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G = D6.1SD16order 192 = 26·3

1st non-split extension by D6 of SD16 acting via SD16/D4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D6.1SD16, D6⋊C8.5C2, C4⋊C4.151D6, C8⋊Dic314C2, (C2×C8).122D6, (C2×Q8).39D6, Q8⋊C410S3, Q82Dic37C2, D63Q8.1C2, C6.30(C2×SD16), C2.16(S3×SD16), C4.55(C4○D12), C12.Q810C2, (C2×Dic3).29D4, (C22×S3).76D4, C22.195(S3×D4), (C6×Q8).28C22, C12.161(C4○D4), C4.86(D42S3), (C2×C12).245C23, (C2×C24).133C22, C2.13(Q16⋊S3), C6.59(C8.C22), C32(C23.47D4), C4⋊Dic3.93C22, C2.16(C23.9D6), C6.24(C22.D4), (S3×C4⋊C4).2C2, (C2×C6).258(C2×D4), (C2×C3⋊C8).37C22, (S3×C2×C4).19C22, (C3×Q8⋊C4)⋊10C2, (C3×C4⋊C4).46C22, (C2×C4).352(C22×S3), SmallGroup(192,364)

Series: Derived Chief Lower central Upper central

C1C2×C12 — D6.1SD16
C1C3C6C2×C6C2×C12S3×C2×C4S3×C4⋊C4 — D6.1SD16
C3C6C2×C12 — D6.1SD16
C1C22C2×C4Q8⋊C4

Generators and relations for D6.1SD16
 G = < a,b,c,d | a6=b2=c8=1, d2=a3, bab=a-1, ac=ca, ad=da, cbc-1=a3b, bd=db, dcd-1=a3c3 >

Subgroups: 296 in 104 conjugacy classes, 39 normal (37 characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×2], C4 [×5], C22, C22 [×4], S3 [×2], C6 [×3], C8 [×2], C2×C4, C2×C4 [×9], Q8 [×2], C23, Dic3 [×3], C12 [×2], C12 [×2], D6 [×2], D6 [×2], C2×C6, C22⋊C4, C4⋊C4, C4⋊C4 [×4], C2×C8, C2×C8, C22×C4 [×2], C2×Q8, C3⋊C8, C24, C4×S3 [×4], C2×Dic3, C2×Dic3 [×2], C2×C12, C2×C12 [×2], C3×Q8 [×2], C22×S3, C22⋊C8, Q8⋊C4, Q8⋊C4, C4.Q8 [×2], C2×C4⋊C4, C22⋊Q8, C2×C3⋊C8, Dic3⋊C4 [×2], C4⋊Dic3 [×2], D6⋊C4, C3×C4⋊C4, C2×C24, S3×C2×C4, S3×C2×C4, C6×Q8, C23.47D4, C12.Q8, C8⋊Dic3, D6⋊C8, Q82Dic3, C3×Q8⋊C4, S3×C4⋊C4, D63Q8, D6.1SD16
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D6 [×3], SD16 [×2], C2×D4, C4○D4 [×2], C22×S3, C22.D4, C2×SD16, C8.C22, C4○D12, S3×D4, D42S3, C23.47D4, C23.9D6, S3×SD16, Q16⋊S3, D6.1SD16

Smallest permutation representation of D6.1SD16
On 96 points
Generators in S96
(1 24 58 29 80 85)(2 17 59 30 73 86)(3 18 60 31 74 87)(4 19 61 32 75 88)(5 20 62 25 76 81)(6 21 63 26 77 82)(7 22 64 27 78 83)(8 23 57 28 79 84)(9 37 42 52 67 95)(10 38 43 53 68 96)(11 39 44 54 69 89)(12 40 45 55 70 90)(13 33 46 56 71 91)(14 34 47 49 72 92)(15 35 48 50 65 93)(16 36 41 51 66 94)
(1 81)(2 63)(3 83)(4 57)(5 85)(6 59)(7 87)(8 61)(9 46)(10 92)(11 48)(12 94)(13 42)(14 96)(15 44)(16 90)(17 21)(18 78)(19 23)(20 80)(22 74)(24 76)(25 58)(26 86)(27 60)(28 88)(29 62)(30 82)(31 64)(32 84)(33 37)(34 68)(35 39)(36 70)(38 72)(40 66)(41 55)(43 49)(45 51)(47 53)(50 89)(52 91)(54 93)(56 95)(65 69)(67 71)(73 77)(75 79)
(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 16 29 51)(2 54 30 11)(3 14 31 49)(4 52 32 9)(5 12 25 55)(6 50 26 15)(7 10 27 53)(8 56 28 13)(17 69 73 39)(18 34 74 72)(19 67 75 37)(20 40 76 70)(21 65 77 35)(22 38 78 68)(23 71 79 33)(24 36 80 66)(41 85 94 58)(42 61 95 88)(43 83 96 64)(44 59 89 86)(45 81 90 62)(46 57 91 84)(47 87 92 60)(48 63 93 82)

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

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

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

33 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C4D4E4F4G4H4I6A6B6C8A8B8C8D12A12B12C12D12E12F24A24B24C24D
order1222223444444444666888812121212121224242424
size111166222448121212242224412124488884444

33 irreducible representations

dim1111111122222222244444
type++++++++++++++--+
imageC1C2C2C2C2C2C2C2S3D4D4D6D6D6C4○D4SD16C4○D12C8.C22D42S3S3×D4S3×SD16Q16⋊S3
kernelD6.1SD16C12.Q8C8⋊Dic3D6⋊C8Q82Dic3C3×Q8⋊C4S3×C4⋊C4D63Q8Q8⋊C4C2×Dic3C22×S3C4⋊C4C2×C8C2×Q8C12D6C4C6C4C22C2C2
# reps1111111111111144411122

Matrix representation of D6.1SD16 in GL6(𝔽73)

72720000
100000
0072000
0007200
000010
000001
,
72720000
010000
0072000
0025100
0000720
0000072
,
7200000
0720000
00466500
0002700
00001261
000060
,
7200000
0720000
0046000
0004600
00005834
0000215

G:=sub<GL(6,GF(73))| [72,1,0,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[72,0,0,0,0,0,72,1,0,0,0,0,0,0,72,25,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,46,0,0,0,0,0,65,27,0,0,0,0,0,0,12,6,0,0,0,0,61,0],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,46,0,0,0,0,0,0,46,0,0,0,0,0,0,58,2,0,0,0,0,34,15] >;

D6.1SD16 in GAP, Magma, Sage, TeX

D_6._1{\rm SD}_{16}
% in TeX

G:=Group("D6.1SD16");
// GroupNames label

G:=SmallGroup(192,364);
// by ID

G=gap.SmallGroup(192,364);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,64,254,219,184,851,438,102,6278]);
// Polycyclic

G:=Group<a,b,c,d|a^6=b^2=c^8=1,d^2=a^3,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=a^3*c^3>;
// generators/relations

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