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## G = M4(2).49D4order 128 = 27

### 13rd non-split extension by M4(2) of D4 acting via D4/C22=C2

p-group, metabelian, nilpotent (class 3), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — M4(2).49D4
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C2×C4○D4 — C2×C8○D4 — M4(2).49D4
 Lower central C1 — C2 — C2×C4 — M4(2).49D4
 Upper central C1 — C22 — C22×C4 — M4(2).49D4
 Jennings C1 — C2 — C2 — C22×C4 — M4(2).49D4

Generators and relations for M4(2).49D4
G = < a,b,c,d | a8=b2=c4=1, d2=a2b, bab=a5, cac-1=a5b, dad-1=ab, bc=cb, dbd-1=a4b, dcd-1=a2bc-1 >

Subgroups: 292 in 156 conjugacy classes, 56 normal (38 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, C4⋊C4, C2×C8, C2×C8, M4(2), M4(2), SD16, Q16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C2×Q8, C4○D4, C4○D4, C4.10D4, D4⋊C4, Q8⋊C4, C2×C4⋊C4, C22×C8, C22×C8, C2×M4(2), C2×M4(2), C8○D4, C2×SD16, C2×Q16, C8.C22, C8.C22, C22×Q8, C2×C4○D4, C4.C42, C22.C42, C2×C4.10D4, C2×Q8⋊C4, C23.36D4, C2×C8○D4, C2×C8.C22, M4(2).49D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C4○D4, C2×C22⋊C4, C4×D4, C22≀C2, C4⋊D4, C22.D4, C23.23D4, D4.3D4, D4.5D4, M4(2).49D4

Smallest permutation representation of M4(2).49D4
On 64 points
Generators in S64
```(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)
(1 49)(2 54)(3 51)(4 56)(5 53)(6 50)(7 55)(8 52)(9 60)(10 57)(11 62)(12 59)(13 64)(14 61)(15 58)(16 63)(17 26)(18 31)(19 28)(20 25)(21 30)(22 27)(23 32)(24 29)(33 47)(34 44)(35 41)(36 46)(37 43)(38 48)(39 45)(40 42)
(1 30 48 13)(2 22 41 57)(3 28 42 11)(4 20 43 63)(5 26 44 9)(6 18 45 61)(7 32 46 15)(8 24 47 59)(10 54 27 35)(12 52 29 33)(14 50 31 39)(16 56 25 37)(17 34 60 53)(19 40 62 51)(21 38 64 49)(23 36 58 55)
(1 58 51 9 5 62 55 13)(2 16 56 57 6 12 52 61)(3 64 53 15 7 60 49 11)(4 14 50 63 8 10 54 59)(17 38 28 42 21 34 32 46)(18 41 25 37 22 45 29 33)(19 36 30 48 23 40 26 44)(20 47 27 35 24 43 31 39)```

`G:=sub<Sym(64)| (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), (1,49)(2,54)(3,51)(4,56)(5,53)(6,50)(7,55)(8,52)(9,60)(10,57)(11,62)(12,59)(13,64)(14,61)(15,58)(16,63)(17,26)(18,31)(19,28)(20,25)(21,30)(22,27)(23,32)(24,29)(33,47)(34,44)(35,41)(36,46)(37,43)(38,48)(39,45)(40,42), (1,30,48,13)(2,22,41,57)(3,28,42,11)(4,20,43,63)(5,26,44,9)(6,18,45,61)(7,32,46,15)(8,24,47,59)(10,54,27,35)(12,52,29,33)(14,50,31,39)(16,56,25,37)(17,34,60,53)(19,40,62,51)(21,38,64,49)(23,36,58,55), (1,58,51,9,5,62,55,13)(2,16,56,57,6,12,52,61)(3,64,53,15,7,60,49,11)(4,14,50,63,8,10,54,59)(17,38,28,42,21,34,32,46)(18,41,25,37,22,45,29,33)(19,36,30,48,23,40,26,44)(20,47,27,35,24,43,31,39)>;`

`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), (1,49)(2,54)(3,51)(4,56)(5,53)(6,50)(7,55)(8,52)(9,60)(10,57)(11,62)(12,59)(13,64)(14,61)(15,58)(16,63)(17,26)(18,31)(19,28)(20,25)(21,30)(22,27)(23,32)(24,29)(33,47)(34,44)(35,41)(36,46)(37,43)(38,48)(39,45)(40,42), (1,30,48,13)(2,22,41,57)(3,28,42,11)(4,20,43,63)(5,26,44,9)(6,18,45,61)(7,32,46,15)(8,24,47,59)(10,54,27,35)(12,52,29,33)(14,50,31,39)(16,56,25,37)(17,34,60,53)(19,40,62,51)(21,38,64,49)(23,36,58,55), (1,58,51,9,5,62,55,13)(2,16,56,57,6,12,52,61)(3,64,53,15,7,60,49,11)(4,14,50,63,8,10,54,59)(17,38,28,42,21,34,32,46)(18,41,25,37,22,45,29,33)(19,36,30,48,23,40,26,44)(20,47,27,35,24,43,31,39) );`

`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)], [(1,49),(2,54),(3,51),(4,56),(5,53),(6,50),(7,55),(8,52),(9,60),(10,57),(11,62),(12,59),(13,64),(14,61),(15,58),(16,63),(17,26),(18,31),(19,28),(20,25),(21,30),(22,27),(23,32),(24,29),(33,47),(34,44),(35,41),(36,46),(37,43),(38,48),(39,45),(40,42)], [(1,30,48,13),(2,22,41,57),(3,28,42,11),(4,20,43,63),(5,26,44,9),(6,18,45,61),(7,32,46,15),(8,24,47,59),(10,54,27,35),(12,52,29,33),(14,50,31,39),(16,56,25,37),(17,34,60,53),(19,40,62,51),(21,38,64,49),(23,36,58,55)], [(1,58,51,9,5,62,55,13),(2,16,56,57,6,12,52,61),(3,64,53,15,7,60,49,11),(4,14,50,63,8,10,54,59),(17,38,28,42,21,34,32,46),(18,41,25,37,22,45,29,33),(19,36,30,48,23,40,26,44),(20,47,27,35,24,43,31,39)]])`

32 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 8A 8B 8C 8D 8E ··· 8J 8K 8L 8M 8N order 1 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 4 4 8 8 8 8 8 ··· 8 8 8 8 8 size 1 1 1 1 2 2 4 4 2 2 2 2 4 4 8 8 8 8 2 2 2 2 4 ··· 4 8 8 8 8

32 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 C2 C4 D4 D4 D4 D4 D4 C4○D4 C4○D4 D4.3D4 D4.5D4 kernel M4(2).49D4 C4.C42 C22.C42 C2×C4.10D4 C2×Q8⋊C4 C23.36D4 C2×C8○D4 C2×C8.C22 C8.C22 C2×C8 M4(2) C2×D4 C2×Q8 C4○D4 C2×C4 C23 C2 C2 # reps 1 1 1 1 1 1 1 1 8 2 2 1 1 2 2 2 2 2

Matrix representation of M4(2).49D4 in GL6(𝔽17)

 1 0 0 0 0 0 0 16 0 0 0 0 0 0 12 12 12 0 0 0 0 0 0 12 0 0 5 12 15 5 0 0 0 10 14 7
,
 16 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 1 0 1 0 0 0 1 1 0 0 0 0 0 15 2 1
,
 0 13 0 0 0 0 13 0 0 0 0 0 0 0 0 16 1 0 0 0 16 1 15 16 0 0 0 1 15 16 0 0 16 16 1 1
,
 0 4 0 0 0 0 4 0 0 0 0 0 0 0 1 0 2 1 0 0 16 1 15 16 0 0 16 0 16 0 0 0 1 1 16 16

`G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,16,0,0,0,0,0,0,12,0,5,0,0,0,12,0,12,10,0,0,12,0,15,14,0,0,0,12,5,7],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,1,1,0,0,0,0,0,1,15,0,0,0,1,0,2,0,0,0,0,0,1],[0,13,0,0,0,0,13,0,0,0,0,0,0,0,0,16,0,16,0,0,16,1,1,16,0,0,1,15,15,1,0,0,0,16,16,1],[0,4,0,0,0,0,4,0,0,0,0,0,0,0,1,16,16,1,0,0,0,1,0,1,0,0,2,15,16,16,0,0,1,16,0,16] >;`

M4(2).49D4 in GAP, Magma, Sage, TeX

`M_4(2)._{49}D_4`
`% in TeX`

`G:=Group("M4(2).49D4");`
`// GroupNames label`

`G:=SmallGroup(128,640);`
`// by ID`

`G=gap.SmallGroup(128,640);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,736,422,2019,1018,521,1411,718,172,2028,1027,124]);`
`// Polycyclic`

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

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