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G = C3⋊C86D4order 192 = 26·3

6th semidirect product of C3⋊C8 and D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C3⋊C86D4, C35(C8⋊D4), C4⋊C4.68D6, C22⋊Q84S3, (C2×C12).78D4, C4.175(S3×D4), (C2×Q8).54D6, C6.D840C2, C6.Q1640C2, C12.155(C2×D4), (C22×C6).95D4, C127D4.13C2, Q82Dic316C2, C6.98(C4⋊D4), (C22×C4).145D6, C12.191(C4○D4), (C6×Q8).48C22, C2.16(D4⋊D6), C4.64(D42S3), C6.117(C8⋊C22), (C2×C12).368C23, C6.91(C8.C22), C23.35(C3⋊D4), (C2×D12).100C22, C4⋊Dic3.147C22, C2.19(C23.14D6), C2.12(Q8.11D6), (C22×C12).172C22, (C3×C22⋊Q8)⋊4C2, (C2×C6).499(C2×D4), (C2×Q82S3)⋊11C2, (C2×C4).56(C3⋊D4), (C2×C3⋊C8).116C22, (C2×C4.Dic3)⋊13C2, (C3×C4⋊C4).115C22, (C2×C4).468(C22×S3), C22.174(C2×C3⋊D4), SmallGroup(192,608)

Series: Derived Chief Lower central Upper central

C1C2×C12 — C3⋊C86D4
C1C3C6C12C2×C12C2×D12C127D4 — C3⋊C86D4
C3C6C2×C12 — C3⋊C86D4
C1C22C22×C4C22⋊Q8

Generators and relations for C3⋊C86D4
 G = < a,b,c,d | a3=b8=c4=d2=1, bab-1=cac-1=a-1, ad=da, cbc-1=b-1, dbd=b5, dcd=c-1 >

Subgroups: 368 in 120 conjugacy classes, 41 normal (39 characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×2], C4 [×4], C22, C22 [×6], S3, C6 [×3], C6, C8 [×3], C2×C4 [×2], C2×C4 [×5], D4 [×4], Q8 [×2], C23, C23, Dic3, C12 [×2], C12 [×3], D6 [×3], C2×C6, C2×C6 [×3], C22⋊C4 [×2], C4⋊C4, C4⋊C4 [×2], C2×C8 [×2], M4(2) [×2], SD16 [×2], C22×C4, C2×D4 [×2], C2×Q8, C3⋊C8 [×2], C3⋊C8, D12 [×2], C2×Dic3, C3⋊D4 [×2], C2×C12 [×2], C2×C12 [×4], C3×Q8 [×2], C22×S3, C22×C6, D4⋊C4, Q8⋊C4, C2.D8, C4⋊D4, C22⋊Q8, C2×M4(2), C2×SD16, C2×C3⋊C8 [×2], C4.Dic3 [×2], C4⋊Dic3, D6⋊C4, Q82S3 [×2], C3×C22⋊C4, C3×C4⋊C4, C3×C4⋊C4, C2×D12, C2×C3⋊D4, C22×C12, C6×Q8, C8⋊D4, C6.Q16, C6.D8, Q82Dic3, C2×C4.Dic3, C127D4, C2×Q82S3, C3×C22⋊Q8, C3⋊C86D4
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×4], C23, D6 [×3], C2×D4 [×2], C4○D4, C3⋊D4 [×2], C22×S3, C4⋊D4, C8⋊C22, C8.C22, S3×D4, D42S3, C2×C3⋊D4, C8⋊D4, C23.14D6, Q8.11D6, D4⋊D6, C3⋊C86D4

Character table of C3⋊C86D4

 class 12A2B2C2D2E34A4B4C4D4E4F6A6B6C6D6E8A8B8C8D12A12B12C12D12E12F12G12H
 size 111142422248824222441212121244448888
ρ1111111111111111111111111111111    trivial
ρ21111-11111-11-1-1111-1-1-111-11-11-1-111-1    linear of order 2
ρ31111-11111-1-11-1111-1-11-1-111-11-11-1-11    linear of order 2
ρ41111111111-1-1111111-1-1-1-11111-1-1-1-1    linear of order 2
ρ511111-11111-1-1-11111111111111-1-1-1-1    linear of order 2
ρ61111-1-1111-1-111111-1-1-111-11-11-11-1-11    linear of order 2
ρ71111-1-1111-11-11111-1-11-1-111-11-1-111-1    linear of order 2
ρ811111-1111111-111111-1-1-1-111111111    linear of order 2
ρ92222-20-122-2-220-1-1-1110000-11-11-111-1    orthogonal lifted from D6
ρ10222220-1222-2-20-1-1-1-1-10000-1-1-1-11111    orthogonal lifted from D6
ρ1122-2-2002-220000-2-22000-22020-200000    orthogonal lifted from D4
ρ122222-202-2-22000222-2-20000-22-220000    orthogonal lifted from D4
ρ13222220-1222220-1-1-1-1-10000-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ1422-2-2002-220000-2-220002-2020-200000    orthogonal lifted from D4
ρ152222202-2-2-2000222220000-2-2-2-20000    orthogonal lifted from D4
ρ162222-20-122-22-20-1-1-1110000-11-111-1-11    orthogonal lifted from D6
ρ172222-20-1-2-22000-1-1-11100001-11-1--3--3-3-3    complex lifted from C3⋊D4
ρ18222220-1-2-2-2000-1-1-1-1-100001111-3--3-3--3    complex lifted from C3⋊D4
ρ19222220-1-2-2-2000-1-1-1-1-100001111--3-3--3-3    complex lifted from C3⋊D4
ρ202222-20-1-2-22000-1-1-11100001-11-1-3-3--3--3    complex lifted from C3⋊D4
ρ2122-2-20022-20000-2-2200-2i002i-20200000    complex lifted from C4○D4
ρ2222-2-20022-20000-2-22002i00-2i-20200000    complex lifted from C4○D4
ρ234-44-4004000000-44-400000000000000    orthogonal lifted from C8⋊C22
ρ2444-4-400-2-44000022-2000000-20200000    orthogonal lifted from S3×D4
ρ254-44-400-20000002-220000000-230230000    orthogonal lifted from D4⋊D6
ρ264-44-400-20000002-220000000230-230000    orthogonal lifted from D4⋊D6
ρ274-4-440040000004-4-400000000000000    symplectic lifted from C8.C22, Schur index 2
ρ2844-4-400-24-4000022-200000020-200000    symplectic lifted from D42S3, Schur index 2
ρ294-4-4400-2000000-222-2-32-3000000000000    complex lifted from Q8.11D6
ρ304-4-4400-2000000-2222-3-2-3000000000000    complex lifted from Q8.11D6

Smallest permutation representation of C3⋊C86D4
On 96 points
Generators in S96
(1 70 57)(2 58 71)(3 72 59)(4 60 65)(5 66 61)(6 62 67)(7 68 63)(8 64 69)(9 96 37)(10 38 89)(11 90 39)(12 40 91)(13 92 33)(14 34 93)(15 94 35)(16 36 95)(17 77 84)(18 85 78)(19 79 86)(20 87 80)(21 73 88)(22 81 74)(23 75 82)(24 83 76)(25 43 54)(26 55 44)(27 45 56)(28 49 46)(29 47 50)(30 51 48)(31 41 52)(32 53 42)
(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 94 86 47)(2 93 87 46)(3 92 88 45)(4 91 81 44)(5 90 82 43)(6 89 83 42)(7 96 84 41)(8 95 85 48)(9 17 31 68)(10 24 32 67)(11 23 25 66)(12 22 26 65)(13 21 27 72)(14 20 28 71)(15 19 29 70)(16 18 30 69)(33 73 56 59)(34 80 49 58)(35 79 50 57)(36 78 51 64)(37 77 52 63)(38 76 53 62)(39 75 54 61)(40 74 55 60)
(1 94)(2 91)(3 96)(4 93)(5 90)(6 95)(7 92)(8 89)(9 59)(10 64)(11 61)(12 58)(13 63)(14 60)(15 57)(16 62)(17 56)(18 53)(19 50)(20 55)(21 52)(22 49)(23 54)(24 51)(25 75)(26 80)(27 77)(28 74)(29 79)(30 76)(31 73)(32 78)(33 68)(34 65)(35 70)(36 67)(37 72)(38 69)(39 66)(40 71)(41 88)(42 85)(43 82)(44 87)(45 84)(46 81)(47 86)(48 83)

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

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

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

Matrix representation of C3⋊C86D4 in GL8(𝔽73)

072000000
172000000
000720000
001720000
0000727200
00001000
0000007272
00000010
,
000720000
007200000
072000000
720000000
000067060
0000666767
0000670670
00006666
,
46507330000
232740660000
664027230000
33750460000
0000306000
0000304300
0000004313
0000004330
,
50463370000
272366400000
406623270000
73346500000
0000431300
0000603000
0000004313
0000006030

G:=sub<GL(8,GF(73))| [0,1,0,0,0,0,0,0,72,72,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,0,0,0,0,72,1,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,72,1,0,0,0,0,0,0,72,0],[0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,0,67,6,67,6,0,0,0,0,0,6,0,6,0,0,0,0,6,67,67,6,0,0,0,0,0,67,0,6],[46,23,66,33,0,0,0,0,50,27,40,7,0,0,0,0,7,40,27,50,0,0,0,0,33,66,23,46,0,0,0,0,0,0,0,0,30,30,0,0,0,0,0,0,60,43,0,0,0,0,0,0,0,0,43,43,0,0,0,0,0,0,13,30],[50,27,40,7,0,0,0,0,46,23,66,33,0,0,0,0,33,66,23,46,0,0,0,0,7,40,27,50,0,0,0,0,0,0,0,0,43,60,0,0,0,0,0,0,13,30,0,0,0,0,0,0,0,0,43,60,0,0,0,0,0,0,13,30] >;

C3⋊C86D4 in GAP, Magma, Sage, TeX

C_3\rtimes C_8\rtimes_6D_4
% in TeX

G:=Group("C3:C8:6D4");
// GroupNames label

G:=SmallGroup(192,608);
// by ID

G=gap.SmallGroup(192,608);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,254,555,184,1123,297,136,6278]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^8=c^4=d^2=1,b*a*b^-1=c*a*c^-1=a^-1,a*d=d*a,c*b*c^-1=b^-1,d*b*d=b^5,d*c*d=c^-1>;
// generators/relations

Export

Character table of C3⋊C86D4 in TeX

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