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G = C42.64D6order 192 = 26·3

64th non-split extension by C42 of D6 acting via D6/C3=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C42.64D6, C3⋊C87D4, C32(C83D4), C4.11(S3×D4), C4⋊D129C2, (C2×D4).48D6, C4.4D42S3, (C2×C12).82D4, C12.25(C2×D4), (C2×Q8).62D6, C2.8(C123D4), C6.17(C41D4), C42.S39C2, (C6×D4).64C22, (C6×Q8).56C22, C2.19(D4⋊D6), C6.120(C8⋊C22), (C4×C12).107C22, (C2×C12).376C23, (C2×D12).101C22, (C2×D4⋊S3)⋊12C2, (C2×C6).507(C2×D4), (C3×C4.4D4)⋊2C2, (C2×Q82S3)⋊13C2, (C2×C4).62(C3⋊D4), (C2×C3⋊C8).122C22, (C2×C4).476(C22×S3), C22.182(C2×C3⋊D4), SmallGroup(192,617)

Series: Derived Chief Lower central Upper central

C1C2×C12 — C42.64D6
C1C3C6C12C2×C12C2×D12C4⋊D12 — C42.64D6
C3C6C2×C12 — C42.64D6
C1C22C42C4.4D4

Generators and relations for C42.64D6
 G = < a,b,c,d | a4=b4=c6=1, d2=cbc-1=b-1, ab=ba, cac-1=a-1b2, dad-1=ab2, bd=db, dcd-1=b-1c-1 >

Subgroups: 528 in 144 conjugacy classes, 43 normal (19 characteristic)
C1, C2, C2 [×2], C2 [×3], C3, C4 [×2], C4 [×3], C22, C22 [×9], S3 [×2], C6, C6 [×2], C6, C8 [×4], C2×C4, C2×C4 [×2], C2×C4, D4 [×10], Q8 [×2], C23 [×3], C12 [×2], C12 [×3], D6 [×6], C2×C6, C2×C6 [×3], C42, C22⋊C4 [×2], C2×C8 [×2], D8 [×4], SD16 [×4], C2×D4, C2×D4 [×4], C2×Q8, C3⋊C8 [×4], D12 [×8], C2×C12, C2×C12 [×2], C2×C12, C3×D4 [×2], C3×Q8 [×2], C22×S3 [×2], C22×C6, C8⋊C4, C4.4D4, C41D4, C2×D8 [×2], C2×SD16 [×2], C2×C3⋊C8 [×2], D4⋊S3 [×4], Q82S3 [×4], C4×C12, C3×C22⋊C4 [×2], C2×D12 [×2], C2×D12 [×2], C6×D4, C6×Q8, C83D4, C42.S3, C4⋊D12, C2×D4⋊S3 [×2], C2×Q82S3 [×2], C3×C4.4D4, C42.64D6
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×6], C23, D6 [×3], C2×D4 [×3], C3⋊D4 [×2], C22×S3, C41D4, C8⋊C22 [×2], S3×D4 [×2], C2×C3⋊D4, C83D4, C123D4, D4⋊D6 [×2], C42.64D6

Character table of C42.64D6

 class 12A2B2C2D2E2F34A4B4C4D4E6A6B6C6D6E8A8B8C8D12A12B12C12D12E12F12G12H
 size 111182424222448222881212121244444488
ρ1111111111111111111111111111111    trivial
ρ21111-1-11111-1-11111-1-11-1-11-1-1-11-1111    linear of order 2
ρ3111111-1111-1-1-1111111-1-11-1-1-11-11-1-1    linear of order 2
ρ41111-1-1-111111-1111-1-11111111111-1-1    linear of order 2
ρ511111-1-111111111111-1-1-1-111111111    linear of order 2
ρ61111-11-1111-1-11111-1-1-111-1-1-1-11-1111    linear of order 2
ρ71111-11111111-1111-1-1-1-1-1-1111111-1-1    linear of order 2
ρ811111-11111-1-1-111111-111-1-1-1-11-11-1-1    linear of order 2
ρ922-2-200022-2000-2-220002-20000-20200    orthogonal lifted from D4
ρ102222200-122-2-2-2-1-1-1-1-10000111-11-111    orthogonal lifted from D6
ρ112222200-122222-1-1-1-1-10000-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ1222220002-2-22-202220000002-2-2-22-200    orthogonal lifted from D4
ρ1322-2-20002-22000-2-2200200-200020-200    orthogonal lifted from D4
ρ1422-2-20002-22000-2-2200-200200020-200    orthogonal lifted from D4
ρ1522220002-2-2-220222000000-222-2-2-200    orthogonal lifted from D4
ρ162222-200-122-2-22-1-1-1110000111-11-1-1-1    orthogonal lifted from D6
ρ172222-200-12222-2-1-1-1110000-1-1-1-1-1-111    orthogonal lifted from D6
ρ1822-2-200022-2000-2-22000-220000-20200    orthogonal lifted from D4
ρ192222000-1-2-2-220-1-1-1-3--300001-1-1111--3-3    complex lifted from C3⋊D4
ρ202222000-1-2-22-20-1-1-1-3--30000-1111-11-3--3    complex lifted from C3⋊D4
ρ212222000-1-2-22-20-1-1-1--3-30000-1111-11--3-3    complex lifted from C3⋊D4
ρ222222000-1-2-2-220-1-1-1--3-300001-1-1111-3--3    complex lifted from C3⋊D4
ρ234-4-440004000004-4-400000000000000    orthogonal lifted from C8⋊C22
ρ244-44-4000400000-44-400000000000000    orthogonal lifted from C8⋊C22
ρ2544-4-4000-2-4400022-2000000000-20200    orthogonal lifted from S3×D4
ρ2644-4-4000-24-400022-200000000020-200    orthogonal lifted from S3×D4
ρ274-44-4000-2000002-22000000023-2300000    orthogonal lifted from D4⋊D6
ρ284-4-44000-200000-22200000023000-23000    orthogonal lifted from D4⋊D6
ρ294-44-4000-2000002-220000000-232300000    orthogonal lifted from D4⋊D6
ρ304-4-44000-200000-222000000-2300023000    orthogonal lifted from D4⋊D6

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

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

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

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

Matrix representation of C42.64D6 in GL6(𝔽73)

3230000
72410000
00006659
0000147
0071400
00596600
,
100000
010000
000010
000001
0072000
0007200
,
100000
3720000
0021333321
0040615212
0033215240
0052123312
,
100000
010000
0033215240
0061406121
0021333321
0012526140

G:=sub<GL(6,GF(73))| [32,72,0,0,0,0,3,41,0,0,0,0,0,0,0,0,7,59,0,0,0,0,14,66,0,0,66,14,0,0,0,0,59,7,0,0],[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],[1,3,0,0,0,0,0,72,0,0,0,0,0,0,21,40,33,52,0,0,33,61,21,12,0,0,33,52,52,33,0,0,21,12,40,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,33,61,21,12,0,0,21,40,33,52,0,0,52,61,33,61,0,0,40,21,21,40] >;

C42.64D6 in GAP, Magma, Sage, TeX

C_4^2._{64}D_6
% in TeX

G:=Group("C4^2.64D6");
// GroupNames label

G:=SmallGroup(192,617);
// by ID

G=gap.SmallGroup(192,617);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,120,254,555,1123,297,136,6278]);
// Polycyclic

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

Export

Character table of C42.64D6 in TeX

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