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## G = M4(2).30D4order 128 = 27

### 11st non-split extension by M4(2) of D4 acting via D4/C4=C2

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — M4(2).30D4
 Chief series C1 — C2 — C22 — C2×C4 — C22×C4 — C22×C8 — C8○2M4(2) — M4(2).30D4
 Lower central C1 — C2 — C2×C4 — M4(2).30D4
 Upper central C1 — C4 — C22×C4 — M4(2).30D4
 Jennings C1 — C2 — C2 — C22×C4 — M4(2).30D4

Generators and relations for M4(2).30D4
G = < a,b,c,d | a8=b2=c4=d2=1, bab=cac-1=a5, dad=a3b, cbc-1=dbd=a4b, dcd=bc-1 >

Subgroups: 300 in 142 conjugacy classes, 54 normal (36 characteristic)
C1, C2, C2 [×5], C4 [×4], C4 [×6], C22 [×3], C22 [×5], C8 [×4], C8 [×2], C2×C4 [×6], C2×C4 [×10], D4 [×8], Q8 [×4], C23, C23 [×2], C42 [×3], C22⋊C4 [×2], C22⋊C4 [×2], C4⋊C4 [×3], C2×C8 [×6], C2×C8, M4(2) [×2], M4(2), D8 [×2], SD16 [×4], Q16 [×2], C22×C4, C22×C4 [×2], C2×D4 [×2], C2×D4 [×2], C2×Q8 [×2], C4○D4 [×8], C4×C8, C8⋊C4, C23⋊C4 [×4], C4≀C2 [×4], C4.Q8, C2.D8, C42⋊C2, C42⋊C2 [×2], C22×C8, C2×M4(2), C2×D8, C2×SD16 [×2], C2×Q16, C4○D8 [×4], C2×C4○D4 [×2], C82M4(2), C23.C23 [×2], C42⋊C22 [×2], C23.25D4, C2×C4○D8, M4(2).30D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×8], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×4], C4○D4 [×2], C2×C22⋊C4, C4×D4 [×2], C4⋊D4 [×2], C4.4D4, C41D4, C24.3C22, M4(2).30D4

Smallest permutation representation of M4(2).30D4
On 32 points
Generators in S32
```(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)
(1 5)(3 7)(9 13)(11 15)(17 21)(19 23)(26 30)(28 32)
(1 22 30 12)(2 19 31 9)(3 24 32 14)(4 21 25 11)(5 18 26 16)(6 23 27 13)(7 20 28 10)(8 17 29 15)
(1 25)(2 32)(3 27)(4 26)(5 29)(6 28)(7 31)(8 30)(9 14)(10 13)(11 16)(12 15)(17 18)(19 20)(21 22)(23 24)```

`G:=sub<Sym(32)| (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), (1,5)(3,7)(9,13)(11,15)(17,21)(19,23)(26,30)(28,32), (1,22,30,12)(2,19,31,9)(3,24,32,14)(4,21,25,11)(5,18,26,16)(6,23,27,13)(7,20,28,10)(8,17,29,15), (1,25)(2,32)(3,27)(4,26)(5,29)(6,28)(7,31)(8,30)(9,14)(10,13)(11,16)(12,15)(17,18)(19,20)(21,22)(23,24)>;`

`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), (1,5)(3,7)(9,13)(11,15)(17,21)(19,23)(26,30)(28,32), (1,22,30,12)(2,19,31,9)(3,24,32,14)(4,21,25,11)(5,18,26,16)(6,23,27,13)(7,20,28,10)(8,17,29,15), (1,25)(2,32)(3,27)(4,26)(5,29)(6,28)(7,31)(8,30)(9,14)(10,13)(11,16)(12,15)(17,18)(19,20)(21,22)(23,24) );`

`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)], [(1,5),(3,7),(9,13),(11,15),(17,21),(19,23),(26,30),(28,32)], [(1,22,30,12),(2,19,31,9),(3,24,32,14),(4,21,25,11),(5,18,26,16),(6,23,27,13),(7,20,28,10),(8,17,29,15)], [(1,25),(2,32),(3,27),(4,26),(5,29),(6,28),(7,31),(8,30),(9,14),(10,13),(11,16),(12,15),(17,18),(19,20),(21,22),(23,24)])`

32 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J ··· 4O 8A 8B 8C 8D 8E ··· 8J order 1 2 2 2 2 2 2 4 4 4 4 4 4 4 4 4 4 ··· 4 8 8 8 8 8 ··· 8 size 1 1 2 2 2 8 8 1 1 2 2 2 4 4 4 4 8 ··· 8 2 2 2 2 4 ··· 4

32 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 4 type + + + + + + + + + image C1 C2 C2 C2 C2 C2 C4 C4 C4 D4 D4 D4 C4○D4 M4(2).30D4 kernel M4(2).30D4 C8○2M4(2) C23.C23 C42⋊C22 C23.25D4 C2×C4○D8 C2×D8 C2×SD16 C2×Q16 C22⋊C4 C2×C8 M4(2) C2×C4 C1 # reps 1 1 2 2 1 1 2 4 2 2 4 2 4 4

Matrix representation of M4(2).30D4 in GL4(𝔽17) generated by

 0 0 12 5 0 0 12 12 14 14 0 0 3 14 0 0
,
 16 0 0 0 0 16 0 0 0 0 1 0 0 0 0 1
,
 0 0 0 13 0 0 4 0 1 0 0 0 0 1 0 0
,
 0 0 3 14 0 0 14 14 3 14 0 0 14 14 0 0
`G:=sub<GL(4,GF(17))| [0,0,14,3,0,0,14,14,12,12,0,0,5,12,0,0],[16,0,0,0,0,16,0,0,0,0,1,0,0,0,0,1],[0,0,1,0,0,0,0,1,0,4,0,0,13,0,0,0],[0,0,3,14,0,0,14,14,3,14,0,0,14,14,0,0] >;`

M4(2).30D4 in GAP, Magma, Sage, TeX

`M_4(2)._{30}D_4`
`% in TeX`

`G:=Group("M4(2).30D4");`
`// GroupNames label`

`G:=SmallGroup(128,708);`
`// by ID`

`G=gap.SmallGroup(128,708);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,288,422,100,2019,1018,2804,172,2028]);`
`// Polycyclic`

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

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