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G = C3⋊C85D4order 192 = 26·3

5th semidirect product of C3⋊C8 and D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C3⋊C85D4, C34(C8⋊D4), C4⋊C4.62D6, (C2×D4).42D6, (C2×C12).74D4, C4.173(S3×D4), C4⋊D4.7S3, C12.151(C2×D4), C6.Q1637C2, (C22×C6).88D4, D4⋊Dic318C2, C6.SD1636C2, C6.96(C4⋊D4), C6.92(C8⋊C22), (C6×D4).58C22, (C22×C4).139D6, C12.186(C4○D4), C4.62(D42S3), C12.48D425C2, (C2×C12).361C23, C23.32(C3⋊D4), C2.13(Q8.14D6), C2.13(D126C22), C6.115(C8.C22), C4⋊Dic3.145C22, C2.17(C23.14D6), (C22×C12).165C22, (C2×Dic6).104C22, (C2×D4.S3)⋊11C2, (C3×C4⋊D4).6C2, (C2×C6).492(C2×D4), (C2×C4).52(C3⋊D4), (C2×C3⋊C8).111C22, (C2×C4.Dic3)⋊12C2, (C3×C4⋊C4).109C22, (C2×C4).461(C22×S3), C22.167(C2×C3⋊D4), SmallGroup(192,601)

Series: Derived Chief Lower central Upper central

C1C2×C12 — C3⋊C85D4
C1C3C6C12C2×C12C2×Dic6C12.48D4 — C3⋊C85D4
C3C6C2×C12 — C3⋊C85D4
C1C22C22×C4C4⋊D4

Generators and relations for C3⋊C85D4
 G = < a,b,c,d | a3=b8=c4=d2=1, bab-1=a-1, ac=ca, ad=da, cbc-1=b-1, dbd=b5, dcd=c-1 >

Subgroups: 320 in 120 conjugacy classes, 41 normal (39 characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×2], C4 [×4], C22, C22 [×6], C6 [×3], C6 [×2], C8 [×3], C2×C4 [×2], C2×C4 [×5], D4 [×4], Q8 [×2], C23, C23, Dic3 [×2], C12 [×2], C12 [×2], C2×C6, C2×C6 [×6], C22⋊C4 [×2], C4⋊C4, C4⋊C4 [×2], C2×C8 [×2], M4(2) [×2], SD16 [×2], C22×C4, C2×D4, C2×D4, C2×Q8, C3⋊C8 [×2], C3⋊C8, Dic6 [×2], C2×Dic3 [×2], C2×C12 [×2], C2×C12 [×3], C3×D4 [×4], C22×C6, C22×C6, D4⋊C4, Q8⋊C4, C2.D8, C4⋊D4, C22⋊Q8, C2×M4(2), C2×SD16, C2×C3⋊C8 [×2], C4.Dic3 [×2], Dic3⋊C4, C4⋊Dic3, D4.S3 [×2], C6.D4, C3×C22⋊C4, C3×C4⋊C4, C2×Dic6, C22×C12, C6×D4, C6×D4, C8⋊D4, C6.Q16, C6.SD16, D4⋊Dic3, C2×C4.Dic3, C12.48D4, C2×D4.S3, C3×C4⋊D4, C3⋊C85D4
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, D126C22, C23.14D6, Q8.14D6, C3⋊C85D4

Character table of C3⋊C85D4

 class 12A2B2C2D2E34A4B4C4D4E4F6A6B6C6D6E6F6G8A8B8C8D12A12B12C12D12E12F
 size 111148222482424222448812121212444488
ρ1111111111111111111111111111111    trivial
ρ211111111111-1-11111111-1-1-1-1111111    linear of order 2
ρ31111-11111-1-1-11111-1-111-11-111-1-11-1-1    linear of order 2
ρ41111-11111-1-11-1111-1-1111-11-11-1-11-1-1    linear of order 2
ρ511111-11111-11111111-1-1-1-1-1-11111-1-1    linear of order 2
ρ611111-11111-1-1-111111-1-111111111-1-1    linear of order 2
ρ71111-1-1111-11-11111-1-1-1-11-11-11-1-1111    linear of order 2
ρ81111-1-1111-111-1111-1-1-1-1-11-111-1-1111    linear of order 2
ρ92222-22-122-2-200-1-1-111-1-10000-111-111    orthogonal lifted from D6
ρ1022-2-2002-2200002-2-200000-202200-200    orthogonal lifted from D4
ρ1122222-2-1222-200-1-1-1-1-1110000-1-1-1-111    orthogonal lifted from D6
ρ122222202-2-2-200022222000000-2-2-2-200    orthogonal lifted from D4
ρ132222-202-2-22000222-2-2000000-222-200    orthogonal lifted from D4
ρ142222-2-2-122-2200-1-1-111110000-111-1-1-1    orthogonal lifted from D6
ρ1522-2-2002-2200002-2-20000020-2200-200    orthogonal lifted from D4
ρ16222222-1222200-1-1-1-1-1-1-10000-1-1-1-1-1-1    orthogonal lifted from S3
ρ17222220-1-2-2-2000-1-1-1-1-1-3--300001111--3-3    complex lifted from C3⋊D4
ρ182222-20-1-2-22000-1-1-111--3-300001-1-11--3-3    complex lifted from C3⋊D4
ρ192222-20-1-2-22000-1-1-111-3--300001-1-11-3--3    complex lifted from C3⋊D4
ρ20222220-1-2-2-2000-1-1-1-1-1--3-300001111-3--3    complex lifted from C3⋊D4
ρ2122-2-20022-200002-2-20000-2i02i0-200200    complex lifted from C4○D4
ρ2222-2-20022-200002-2-200002i0-2i0-200200    complex lifted from C4○D4
ρ234-4-44004000000-44-400000000000000    orthogonal lifted from C8⋊C22
ρ2444-4-400-2-440000-22200000000-200200    orthogonal lifted from S3×D4
ρ254-44-4004000000-4-4400000000000000    symplectic lifted from C8.C22, Schur index 2
ρ2644-4-400-24-40000-22200000000200-200    symplectic lifted from D42S3, Schur index 2
ρ274-44-400-200000022-2000000000-2323000    symplectic lifted from Q8.14D6, Schur index 2
ρ284-44-400-200000022-200000000023-23000    symplectic lifted from Q8.14D6, Schur index 2
ρ294-4-4400-20000002-222-3-2-3000000000000    complex lifted from D126C22
ρ304-4-4400-20000002-22-2-32-3000000000000    complex lifted from D126C22

Smallest permutation representation of C3⋊C85D4
On 96 points
Generators in S96
(1 41 50)(2 51 42)(3 43 52)(4 53 44)(5 45 54)(6 55 46)(7 47 56)(8 49 48)(9 80 86)(10 87 73)(11 74 88)(12 81 75)(13 76 82)(14 83 77)(15 78 84)(16 85 79)(17 26 91)(18 92 27)(19 28 93)(20 94 29)(21 30 95)(22 96 31)(23 32 89)(24 90 25)(33 63 66)(34 67 64)(35 57 68)(36 69 58)(37 59 70)(38 71 60)(39 61 72)(40 65 62)
(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 37 95 15)(2 36 96 14)(3 35 89 13)(4 34 90 12)(5 33 91 11)(6 40 92 10)(7 39 93 9)(8 38 94 16)(17 74 45 63)(18 73 46 62)(19 80 47 61)(20 79 48 60)(21 78 41 59)(22 77 42 58)(23 76 43 57)(24 75 44 64)(25 81 53 67)(26 88 54 66)(27 87 55 65)(28 86 56 72)(29 85 49 71)(30 84 50 70)(31 83 51 69)(32 82 52 68)
(1 15)(2 12)(3 9)(4 14)(5 11)(6 16)(7 13)(8 10)(17 63)(18 60)(19 57)(20 62)(21 59)(22 64)(23 61)(24 58)(25 69)(26 66)(27 71)(28 68)(29 65)(30 70)(31 67)(32 72)(33 91)(34 96)(35 93)(36 90)(37 95)(38 92)(39 89)(40 94)(41 78)(42 75)(43 80)(44 77)(45 74)(46 79)(47 76)(48 73)(49 87)(50 84)(51 81)(52 86)(53 83)(54 88)(55 85)(56 82)

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

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

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

Matrix representation of C3⋊C85D4 in GL8(𝔽73)

072000000
172000000
006400000
81280000
0000727200
00001000
000000072
000000172
,
10000000
172000000
28072280000
00010000
00000044
000000465
00006167061
0000676610
,
00010000
657271650000
81100000
720000000
0000431300
0000603000
000017174360
00000171330
,
00010000
657271650000
00100000
10000000
0000431300
0000603000
0000003013
0000006043

G:=sub<GL(8,GF(73))| [0,1,0,8,0,0,0,0,72,72,0,1,0,0,0,0,0,0,64,2,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,72,1,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72],[1,1,28,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,28,1,0,0,0,0,0,0,0,0,0,0,61,67,0,0,0,0,0,0,67,6,0,0,0,0,4,4,0,61,0,0,0,0,4,65,61,0],[0,65,8,72,0,0,0,0,0,72,1,0,0,0,0,0,0,71,1,0,0,0,0,0,1,65,0,0,0,0,0,0,0,0,0,0,43,60,17,0,0,0,0,0,13,30,17,17,0,0,0,0,0,0,43,13,0,0,0,0,0,0,60,30],[0,65,0,1,0,0,0,0,0,72,0,0,0,0,0,0,0,71,1,0,0,0,0,0,1,65,0,0,0,0,0,0,0,0,0,0,43,60,0,0,0,0,0,0,13,30,0,0,0,0,0,0,0,0,30,60,0,0,0,0,0,0,13,43] >;

C3⋊C85D4 in GAP, Magma, Sage, TeX

C_3\rtimes C_8\rtimes_5D_4
% in TeX

G:=Group("C3:C8:5D4");
// GroupNames label

G:=SmallGroup(192,601);
// by ID

G=gap.SmallGroup(192,601);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,253,254,555,1123,297,136,6278]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^8=c^4=d^2=1,b*a*b^-1=a^-1,a*c=c*a,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⋊C85D4 in TeX

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