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G = Q8×Dic3order 96 = 25·3

Direct product of Q8 and Dic3

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Q8×Dic3, C33(C4×Q8), (C3×Q8)⋊3C4, C2.3(S3×Q8), (C2×C4).56D6, (C2×Q8).7S3, (C6×Q8).5C2, C6.16(C2×Q8), C12.14(C2×C4), C4.4(C2×Dic3), C6.35(C4○D4), C4⋊Dic3.12C2, (C2×C6).57C23, C6.26(C22×C4), (C4×Dic3).4C2, (C2×C12).39C22, C2.3(Q83S3), C2.7(C22×Dic3), C22.26(C22×S3), (C2×Dic3).40C22, SmallGroup(96,152)

Series: Derived Chief Lower central Upper central

C1C6 — Q8×Dic3
C1C3C6C2×C6C2×Dic3C4×Dic3 — Q8×Dic3
C3C6 — Q8×Dic3
C1C22C2×Q8

Generators and relations for Q8×Dic3
 G = < a,b,c,d | a4=c6=1, b2=a2, d2=c3, bab-1=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 114 in 70 conjugacy classes, 51 normal (14 characteristic)
C1, C2 [×3], C3, C4 [×6], C4 [×5], C22, C6 [×3], C2×C4 [×3], C2×C4 [×4], Q8 [×4], Dic3 [×2], Dic3 [×3], C12 [×6], C2×C6, C42 [×3], C4⋊C4 [×3], C2×Q8, C2×Dic3, C2×Dic3 [×3], C2×C12 [×3], C3×Q8 [×4], C4×Q8, C4×Dic3 [×3], C4⋊Dic3 [×3], C6×Q8, Q8×Dic3
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], Q8 [×2], C23, Dic3 [×4], D6 [×3], C22×C4, C2×Q8, C4○D4, C2×Dic3 [×6], C22×S3, C4×Q8, S3×Q8, Q83S3, C22×Dic3, Q8×Dic3

Character table of Q8×Dic3

 class 12A2B2C34A4B4C4D4E4F4G4H4I4J4K4L4M4N4O4P6A6B6C12A12B12C12D12E12F
 size 111122222223333666666222444444
ρ1111111111111111111111111111111    trivial
ρ2111111-1-11-1-1-1-1-1-111-1-111111-1-1-11-11    linear of order 2
ρ3111111-1-11-1-11111-1-111-1-1111-1-1-11-11    linear of order 2
ρ411111111111-1-1-1-1-1-1-1-1-1-1111111111    linear of order 2
ρ511111-111-1-1-11111-11-1-1-1111111-1-1-1-1    linear of order 2
ρ611111-1-1-1-111-1-1-1-1-1111-11111-1-11-11-1    linear of order 2
ρ711111-1-1-1-11111111-1-1-11-1111-1-11-11-1    linear of order 2
ρ811111-111-1-1-1-1-1-1-11-1111-111111-1-1-1-1    linear of order 2
ρ911-1-111-11-11-1-i-iiiii-ii-i-i-1-11-1111-1-1    linear of order 4
ρ1011-1-1111-1-1-11ii-i-iiii-i-i-i-1-111-1-111-1    linear of order 4
ρ1111-1-111-11-11-1ii-i-i-i-ii-iii-1-11-1111-1-1    linear of order 4
ρ1211-1-1111-1-1-11-i-iii-i-i-iiii-1-111-1-111-1    linear of order 4
ρ1311-1-11-1-111-11-i-iii-iii-ii-i-1-11-11-1-111    linear of order 4
ρ1411-1-11-11-111-1ii-i-i-ii-iii-i-1-111-11-1-11    linear of order 4
ρ1511-1-11-1-111-11ii-i-ii-i-ii-ii-1-11-11-1-111    linear of order 4
ρ1611-1-11-11-111-1-i-iiii-ii-i-ii-1-111-11-1-11    linear of order 4
ρ172222-1-222-2-2-20000000000-1-1-1-1-11111    orthogonal lifted from D6
ρ182222-12222220000000000-1-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ192222-1-2-2-2-2220000000000-1-1-111-11-11    orthogonal lifted from D6
ρ202222-12-2-22-2-20000000000-1-1-1111-11-1    orthogonal lifted from D6
ρ2122-2-2-1-2-222-22000000000011-11-111-1-1    symplectic lifted from Dic3, Schur index 2
ρ2222-2-2-122-2-2-22000000000011-1-111-1-11    symplectic lifted from Dic3, Schur index 2
ρ232-22-220000002-22-2000000-22-2000000    symplectic lifted from Q8, Schur index 2
ρ242-22-22000000-22-22000000-22-2000000    symplectic lifted from Q8, Schur index 2
ρ2522-2-2-1-22-222-2000000000011-1-11-111-1    symplectic lifted from Dic3, Schur index 2
ρ2622-2-2-12-22-22-2000000000011-11-1-1-111    symplectic lifted from Dic3, Schur index 2
ρ272-2-222000000-2i2i2i-2i0000002-2-2000000    complex lifted from C4○D4
ρ282-2-2220000002i-2i-2i2i0000002-2-2000000    complex lifted from C4○D4
ρ294-4-44-20000000000000000-222000000    orthogonal lifted from Q83S3, Schur index 2
ρ304-44-4-200000000000000002-22000000    symplectic lifted from S3×Q8, Schur index 2

Smallest permutation representation of Q8×Dic3
Regular action on 96 points
Generators in S96
(1 28 16 20)(2 29 17 21)(3 30 18 22)(4 25 13 23)(5 26 14 24)(6 27 15 19)(7 79 91 86)(8 80 92 87)(9 81 93 88)(10 82 94 89)(11 83 95 90)(12 84 96 85)(31 43 39 50)(32 44 40 51)(33 45 41 52)(34 46 42 53)(35 47 37 54)(36 48 38 49)(55 74 62 67)(56 75 63 68)(57 76 64 69)(58 77 65 70)(59 78 66 71)(60 73 61 72)
(1 40 16 32)(2 41 17 33)(3 42 18 34)(4 37 13 35)(5 38 14 36)(6 39 15 31)(7 74 91 67)(8 75 92 68)(9 76 93 69)(10 77 94 70)(11 78 95 71)(12 73 96 72)(19 50 27 43)(20 51 28 44)(21 52 29 45)(22 53 30 46)(23 54 25 47)(24 49 26 48)(55 86 62 79)(56 87 63 80)(57 88 64 81)(58 89 65 82)(59 90 66 83)(60 85 61 84)
(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 55 4 58)(2 60 5 57)(3 59 6 56)(7 54 10 51)(8 53 11 50)(9 52 12 49)(13 65 16 62)(14 64 17 61)(15 63 18 66)(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 80 34 83)(32 79 35 82)(33 84 36 81)(37 89 40 86)(38 88 41 85)(39 87 42 90)(43 92 46 95)(44 91 47 94)(45 96 48 93)

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

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

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

Q8×Dic3 is a maximal subgroup of
Dic37SD16  C3⋊Q16⋊C4  Dic34Q16  Q82Dic6  Q83Dic6  Q8.3Dic6  Q8.4Dic6  Q83(C4×S3)  Dic35SD16  SD16⋊Dic3  (C3×Q8).D4  Dic33Q16  Q16⋊Dic3  (C2×Q16)⋊S3  Q86Dic6  Q87Dic6  C4×S3×Q8  C42.125D6  C4×Q83S3  C42.126D6  (Q8×Dic3)⋊C2  C4⋊C4.187D6  C6.152- 1+4  C6.1182+ 1+4  C6.212- 1+4  C6.232- 1+4  C6.772- 1+4  C6.242- 1+4  C42.139D6  C42.234D6  C42.143D6  C42.144D6  Dic68Q8  C42.241D6  C42.176D6  C42.177D6  C6.422- 1+4  C6.452- 1+4  C6.1442+ 1+4  C6.1072- 1+4  C6.1482+ 1+4  C62.13C23  Dic156Q8
Q8×Dic3 is a maximal quotient of
C4⋊C45Dic3  C4⋊C46Dic3  C42.210D6  (C6×Q8)⋊7C4  C62.13C23  Dic156Q8

Matrix representation of Q8×Dic3 in GL4(𝔽13) generated by

121100
1100
00120
00012
,
5000
8800
0010
0001
,
12000
01200
00112
0010
,
8000
0800
0092
00114
G:=sub<GL(4,GF(13))| [12,1,0,0,11,1,0,0,0,0,12,0,0,0,0,12],[5,8,0,0,0,8,0,0,0,0,1,0,0,0,0,1],[12,0,0,0,0,12,0,0,0,0,1,1,0,0,12,0],[8,0,0,0,0,8,0,0,0,0,9,11,0,0,2,4] >;

Q8×Dic3 in GAP, Magma, Sage, TeX

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

G:=Group("Q8xDic3");
// GroupNames label

G:=SmallGroup(96,152);
// by ID

G=gap.SmallGroup(96,152);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-3,48,103,188,86,2309]);
// Polycyclic

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

Export

Character table of Q8×Dic3 in TeX

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