511111111-1-1-1-1111111-111-1-1-1-1-111-1-1    linear of order 2
ρ611111111-1-1-11-1111111-1-11-1-1-1-111-1-1    linear of order 2
ρ71111-111111-1-1-1111-1-11111111111-1-1    linear of order 2
ρ81111-111111-111111-1-1-1-1-1-1111111-1-1    linear of order 2
ρ92222-2-122-2-2200-1-1-11100001111-1-1-1-1    orthogonal lifted from D6
ρ1022222-122-2-2-200-1-1-1-1-100001111-1-111    orthogonal lifted from D6
ρ11222202-2-22-20002220000002-2-22-2-200    orthogonal lifted from D4
ρ122222-2-12222-200-1-1-1110000-1-1-1-1-1-111    orthogonal lifted from D6
ρ1322222-12222200-1-1-1-1-10000-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ14222202-2-2-22000222000000-222-2-2-200    orthogonal lifted from D4
ρ1522220-1-2-2-22000-1-1-1-3--300001-1-1111-3--3    complex lifted from C3⋊D4
ρ1622220-1-2-22-2000-1-1-1--3-30000-111-111-3--3    complex lifted from C3⋊D4
ρ1722220-1-2-22-2000-1-1-1-3--30000-111-111--3-3    complex lifted from C3⋊D4
ρ1822220-1-2-2-22000-1-1-1--3-300001-1-1111--3-3    complex lifted from C3⋊D4
ρ192-22-2022-2000002-2-20002i-2i000002-200    complex lifted from C4○D4
ρ202-22-202-22000002-2-200-2i002i0000-2200    complex lifted from C4○D4
ρ212-22-202-22000002-2-2002i00-2i0000-2200    complex lifted from C4○D4
ρ222-22-2022-2000002-2-2000-2i2i000002-200    complex lifted from C4○D4
ρ2344-4-40-2000000022-2000000023-2300000    orthogonal lifted from D4⋊D6
ρ2444-4-4040000000-4-4400000000000000    orthogonal lifted from C8⋊C22
ρ2544-4-40-2000000022-20000000-232300000    orthogonal lifted from D4⋊D6
ρ264-44-40-2-4400000-22200000000002-200    symplectic lifted from D42S3, Schur index 2
ρ274-4-44040000000-44-400000000000000    symplectic lifted from C8.C22, Schur index 2
ρ284-4-440-200000002-220000002300-230000    symplectic lifted from Q8.14D6, Schur index 2
ρ294-44-40-24-400000-2220000000000-2200    symplectic lifted from D42S3, Schur index 2
ρ304-4-440-200000002-22000000-2300230000    symplectic lifted from Q8.14D6, Schur index 2

Smallest permutation representation of C42.62D6
On 96 points
Generators in S96
(1 17 12 60)(2 15 10 58)(3 13 11 56)(4 18 9 55)(5 16 7 59)(6 14 8 57)(19 46 34 43)(20 65 35 62)(21 48 36 45)(22 61 31 64)(23 44 32 47)(24 63 33 66)(25 80 86 68)(26 77 87 50)(27 82 88 70)(28 73 89 52)(29 84 90 72)(30 75 85 54)(37 51 94 78)(38 71 95 83)(39 53 96 74)(40 67 91 79)(41 49 92 76)(42 69 93 81)
(1 19 8 22)(2 23 9 20)(3 21 7 24)(4 35 10 32)(5 33 11 36)(6 31 12 34)(13 48 59 63)(14 64 60 43)(15 44 55 65)(16 66 56 45)(17 46 57 61)(18 62 58 47)(25 28 41 38)(26 39 42 29)(27 30 37 40)(49 71 80 73)(50 74 81 72)(51 67 82 75)(52 76 83 68)(53 69 84 77)(54 78 79 70)(85 94 91 88)(86 89 92 95)(87 96 93 90)
(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)(49 50 51 52 53 54)(55 56 57 58 59 60)(61 62 63 64 65 66)(67 68 69 70 71 72)(73 74 75 76 77 78)(79 80 81 82 83 84)(85 86 87 88 89 90)(91 92 93 94 95 96)
(1 54 12 75)(2 50 10 77)(3 52 11 73)(4 69 9 81)(5 71 7 83)(6 67 8 79)(13 89 56 28)(14 40 57 91)(15 87 58 26)(16 38 59 95)(17 85 60 30)(18 42 55 93)(19 70 34 82)(20 74 35 53)(21 68 36 80)(22 78 31 51)(23 72 32 84)(24 76 33 49)(25 48 86 45)(27 46 88 43)(29 44 90 47)(37 61 94 64)(39 65 96 62)(41 63 92 66)

G:=sub<Sym(96)| (1,17,12,60)(2,15,10,58)(3,13,11,56)(4,18,9,55)(5,16,7,59)(6,14,8,57)(19,46,34,43)(20,65,35,62)(21,48,36,45)(22,61,31,64)(23,44,32,47)(24,63,33,66)(25,80,86,68)(26,77,87,50)(27,82,88,70)(28,73,89,52)(29,84,90,72)(30,75,85,54)(37,51,94,78)(38,71,95,83)(39,53,96,74)(40,67,91,79)(41,49,92,76)(42,69,93,81), (1,19,8,22)(2,23,9,20)(3,21,7,24)(4,35,10,32)(5,33,11,36)(6,31,12,34)(13,48,59,63)(14,64,60,43)(15,44,55,65)(16,66,56,45)(17,46,57,61)(18,62,58,47)(25,28,41,38)(26,39,42,29)(27,30,37,40)(49,71,80,73)(50,74,81,72)(51,67,82,75)(52,76,83,68)(53,69,84,77)(54,78,79,70)(85,94,91,88)(86,89,92,95)(87,96,93,90), (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)(49,50,51,52,53,54)(55,56,57,58,59,60)(61,62,63,64,65,66)(67,68,69,70,71,72)(73,74,75,76,77,78)(79,80,81,82,83,84)(85,86,87,88,89,90)(91,92,93,94,95,96), (1,54,12,75)(2,50,10,77)(3,52,11,73)(4,69,9,81)(5,71,7,83)(6,67,8,79)(13,89,56,28)(14,40,57,91)(15,87,58,26)(16,38,59,95)(17,85,60,30)(18,42,55,93)(19,70,34,82)(20,74,35,53)(21,68,36,80)(22,78,31,51)(23,72,32,84)(24,76,33,49)(25,48,86,45)(27,46,88,43)(29,44,90,47)(37,61,94,64)(39,65,96,62)(41,63,92,66)>;

G:=Group( (1,17,12,60)(2,15,10,58)(3,13,11,56)(4,18,9,55)(5,16,7,59)(6,14,8,57)(19,46,34,43)(20,65,35,62)(21,48,36,45)(22,61,31,64)(23,44,32,47)(24,63,33,66)(25,80,86,68)(26,77,87,50)(27,82,88,70)(28,73,89,52)(29,84,90,72)(30,75,85,54)(37,51,94,78)(38,71,95,83)(39,53,96,74)(40,67,91,79)(41,49,92,76)(42,69,93,81), (1,19,8,22)(2,23,9,20)(3,21,7,24)(4,35,10,32)(5,33,11,36)(6,31,12,34)(13,48,59,63)(14,64,60,43)(15,44,55,65)(16,66,56,45)(17,46,57,61)(18,62,58,47)(25,28,41,38)(26,39,42,29)(27,30,37,40)(49,71,80,73)(50,74,81,72)(51,67,82,75)(52,76,83,68)(53,69,84,77)(54,78,79,70)(85,94,91,88)(86,89,92,95)(87,96,93,90), (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)(49,50,51,52,53,54)(55,56,57,58,59,60)(61,62,63,64,65,66)(67,68,69,70,71,72)(73,74,75,76,77,78)(79,80,81,82,83,84)(85,86,87,88,89,90)(91,92,93,94,95,96), (1,54,12,75)(2,50,10,77)(3,52,11,73)(4,69,9,81)(5,71,7,83)(6,67,8,79)(13,89,56,28)(14,40,57,91)(15,87,58,26)(16,38,59,95)(17,85,60,30)(18,42,55,93)(19,70,34,82)(20,74,35,53)(21,68,36,80)(22,78,31,51)(23,72,32,84)(24,76,33,49)(25,48,86,45)(27,46,88,43)(29,44,90,47)(37,61,94,64)(39,65,96,62)(41,63,92,66) );

G=PermutationGroup([(1,17,12,60),(2,15,10,58),(3,13,11,56),(4,18,9,55),(5,16,7,59),(6,14,8,57),(19,46,34,43),(20,65,35,62),(21,48,36,45),(22,61,31,64),(23,44,32,47),(24,63,33,66),(25,80,86,68),(26,77,87,50),(27,82,88,70),(28,73,89,52),(29,84,90,72),(30,75,85,54),(37,51,94,78),(38,71,95,83),(39,53,96,74),(40,67,91,79),(41,49,92,76),(42,69,93,81)], [(1,19,8,22),(2,23,9,20),(3,21,7,24),(4,35,10,32),(5,33,11,36),(6,31,12,34),(13,48,59,63),(14,64,60,43),(15,44,55,65),(16,66,56,45),(17,46,57,61),(18,62,58,47),(25,28,41,38),(26,39,42,29),(27,30,37,40),(49,71,80,73),(50,74,81,72),(51,67,82,75),(52,76,83,68),(53,69,84,77),(54,78,79,70),(85,94,91,88),(86,89,92,95),(87,96,93,90)], [(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),(49,50,51,52,53,54),(55,56,57,58,59,60),(61,62,63,64,65,66),(67,68,69,70,71,72),(73,74,75,76,77,78),(79,80,81,82,83,84),(85,86,87,88,89,90),(91,92,93,94,95,96)], [(1,54,12,75),(2,50,10,77),(3,52,11,73),(4,69,9,81),(5,71,7,83),(6,67,8,79),(13,89,56,28),(14,40,57,91),(15,87,58,26),(16,38,59,95),(17,85,60,30),(18,42,55,93),(19,70,34,82),(20,74,35,53),(21,68,36,80),(22,78,31,51),(23,72,32,84),(24,76,33,49),(25,48,86,45),(27,46,88,43),(29,44,90,47),(37,61,94,64),(39,65,96,62),(41,63,92,66)])

Matrix representation of C42.62D6 in GL6(𝔽73)

2130000
23520000
00006659
0000147
0071400
00596600
,
100000
010000
000010
000001
0072000
0007200
,
100000
59720000
00727200
001000
000011
0000720
,
17650000
18560000
00693693
007474
00693470
00746669

G:=sub<GL(6,GF(73))| [21,23,0,0,0,0,3,52,0,0,0,0,0,0,0,0,7,59,0,0,0,0,14,66,0,0,66,14,0,0,0,0,59,7,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,1,0,0,0,0,0,0,1,0,0],[1,59,0,0,0,0,0,72,0,0,0,0,0,0,72,1,0,0,0,0,72,0,0,0,0,0,0,0,1,72,0,0,0,0,1,0],[17,18,0,0,0,0,65,56,0,0,0,0,0,0,69,7,69,7,0,0,3,4,3,4,0,0,69,7,4,66,0,0,3,4,70,69] >;

C42.62D6 in GAP, Magma, Sage, TeX

C_4^2._{62}D_6
% in TeX

G:=Group("C4^2.62D6");
// GroupNames label

G:=SmallGroup(192,614);
// by ID

G=gap.SmallGroup(192,614);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,477,64,590,471,438,102,6278]);
// Polycyclic

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

Export

Character table of C42.62D6 in TeX

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