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G = Dic3⋊Q8order 96 = 25·3

2nd semidirect product of Dic3 and Q8 acting via Q8/C4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Dic32Q8, C12.21D4, C33(C4⋊Q8), C2.8(S3×Q8), C6.56(C2×D4), (C2×C4).55D6, (C6×Q8).4C2, (C2×Q8).6S3, C6.15(C2×Q8), C4.10(C3⋊D4), Dic3⋊C4.6C2, (C2×C6).56C23, (C2×Dic6).9C2, (C4×Dic3).3C2, (C2×C12).63C22, C22.63(C22×S3), (C2×Dic3).20C22, C2.20(C2×C3⋊D4), SmallGroup(96,151)

Series: Derived Chief Lower central Upper central

C1C2×C6 — Dic3⋊Q8
C1C3C6C2×C6C2×Dic3C4×Dic3 — Dic3⋊Q8
C3C2×C6 — Dic3⋊Q8
C1C22C2×Q8

Generators and relations for Dic3⋊Q8
 G = < a,b,c,d | a6=c4=1, b2=a3, d2=c2, bab-1=a-1, ac=ca, ad=da, cbc-1=a3b, bd=db, dcd-1=c-1 >

Subgroups: 130 in 68 conjugacy classes, 37 normal (13 characteristic)
C1, C2, C2 [×2], C3, C4 [×2], C4 [×8], C22, C6, C6 [×2], C2×C4, C2×C4 [×2], C2×C4 [×4], Q8 [×4], Dic3 [×4], Dic3 [×2], C12 [×2], C12 [×2], C2×C6, C42, C4⋊C4 [×4], C2×Q8, C2×Q8, Dic6 [×2], C2×Dic3 [×4], C2×C12, C2×C12 [×2], C3×Q8 [×2], C4⋊Q8, C4×Dic3, Dic3⋊C4 [×4], C2×Dic6, C6×Q8, Dic3⋊Q8
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], Q8 [×4], C23, D6 [×3], C2×D4, C2×Q8 [×2], C3⋊D4 [×2], C22×S3, C4⋊Q8, S3×Q8 [×2], C2×C3⋊D4, Dic3⋊Q8

Character table of Dic3⋊Q8

 class 12A2B2C34A4B4C4D4E4F4G4H4I4J6A6B6C12A12B12C12D12E12F
 size 11112224466661212222444444
ρ1111111111111111111111111    trivial
ρ211111-1-11-11-11-11-1111-11-1-1-11    linear of order 2
ρ311111-1-1-111-11-1-11111-1-11-11-1    linear of order 2
ρ41111111-1-11111-1-11111-1-11-1-1    linear of order 2
ρ511111-1-11-1-11-11-11111-11-1-1-11    linear of order 2
ρ6111111111-1-1-1-1-1-1111111111    linear of order 2
ρ71111111-1-1-1-1-1-1111111-1-11-1-1    linear of order 2
ρ811111-1-1-11-11-111-1111-1-11-11-1    linear of order 2
ρ92-22-22-22000000002-2-2200-200    orthogonal lifted from D4
ρ102222-1-2-2-22000000-1-1-111-11-11    orthogonal lifted from D6
ρ112222-12222000000-1-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ122222-1-2-22-2000000-1-1-11-1111-1    orthogonal lifted from D6
ρ132-22-222-2000000002-2-2-200200    orthogonal lifted from D4
ρ142222-122-2-2000000-1-1-1-111-111    orthogonal lifted from D6
ρ1522-2-2200000-20200-22-2000000    symplectic lifted from Q8, Schur index 2
ρ162-2-2220000-202000-2-22000000    symplectic lifted from Q8, Schur index 2
ρ172-2-222000020-2000-2-22000000    symplectic lifted from Q8, Schur index 2
ρ1822-2-220000020-200-22-2000000    symplectic lifted from Q8, Schur index 2
ρ192-22-2-1-2200000000-111-1--3--31-3-3    complex lifted from C3⋊D4
ρ202-22-2-12-200000000-1111-3--3-1-3--3    complex lifted from C3⋊D4
ρ212-22-2-12-200000000-1111--3-3-1--3-3    complex lifted from C3⋊D4
ρ222-22-2-1-2200000000-111-1-3-31--3--3    complex lifted from C3⋊D4
ρ2344-4-4-200000000002-22000000    symplectic lifted from S3×Q8, Schur index 2
ρ244-4-44-2000000000022-2000000    symplectic lifted from S3×Q8, Schur index 2

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

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

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

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

Dic3⋊Q8 is a maximal subgroup of
(C2×C12).D4  D12.4D4  Dic3.1Q16  Dic3⋊Q16  (C2×Q8).36D6  Dic3⋊SD16  Dic33SD16  C24.31D4  C2415D4  Dic33Q16  C24.26D4  C24.37D4  D12.40D4  2- 1+4.2S3  Dic610Q8  C42.122D6  C42.232D6  C42.134D6  (Q8×Dic3)⋊C2  C6.752- 1+4  C6.152- 1+4  D1222D4  Dic621D4  C6.522+ 1+4  C6.222- 1+4  C6.252- 1+4  C42.233D6  C42.137D6  C42.138D6  C42.139D6  C42.140D6  C42.141D6  Dic68Q8  Dic69Q8  S3×C4⋊Q8  C42.171D6  C42.174D6  C42.180D6  Q8×C3⋊D4  C6.442- 1+4  C6.1042- 1+4  C6.1052- 1+4  (C2×C12)⋊17D4  Dic9⋊Q8  C62.9C23  Dic3⋊Dic6  C62.43C23  C62.259C23  Dic151Q8  Dic158Q8  C60.48D4  Dic154Q8
Dic3⋊Q8 is a maximal quotient of
C4.(D6⋊C4)  (C4×Dic3)⋊8C4  Dic3⋊(C4⋊C4)  (C2×Dic3)⋊Q8  (C2×C4).44D12  (C2×Dic3).Q8  C42.68D6  C42.215D6  C12.17D8  C12.SD16  C42.76D6  (C6×Q8)⋊7C4  C22.52(S3×Q8)  Dic9⋊Q8  C62.9C23  Dic3⋊Dic6  C62.43C23  C62.259C23  Dic151Q8  Dic158Q8  C60.48D4  Dic154Q8

Matrix representation of Dic3⋊Q8 in GL6(𝔽13)

900000
030000
0012000
0001200
0000120
0000012
,
010000
100000
0091000
0010400
000050
000008
,
100000
010000
000100
0012000
000001
000010
,
100000
010000
004300
003900
000010
000001

G:=sub<GL(6,GF(13))| [9,0,0,0,0,0,0,3,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,9,10,0,0,0,0,10,4,0,0,0,0,0,0,5,0,0,0,0,0,0,8],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,3,0,0,0,0,3,9,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

Dic3⋊Q8 in GAP, Magma, Sage, TeX

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

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

G:=SmallGroup(96,151);
// by ID

G=gap.SmallGroup(96,151);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-3,96,55,362,116,50,2309]);
// Polycyclic

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

Export

Character table of Dic3⋊Q8 in TeX

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