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## G = C24.50D4order 128 = 27

### 5th non-split extension by C24 of D4 acting via D4/C2=C22

p-group, metabelian, nilpotent (class 2), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C23 — C24.50D4
 Chief series C1 — C2 — C22 — C23 — C24 — C25 — D4×C23 — C24.50D4
 Lower central C1 — C23 — C24.50D4
 Upper central C1 — C24 — C24.50D4
 Jennings C1 — C24 — C24.50D4

Generators and relations for C24.50D4
G = < a,b,c,d,e,f | a2=b2=c2=d2=e4=1, f2=c, eae-1=ab=ba, ac=ca, ad=da, faf-1=abd, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=ce-1 >

Subgroups: 1268 in 584 conjugacy classes, 124 normal (8 characteristic)
C1, C2, C2 [×14], C2 [×8], C4 [×12], C22, C22 [×34], C22 [×72], C2×C4 [×4], C2×C4 [×52], D4 [×32], C23, C23 [×22], C23 [×104], C22⋊C4 [×32], C22×C4 [×14], C22×C4 [×28], C2×D4 [×56], C24, C24 [×12], C24 [×24], C2.C42 [×4], C2×C22⋊C4 [×24], C23×C4, C23×C4 [×4], C22×D4 [×4], C22×D4 [×12], C25 [×2], C2×C2.C42 [×2], C22×C22⋊C4 [×4], D4×C23, C24.50D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×20], C23, C22⋊C4 [×12], C22×C4, C2×D4 [×10], C4○D4 [×4], C2×C22⋊C4 [×3], C4×D4 [×4], C22≀C2 [×8], C4⋊D4 [×12], C22.D4 [×4], C4.4D4 [×2], C41D4 [×2], C243C4, C23.23D4 [×4], C24.3C22 [×2], C232D4 [×4], C23.10D4 [×4], C24.50D4

Smallest permutation representation of C24.50D4
On 64 points
Generators in S64
```(1 45)(2 38)(3 47)(4 40)(5 12)(6 13)(7 10)(8 15)(9 19)(11 17)(14 20)(16 18)(21 31)(22 34)(23 29)(24 36)(25 33)(26 32)(27 35)(28 30)(37 43)(39 41)(42 48)(44 46)(49 59)(50 64)(51 57)(52 62)(53 63)(54 60)(55 61)(56 58)
(1 43)(2 44)(3 41)(4 42)(5 18)(6 19)(7 20)(8 17)(9 13)(10 14)(11 15)(12 16)(21 25)(22 26)(23 27)(24 28)(29 35)(30 36)(31 33)(32 34)(37 45)(38 46)(39 47)(40 48)(49 53)(50 54)(51 55)(52 56)(57 61)(58 62)(59 63)(60 64)
(1 9)(2 10)(3 11)(4 12)(5 40)(6 37)(7 38)(8 39)(13 43)(14 44)(15 41)(16 42)(17 47)(18 48)(19 45)(20 46)(21 51)(22 52)(23 49)(24 50)(25 55)(26 56)(27 53)(28 54)(29 59)(30 60)(31 57)(32 58)(33 61)(34 62)(35 63)(36 64)
(1 51)(2 52)(3 49)(4 50)(5 36)(6 33)(7 34)(8 35)(9 21)(10 22)(11 23)(12 24)(13 25)(14 26)(15 27)(16 28)(17 29)(18 30)(19 31)(20 32)(37 61)(38 62)(39 63)(40 64)(41 53)(42 54)(43 55)(44 56)(45 57)(46 58)(47 59)(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 10 9 2)(3 12 11 4)(5 35 40 63)(6 62 37 34)(7 33 38 61)(8 64 39 36)(13 44 43 14)(15 42 41 16)(17 60 47 30)(18 29 48 59)(19 58 45 32)(20 31 46 57)(21 52 51 22)(23 50 49 24)(25 56 55 26)(27 54 53 28)```

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

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

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

44 conjugacy classes

 class 1 2A ··· 2O 2P ··· 2W 4A ··· 4T order 1 2 ··· 2 2 ··· 2 4 ··· 4 size 1 1 ··· 1 4 ··· 4 4 ··· 4

44 irreducible representations

 dim 1 1 1 1 1 2 2 2 type + + + + + + image C1 C2 C2 C2 C4 D4 D4 C4○D4 kernel C24.50D4 C2×C2.C42 C22×C22⋊C4 D4×C23 C22×D4 C22×C4 C24 C23 # reps 1 2 4 1 8 12 8 8

Matrix representation of C24.50D4 in GL8(𝔽5)

 0 4 0 0 0 0 0 0 4 0 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 4 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 2 4
,
 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 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 4
,
 4 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 4 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 1
,
 4 0 0 0 0 0 0 0 0 4 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 4 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 3 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 2 2 0 0 0 0 0 0 1 3 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 4 1 0 0 0 0 0 0 3 1
,
 2 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 2 2 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 3 1 0 0 0 0 0 0 0 0 1 4 0 0 0 0 0 0 0 4

`G:=sub<GL(8,GF(5))| [0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,4,4,0,0,0,0,0,0,0,0,1,2,0,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,2,1,0,0,0,0,0,0,2,3,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,3,0,0,0,0,0,0,1,1],[2,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,2,3,0,0,0,0,0,0,0,0,4,3,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,4,4] >;`

C24.50D4 in GAP, Magma, Sage, TeX

`C_2^4._{50}D_4`
`% in TeX`

`G:=Group("C2^4.50D4");`
`// GroupNames label`

`G:=SmallGroup(128,170);`
`// by ID`

`G=gap.SmallGroup(128,170);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,2,2,448,141,422,387]);`
`// Polycyclic`

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

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