Copied to
clipboard

## G = M4(2).16D6order 192 = 26·3

### 16th non-split extension by M4(2) of D6 acting via D6/C3=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C12 — M4(2).16D6
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C2×Dic6 — Q8.14D6 — M4(2).16D6
 Lower central C3 — C6 — C2×C12 — M4(2).16D6
 Upper central C1 — C2 — C2×C4 — C8.C22

Generators and relations for M4(2).16D6
G = < a,b,c,d | a8=b2=c6=1, d2=a6, bab=a5, cac-1=a3, dad-1=a3b, cbc-1=dbd-1=a4b, dcd-1=a2c-1 >

Subgroups: 240 in 100 conjugacy classes, 37 normal (all characteristic)
C1, C2, C2 [×2], C3, C4 [×2], C4 [×3], C22, C22, C6, C6 [×2], C8 [×5], C2×C4, C2×C4 [×3], D4, D4, Q8, Q8 [×4], Dic3, C12 [×2], C12 [×2], C2×C6, C2×C6, C2×C8 [×2], M4(2), M4(2) [×3], SD16 [×2], Q16 [×4], C2×Q8, C2×Q8, C4○D4, C3⋊C8 [×2], C3⋊C8 [×2], C24, Dic6 [×2], C2×Dic3, C2×C12, C2×C12 [×2], C3×D4, C3×D4, C3×Q8, C3×Q8 [×2], C4.10D4 [×2], C8.C4, C8○D4, C2×Q16, C8.C22, C8.C22, C2×C3⋊C8, C2×C3⋊C8, C4.Dic3 [×2], C4.Dic3, D4.S3, C3⋊Q16 [×3], C3×M4(2), C3×SD16, C3×Q16, C2×Dic6, C6×Q8, C3×C4○D4, D4.5D4, C12.53D4, C12.47D4, C12.10D4, C2×C3⋊Q16, D4.Dic3, Q8.14D6, C3×C8.C22, M4(2).16D6
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×4], C23, D6 [×3], C2×D4 [×2], C4○D4, C3⋊D4 [×2], C22×S3, C4⋊D4, S3×D4, D42S3, C2×C3⋊D4, D4.5D4, C23.14D6, M4(2).16D6

Character table of M4(2).16D6

 class 1 2A 2B 2C 3 4A 4B 4C 4D 4E 6A 6B 6C 8A 8B 8C 8D 8E 8F 8G 12A 12B 12C 12D 12E 24A 24B size 1 1 2 4 2 2 2 4 8 24 2 4 8 6 6 8 12 12 12 24 4 4 8 8 8 8 8 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 -1 1 1 1 -1 1 1 1 1 -1 1 1 -1 -1 1 -1 -1 1 1 1 1 -1 -1 -1 linear of order 2 ρ3 1 1 1 -1 1 1 1 -1 1 -1 1 1 -1 -1 -1 -1 1 -1 1 1 1 1 1 1 -1 -1 -1 linear of order 2 ρ4 1 1 1 1 1 1 1 1 1 -1 1 1 1 -1 -1 1 -1 -1 -1 -1 1 1 1 1 1 1 1 linear of order 2 ρ5 1 1 1 -1 1 1 1 -1 -1 -1 1 1 -1 1 1 1 -1 1 -1 1 1 1 -1 -1 -1 1 1 linear of order 2 ρ6 1 1 1 1 1 1 1 1 -1 -1 1 1 1 1 1 -1 1 1 1 -1 1 1 -1 -1 1 -1 -1 linear of order 2 ρ7 1 1 1 -1 1 1 1 -1 -1 1 1 1 -1 -1 -1 1 1 -1 1 -1 1 1 -1 -1 -1 1 1 linear of order 2 ρ8 1 1 1 1 1 1 1 1 -1 1 1 1 1 -1 -1 -1 -1 -1 -1 1 1 1 -1 -1 1 -1 -1 linear of order 2 ρ9 2 2 -2 2 2 -2 2 -2 0 0 2 -2 2 0 0 0 0 0 0 0 -2 2 0 0 -2 0 0 orthogonal lifted from D4 ρ10 2 2 2 2 -1 2 2 2 2 0 -1 -1 -1 0 0 2 0 0 0 0 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ11 2 2 2 2 -1 2 2 2 -2 0 -1 -1 -1 0 0 -2 0 0 0 0 -1 -1 1 1 -1 1 1 orthogonal lifted from D6 ρ12 2 2 -2 -2 2 -2 2 2 0 0 2 -2 -2 0 0 0 0 0 0 0 -2 2 0 0 2 0 0 orthogonal lifted from D4 ρ13 2 2 -2 0 2 2 -2 0 0 0 2 -2 0 -2 -2 0 0 2 0 0 2 -2 0 0 0 0 0 orthogonal lifted from D4 ρ14 2 2 2 -2 -1 2 2 -2 2 0 -1 -1 1 0 0 -2 0 0 0 0 -1 -1 -1 -1 1 1 1 orthogonal lifted from D6 ρ15 2 2 2 -2 -1 2 2 -2 -2 0 -1 -1 1 0 0 2 0 0 0 0 -1 -1 1 1 1 -1 -1 orthogonal lifted from D6 ρ16 2 2 -2 0 2 2 -2 0 0 0 2 -2 0 2 2 0 0 -2 0 0 2 -2 0 0 0 0 0 orthogonal lifted from D4 ρ17 2 2 -2 -2 -1 -2 2 2 0 0 -1 1 1 0 0 0 0 0 0 0 1 -1 -√-3 √-3 -1 √-3 -√-3 complex lifted from C3⋊D4 ρ18 2 2 -2 2 -1 -2 2 -2 0 0 -1 1 -1 0 0 0 0 0 0 0 1 -1 √-3 -√-3 1 √-3 -√-3 complex lifted from C3⋊D4 ρ19 2 2 -2 2 -1 -2 2 -2 0 0 -1 1 -1 0 0 0 0 0 0 0 1 -1 -√-3 √-3 1 -√-3 √-3 complex lifted from C3⋊D4 ρ20 2 2 -2 -2 -1 -2 2 2 0 0 -1 1 1 0 0 0 0 0 0 0 1 -1 √-3 -√-3 -1 -√-3 √-3 complex lifted from C3⋊D4 ρ21 2 2 2 0 2 -2 -2 0 0 0 2 2 0 0 0 0 2i 0 -2i 0 -2 -2 0 0 0 0 0 complex lifted from C4○D4 ρ22 2 2 2 0 2 -2 -2 0 0 0 2 2 0 0 0 0 -2i 0 2i 0 -2 -2 0 0 0 0 0 complex lifted from C4○D4 ρ23 4 4 -4 0 -2 4 -4 0 0 0 -2 2 0 0 0 0 0 0 0 0 -2 2 0 0 0 0 0 orthogonal lifted from S3×D4 ρ24 4 4 4 0 -2 -4 -4 0 0 0 -2 -2 0 0 0 0 0 0 0 0 2 2 0 0 0 0 0 symplectic lifted from D4⋊2S3, Schur index 2 ρ25 4 -4 0 0 4 0 0 0 0 0 -4 0 0 2√2 -2√2 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from D4.5D4, Schur index 2 ρ26 4 -4 0 0 4 0 0 0 0 0 -4 0 0 -2√2 2√2 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from D4.5D4, Schur index 2 ρ27 8 -8 0 0 -4 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 symplectic faithful, Schur index 2

Smallest permutation representation of M4(2).16D6
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 17)(2 22)(3 19)(4 24)(5 21)(6 18)(7 23)(8 20)(9 86)(10 83)(11 88)(12 85)(13 82)(14 87)(15 84)(16 81)(25 51)(26 56)(27 53)(28 50)(29 55)(30 52)(31 49)(32 54)(33 91)(34 96)(35 93)(36 90)(37 95)(38 92)(39 89)(40 94)(41 57)(42 62)(43 59)(44 64)(45 61)(46 58)(47 63)(48 60)(65 79)(66 76)(67 73)(68 78)(69 75)(70 80)(71 77)(72 74)
(1 32 71)(2 27 72 4 25 66)(3 30 65 7 26 69)(5 28 67)(6 31 68 8 29 70)(9 91 63 15 93 61)(10 94 64)(11 89 57 13 95 59)(12 92 58 16 96 62)(14 90 60)(17 50 77 21 54 73)(18 53 78 24 55 76)(19 56 79)(20 51 80 22 49 74)(23 52 75)(33 43 84 39 45 82)(34 46 85)(35 41 86 37 47 88)(36 44 87 40 48 83)(38 42 81)
(1 13 7 11 5 9 3 15)(2 81 8 87 6 85 4 83)(10 18 16 24 14 22 12 20)(17 86 23 84 21 82 19 88)(25 42 31 48 29 46 27 44)(26 61 32 59 30 57 28 63)(33 73 39 79 37 77 35 75)(34 66 40 72 38 70 36 68)(41 54 47 52 45 50 43 56)(49 64 55 62 53 60 51 58)(65 91 71 89 69 95 67 93)(74 96 80 94 78 92 76 90)```

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

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

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

Matrix representation of M4(2).16D6 in GL8(𝔽73)

 0 0 1 72 0 0 0 0 1 1 2 1 0 0 0 0 48 48 72 0 0 0 0 0 49 48 72 0 0 0 0 0 0 0 0 0 0 0 27 0 0 0 0 0 0 0 0 46 0 0 0 0 0 27 0 0 0 0 0 0 27 0 0 0
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 27 0 0 0 0 0 0 46 0 0 0 0 0 0 0 0 0 0 27 0 0 0 0 0 0 46 0
,
 0 72 0 0 0 0 0 0 1 72 0 0 0 0 0 0 49 50 1 1 0 0 0 0 48 49 72 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0
,
 46 46 19 46 0 0 0 0 0 0 46 27 0 0 0 0 0 0 27 0 0 0 0 0 0 27 27 0 0 0 0 0 0 0 0 0 0 0 0 22 0 0 0 0 0 0 22 0 0 0 0 0 63 0 0 0 0 0 0 0 0 10 0 0

`G:=sub<GL(8,GF(73))| [0,1,48,49,0,0,0,0,0,1,48,48,0,0,0,0,1,2,72,72,0,0,0,0,72,1,0,0,0,0,0,0,0,0,0,0,0,0,0,27,0,0,0,0,0,0,27,0,0,0,0,0,27,0,0,0,0,0,0,0,0,46,0,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,46,0,0,0,0,0,0,27,0,0,0,0,0,0,0,0,0,0,46,0,0,0,0,0,0,27,0],[0,1,49,48,0,0,0,0,72,72,50,49,0,0,0,0,0,0,1,72,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[46,0,0,0,0,0,0,0,46,0,0,27,0,0,0,0,19,46,27,27,0,0,0,0,46,27,0,0,0,0,0,0,0,0,0,0,0,0,63,0,0,0,0,0,0,0,0,10,0,0,0,0,0,22,0,0,0,0,0,0,22,0,0,0] >;`

M4(2).16D6 in GAP, Magma, Sage, TeX

`M_4(2)._{16}D_6`
`% in TeX`

`G:=Group("M4(2).16D6");`
`// GroupNames label`

`G:=SmallGroup(192,763);`
`// by ID`

`G=gap.SmallGroup(192,763);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,253,254,219,184,1123,297,136,1684,851,438,102,6278]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^8=b^2=c^6=1,d^2=a^6,b*a*b=a^5,c*a*c^-1=a^3,d*a*d^-1=a^3*b,c*b*c^-1=d*b*d^-1=a^4*b,d*c*d^-1=a^2*c^-1>;`
`// generators/relations`

Export

׿
×
𝔽