Copied to
clipboard

G = D6.2Q16order 192 = 26·3

2nd non-split extension by D6 of Q16 acting via Q16/Q8=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D6.2Q16, C2.D84S3, D6⋊C8.7C2, C4⋊C4.49D6, (C2×C8).28D6, C6.24(C2×Q16), C2.14(S3×Q16), C6.Q1622C2, C4.D12.7C2, C4.81(C4○D12), C12.39(C4○D4), C6.SD1619C2, C2.22(D8⋊S3), C6.41(C8⋊C22), C2.Dic1214C2, (C2×Dic3).49D4, (C22×S3).86D4, C22.230(S3×D4), (C2×C24).170C22, (C2×C12).300C23, C4.29(Q83S3), C33(C23.48D4), C2.16(D6.D4), C4⋊Dic3.125C22, (C2×Dic6).89C22, C6.46(C22.D4), (S3×C4⋊C4).9C2, (C3×C2.D8)⋊12C2, (C2×C6).305(C2×D4), (C2×C3⋊C8).70C22, (S3×C2×C4).39C22, (C3×C4⋊C4).93C22, (C2×C4).403(C22×S3), SmallGroup(192,443)

Series: Derived Chief Lower central Upper central

C1C2×C12 — D6.2Q16
C1C3C6C2×C6C2×C12S3×C2×C4S3×C4⋊C4 — D6.2Q16
C3C6C2×C12 — D6.2Q16
C1C22C2×C4C2.D8

Generators and relations for D6.2Q16
 G = < a,b,c,d | a6=b2=c8=1, d2=a3c4, bab=a-1, ac=ca, ad=da, cbc-1=a3b, bd=db, dcd-1=c-1 >

Subgroups: 304 in 104 conjugacy classes, 39 normal (37 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C8, C2×C4, C2×C4, Q8, C23, Dic3, C12, C12, D6, D6, C2×C6, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C2×Q8, C3⋊C8, C24, Dic6, C4×S3, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×S3, C22⋊C8, Q8⋊C4, C2.D8, C2.D8, C2×C4⋊C4, C22⋊Q8, C2×C3⋊C8, Dic3⋊C4, C4⋊Dic3, C4⋊Dic3, D6⋊C4, C3×C4⋊C4, C2×C24, C2×Dic6, S3×C2×C4, S3×C2×C4, C23.48D4, C6.Q16, C6.SD16, C2.Dic12, D6⋊C8, C3×C2.D8, S3×C4⋊C4, C4.D12, D6.2Q16
Quotients: C1, C2, C22, S3, D4, C23, D6, Q16, C2×D4, C4○D4, C22×S3, C22.D4, C2×Q16, C8⋊C22, C4○D12, S3×D4, Q83S3, C23.48D4, D6.D4, D8⋊S3, S3×Q16, D6.2Q16

Smallest permutation representation of D6.2Q16
On 96 points
Generators in S96
(1 42 19 73 49 62)(2 43 20 74 50 63)(3 44 21 75 51 64)(4 45 22 76 52 57)(5 46 23 77 53 58)(6 47 24 78 54 59)(7 48 17 79 55 60)(8 41 18 80 56 61)(9 35 32 65 87 89)(10 36 25 66 88 90)(11 37 26 67 81 91)(12 38 27 68 82 92)(13 39 28 69 83 93)(14 40 29 70 84 94)(15 33 30 71 85 95)(16 34 31 72 86 96)
(1 58)(2 24)(3 60)(4 18)(5 62)(6 20)(7 64)(8 22)(9 93)(10 29)(11 95)(12 31)(13 89)(14 25)(15 91)(16 27)(17 75)(19 77)(21 79)(23 73)(26 71)(28 65)(30 67)(32 69)(33 81)(34 38)(35 83)(36 40)(37 85)(39 87)(41 45)(42 53)(43 47)(44 55)(46 49)(48 51)(50 54)(52 56)(57 80)(59 74)(61 76)(63 78)(66 94)(68 96)(70 90)(72 92)(82 86)(84 88)
(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 13 77 65)(2 12 78 72)(3 11 79 71)(4 10 80 70)(5 9 73 69)(6 16 74 68)(7 15 75 67)(8 14 76 66)(17 30 64 91)(18 29 57 90)(19 28 58 89)(20 27 59 96)(21 26 60 95)(22 25 61 94)(23 32 62 93)(24 31 63 92)(33 51 81 48)(34 50 82 47)(35 49 83 46)(36 56 84 45)(37 55 85 44)(38 54 86 43)(39 53 87 42)(40 52 88 41)

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

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

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

33 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C4D4E4F4G4H4I6A6B6C8A8B8C8D12A12B12C12D12E12F24A24B24C24D
order1222223444444444666888812121212121224242424
size111166222448121212242224412124488884444

33 irreducible representations

dim111111112222222244444
type+++++++++++++-+++-
imageC1C2C2C2C2C2C2C2S3D4D4D6D6C4○D4Q16C4○D12C8⋊C22Q83S3S3×D4D8⋊S3S3×Q16
kernelD6.2Q16C6.Q16C6.SD16C2.Dic12D6⋊C8C3×C2.D8S3×C4⋊C4C4.D12C2.D8C2×Dic3C22×S3C4⋊C4C2×C8C12D6C4C6C4C22C2C2
# reps111111111112144411122

Matrix representation of D6.2Q16 in GL4(𝔽73) generated by

1100
72000
0010
0001
,
1100
07200
00720
00072
,
306000
134300
005148
00063
,
46000
04600
00734
002066
G:=sub<GL(4,GF(73))| [1,72,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,1,72,0,0,0,0,72,0,0,0,0,72],[30,13,0,0,60,43,0,0,0,0,51,0,0,0,48,63],[46,0,0,0,0,46,0,0,0,0,7,20,0,0,34,66] >;

D6.2Q16 in GAP, Magma, Sage, TeX

D_6._2Q_{16}
% in TeX

G:=Group("D6.2Q16");
// GroupNames label

G:=SmallGroup(192,443);
// by ID

G=gap.SmallGroup(192,443);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,64,926,219,268,851,102,6278]);
// Polycyclic

G:=Group<a,b,c,d|a^6=b^2=c^8=1,d^2=a^3*c^4,b*a*b=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

׿
×
𝔽