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G = Q82Dic3order 96 = 25·3

1st semidirect product of Q8 and Dic3 acting via Dic3/C6=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C12.9D4, C6.5Q16, Q82Dic3, C6.8SD16, (C3×Q8)⋊1C4, C12.8(C2×C4), (C2×C4).40D6, (C2×C6).34D4, (C6×Q8).1C2, (C2×Q8).3S3, C33(Q8⋊C4), C4.2(C2×Dic3), C4.14(C3⋊D4), C4⋊Dic3.10C2, C2.3(C3⋊Q16), C6.16(C22⋊C4), (C2×C12).18C22, C2.3(Q82S3), C2.6(C6.D4), C22.18(C3⋊D4), (C2×C3⋊C8).5C2, SmallGroup(96,42)

Series: Derived Chief Lower central Upper central

C1C12 — Q82Dic3
C1C3C6C2×C6C2×C12C4⋊Dic3 — Q82Dic3
C3C6C12 — Q82Dic3
C1C22C2×C4C2×Q8

Generators and relations for Q82Dic3
 G = < a,b,c,d | a4=c6=1, b2=a2, d2=c3, bab-1=dad-1=a-1, ac=ca, bc=cb, dbd-1=a-1b, dcd-1=c-1 >

2C4
2C4
12C4
2Q8
2C2×C4
6C2×C4
6C8
2C12
2C12
4Dic3
3C4⋊C4
3C2×C8
2C3×Q8
2C3⋊C8
2C2×Dic3
2C2×C12
3Q8⋊C4

Character table of Q82Dic3

 class 12A2B2C34A4B4C4D4E4F6A6B6C8A8B8C8D12A12B12C12D12E12F
 size 11112224412122226666444444
ρ1111111111111111111111111    trivial
ρ21111111-1-111111-1-1-1-11-1-11-1-1    linear of order 2
ρ3111111111-1-1111-1-1-1-1111111    linear of order 2
ρ41111111-1-1-1-111111111-1-11-1-1    linear of order 2
ρ511-1-11-111-1i-i-1-11-ii-ii-1111-1-1    linear of order 4
ρ611-1-11-111-1-ii-1-11i-ii-i-1111-1-1    linear of order 4
ρ711-1-11-11-11-ii-1-11-ii-ii-1-1-1111    linear of order 4
ρ811-1-11-11-11i-i-1-11i-ii-i-1-1-1111    linear of order 4
ρ92222-1222200-1-1-10000-1-1-1-1-1-1    orthogonal lifted from S3
ρ1022-2-222-20000-2-220000200-200    orthogonal lifted from D4
ρ1122222-2-200002220000-200-200    orthogonal lifted from D4
ρ122222-122-2-200-1-1-10000-111-111    orthogonal lifted from D6
ρ1322-2-2-1-222-20011-100001-1-1-111    symplectic lifted from Dic3, Schur index 2
ρ1422-2-2-1-22-220011-10000111-1-1-1    symplectic lifted from Dic3, Schur index 2
ρ152-2-222000000-22-2-222-2000000    symplectic lifted from Q16, Schur index 2
ρ162-2-222000000-22-22-2-22000000    symplectic lifted from Q16, Schur index 2
ρ1722-2-2-12-2000011-10000-1--3-31-3--3    complex lifted from C3⋊D4
ρ1822-2-2-12-2000011-10000-1-3--31--3-3    complex lifted from C3⋊D4
ρ192222-1-2-20000-1-1-100001--3-31--3-3    complex lifted from C3⋊D4
ρ202-22-220000002-2-2-2-2--2--2000000    complex lifted from SD16
ρ212-22-220000002-2-2--2--2-2-2000000    complex lifted from SD16
ρ222222-1-2-20000-1-1-100001-3--31-3--3    complex lifted from C3⋊D4
ρ234-44-4-2000000-2220000000000    orthogonal lifted from Q82S3
ρ244-4-44-20000002-220000000000    symplectic lifted from C3⋊Q16, Schur index 2

Smallest permutation representation of Q82Dic3
Regular action on 96 points
Generators in S96
(1 28 23 7)(2 29 24 8)(3 30 19 9)(4 25 20 10)(5 26 21 11)(6 27 22 12)(13 94 81 89)(14 95 82 90)(15 96 83 85)(16 91 84 86)(17 92 79 87)(18 93 80 88)(31 47 51 39)(32 48 52 40)(33 43 53 41)(34 44 54 42)(35 45 49 37)(36 46 50 38)(55 68 78 63)(56 69 73 64)(57 70 74 65)(58 71 75 66)(59 72 76 61)(60 67 77 62)
(1 44 23 42)(2 45 24 37)(3 46 19 38)(4 47 20 39)(5 48 21 40)(6 43 22 41)(7 54 28 34)(8 49 29 35)(9 50 30 36)(10 51 25 31)(11 52 26 32)(12 53 27 33)(13 67 81 62)(14 68 82 63)(15 69 83 64)(16 70 84 65)(17 71 79 66)(18 72 80 61)(55 90 78 95)(56 85 73 96)(57 86 74 91)(58 87 75 92)(59 88 76 93)(60 89 77 94)
(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 65 4 62)(2 64 5 61)(3 63 6 66)(7 57 10 60)(8 56 11 59)(9 55 12 58)(13 54 16 51)(14 53 17 50)(15 52 18 49)(19 68 22 71)(20 67 23 70)(21 72 24 69)(25 77 28 74)(26 76 29 73)(27 75 30 78)(31 81 34 84)(32 80 35 83)(33 79 36 82)(37 85 40 88)(38 90 41 87)(39 89 42 86)(43 92 46 95)(44 91 47 94)(45 96 48 93)

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

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

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

Q82Dic3 is a maximal subgroup of
Q82Dic6  Dic3.1Q16  Q83Dic6  (C2×C8).D6  Q8.3Dic6  (C2×Q8).36D6  Q8.4Dic6  Q8⋊C4⋊S3  S3×Q8⋊C4  (S3×Q8)⋊C4  Q87(C4×S3)  C4⋊C4.150D6  D6.1SD16  D6.Q16  D6⋊C8.C2  C8⋊Dic3⋊C2  Q84Dic6  Q85Dic6  Q8.5Dic6  C4×Q82S3  C42.56D6  C4×C3⋊Q16  C42.59D6  (C2×Q8).49D6  (C2×C6).Q16  (C2×Q8).51D6  C3⋊C824D4  C3⋊C86D4  C3⋊C8.29D4  C3⋊C8.6D4  C42.61D6  C42.62D6  C42.213D6  D12.23D4  C12.9Q16  C42.77D6  C125SD16  C12⋊Q16  Dic3×SD16  Dic33SD16  SD16⋊Dic3  (C3×D4).D4  D68SD16  C2414D4  D127D4  C248D4  Dic3×Q16  Dic33Q16  Q16⋊Dic3  (C2×Q16)⋊S3  D65Q16  D12.17D4  D63Q16  C24.36D4  (C6×Q8)⋊6C4  (C3×Q8)⋊13D4  (C2×C6)⋊8Q16  C4○D43Dic3  C4○D44Dic3  (C3×D4)⋊14D4  (C3×D4).32D4  Q82Dic9  Q8⋊Dic9  Dic6⋊Dic3  C6.Dic12  C62.117D4  C6.GL2(𝔽3)  C30.Q16  C6.Dic20  Q82Dic15  Dic102Dic3
Q82Dic3 is a maximal quotient of
C12.C42  C12.26Q16  (C6×Q8)⋊C4  C12.5Q16  C12.10D8  Q82Dic9  Dic6⋊Dic3  C6.Dic12  C62.117D4  C30.Q16  C6.Dic20  Q82Dic15  Dic102Dic3

Matrix representation of Q82Dic3 in GL5(𝔽73)

10000
00100
072000
000720
000072
,
10000
0656400
064800
0003013
0006043
,
720000
01000
00100
000721
000720
,
460000
029600
064400
000518
0002368

G:=sub<GL(5,GF(73))| [1,0,0,0,0,0,0,72,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,72],[1,0,0,0,0,0,65,64,0,0,0,64,8,0,0,0,0,0,30,60,0,0,0,13,43],[72,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,72,72,0,0,0,1,0],[46,0,0,0,0,0,29,6,0,0,0,6,44,0,0,0,0,0,5,23,0,0,0,18,68] >;

Q82Dic3 in GAP, Magma, Sage, TeX

Q_8\rtimes_2{\rm Dic}_3
% in TeX

G:=Group("Q8:2Dic3");
// GroupNames label

G:=SmallGroup(96,42);
// by ID

G=gap.SmallGroup(96,42);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-3,24,121,103,579,297,69,2309]);
// Polycyclic

G:=Group<a,b,c,d|a^4=c^6=1,b^2=a^2,d^2=c^3,b*a*b^-1=d*a*d^-1=a^-1,a*c=c*a,b*c=c*b,d*b*d^-1=a^-1*b,d*c*d^-1=c^-1>;
// generators/relations

Export

Subgroup lattice of Q82Dic3 in TeX
Character table of Q82Dic3 in TeX

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