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## G = D4.M4(2)  order 128 = 27

### 2nd non-split extension by D4 of M4(2) acting via M4(2)/C8=C2

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — D4.M4(2)
 Chief series C1 — C2 — C22 — C2×C4 — C42 — C4×C8 — C8×D4 — D4.M4(2)
 Lower central C1 — C2 — C2×C4 — D4.M4(2)
 Upper central C1 — C2×C4 — C4×C8 — D4.M4(2)
 Jennings C1 — C22 — C22 — C42 — D4.M4(2)

Generators and relations for D4.M4(2)
G = < a,b,c,d | a4=b2=c8=1, d2=a2, bab=cac-1=dad-1=a-1, cbc-1=dbd-1=ab, dcd-1=a2c5 >

Subgroups: 152 in 84 conjugacy classes, 42 normal (40 characteristic)
C1, C2 [×3], C2 [×2], C4 [×4], C4 [×4], C22, C22 [×4], C8 [×8], C2×C4 [×3], C2×C4 [×5], D4 [×2], D4, Q8 [×2], C23, C42, C42, C22⋊C4, C4⋊C4 [×2], C4⋊C4, C2×C8 [×4], C2×C8 [×5], SD16 [×2], C22×C4, C2×D4, C2×Q8, C4×C8 [×3], C8⋊C4, C22⋊C8, D4⋊C4, Q8⋊C4, C4⋊C8 [×2], C4⋊C8, C4.Q8, C4×D4, C4×Q8, C22×C8, C2×SD16, C8⋊C8, D4⋊C8, Q8⋊C8, C81C8, C8×D4, C4×SD16, C84Q8, D4.M4(2)
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], C23, M4(2) [×2], C22×C4, C2×D4, C4○D4, C4×D4, C2×M4(2), C8○D4, C8⋊C22, C8.C22, C89D4, SD16⋊C4, C8○D8, D4.M4(2)

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

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

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

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

38 conjugacy classes

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

38 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 4 4 type + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 C2 C4 C4 C4 C4 D4 C4○D4 M4(2) C8○D4 C8○D8 C8⋊C22 C8.C22 kernel D4.M4(2) C8⋊C8 D4⋊C8 Q8⋊C8 C8⋊1C8 C8×D4 C4×SD16 C8⋊4Q8 D4⋊C4 Q8⋊C4 C4.Q8 C2×SD16 C2×C8 C2×C4 D4 C4 C2 C4 C4 # reps 1 1 1 1 1 1 1 1 2 2 2 2 2 2 4 4 8 1 1

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

 0 1 0 0 16 0 0 0 0 0 16 0 0 0 0 16
,
 0 16 0 0 16 0 0 0 0 0 1 0 0 0 3 16
,
 10 7 0 0 7 7 0 0 0 0 5 8 0 0 8 12
,
 5 12 0 0 12 12 0 0 0 0 10 16 0 0 14 7
`G:=sub<GL(4,GF(17))| [0,16,0,0,1,0,0,0,0,0,16,0,0,0,0,16],[0,16,0,0,16,0,0,0,0,0,1,3,0,0,0,16],[10,7,0,0,7,7,0,0,0,0,5,8,0,0,8,12],[5,12,0,0,12,12,0,0,0,0,10,14,0,0,16,7] >;`

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

`D_4.M_4(2)`
`% in TeX`

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

`G:=SmallGroup(128,317);`
`// by ID`

`G=gap.SmallGroup(128,317);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,2102,100,1123,570,136,172]);`
`// Polycyclic`

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

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