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## G = C24.133D4order 128 = 27

### 2nd non-split extension by C24 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 — C24.133D4
 Chief series C1 — C2 — C22 — C23 — C22×C4 — C23×C4 — C23×C8 — C24.133D4
 Lower central C1 — C2 — C2×C4 — C24.133D4
 Upper central C1 — C23 — C23×C4 — C24.133D4
 Jennings C1 — C2 — C2 — C22×C4 — C24.133D4

Generators and relations for C24.133D4
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e4=d, f2=dc=cd, faf-1=ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, bf=fb, ce=ec, cf=fc, de=ed, df=fd, fef-1=e3 >

Subgroups: 316 in 168 conjugacy classes, 76 normal (18 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×2], C4 [×2], C4 [×6], C22, C22 [×10], C22 [×12], C8 [×4], C8 [×2], C2×C4 [×2], C2×C4 [×6], C2×C4 [×22], C23, C23 [×6], C23 [×4], C22⋊C4 [×4], C4⋊C4 [×8], C2×C8 [×8], C2×C8 [×10], C22×C4 [×2], C22×C4 [×4], C22×C4 [×8], C24, C2.C42 [×2], C4.Q8 [×4], C2×C22⋊C4 [×2], C2×C4⋊C4 [×4], C22×C8 [×2], C22×C8 [×4], C22×C8 [×4], C23×C4, C22.4Q16 [×2], C23.7Q8 [×2], C2×C4.Q8 [×2], C23×C8, C24.133D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×6], Q8 [×2], C23, C22⋊C4 [×4], C4⋊C4 [×4], SD16 [×4], C22×C4, C2×D4 [×3], C2×Q8, C4○D4 [×2], C4.Q8 [×4], C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C4⋊D4 [×2], C22⋊Q8 [×2], C2×SD16 [×2], C4○D8 [×2], C23.7Q8, C2×C4.Q8, C23.25D4, C88D4 [×4], C24.133D4

Smallest permutation representation of C24.133D4
On 64 points
Generators in S64
```(1 43)(2 44)(3 45)(4 46)(5 47)(6 48)(7 41)(8 42)(9 31)(10 32)(11 25)(12 26)(13 27)(14 28)(15 29)(16 30)(17 63)(18 64)(19 57)(20 58)(21 59)(22 60)(23 61)(24 62)(33 50)(34 51)(35 52)(36 53)(37 54)(38 55)(39 56)(40 49)
(1 56)(2 49)(3 50)(4 51)(5 52)(6 53)(7 54)(8 55)(9 59)(10 60)(11 61)(12 62)(13 63)(14 64)(15 57)(16 58)(17 27)(18 28)(19 29)(20 30)(21 31)(22 32)(23 25)(24 26)(33 45)(34 46)(35 47)(36 48)(37 41)(38 42)(39 43)(40 44)
(1 43)(2 44)(3 45)(4 46)(5 47)(6 48)(7 41)(8 42)(9 21)(10 22)(11 23)(12 24)(13 17)(14 18)(15 19)(16 20)(25 61)(26 62)(27 63)(28 64)(29 57)(30 58)(31 59)(32 60)(33 50)(34 51)(35 52)(36 53)(37 54)(38 55)(39 56)(40 49)
(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)(25 29)(26 30)(27 31)(28 32)(33 37)(34 38)(35 39)(36 40)(41 45)(42 46)(43 47)(44 48)(49 53)(50 54)(51 55)(52 56)(57 61)(58 62)(59 63)(60 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 24 47 16)(2 19 48 11)(3 22 41 14)(4 17 42 9)(5 20 43 12)(6 23 44 15)(7 18 45 10)(8 21 46 13)(25 40 57 53)(26 35 58 56)(27 38 59 51)(28 33 60 54)(29 36 61 49)(30 39 62 52)(31 34 63 55)(32 37 64 50)```

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

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

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

44 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 4A ··· 4H 4I ··· 4P 8A ··· 8P order 1 2 ··· 2 2 2 2 2 4 ··· 4 4 ··· 4 8 ··· 8 size 1 1 ··· 1 2 2 2 2 2 ··· 2 8 ··· 8 2 ··· 2

44 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 type + + + + + + + - + image C1 C2 C2 C2 C2 C4 D4 D4 Q8 D4 C4○D4 SD16 C4○D8 kernel C24.133D4 C22.4Q16 C23.7Q8 C2×C4.Q8 C23×C8 C22×C8 C2×C8 C22×C4 C22×C4 C24 C2×C4 C23 C22 # reps 1 2 2 2 1 8 4 1 2 1 4 8 8

Matrix representation of C24.133D4 in GL5(𝔽17)

 1 0 0 0 0 0 1 0 0 0 0 10 16 0 0 0 0 0 16 0 0 0 0 0 1
,
 1 0 0 0 0 0 16 0 0 0 0 0 16 0 0 0 0 0 16 0 0 0 0 0 16
,
 16 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 16 0 0 0 0 0 16
,
 1 0 0 0 0 0 16 0 0 0 0 0 16 0 0 0 0 0 1 0 0 0 0 0 1
,
 16 0 0 0 0 0 2 0 0 0 0 4 8 0 0 0 0 0 16 0 0 0 0 0 16
,
 4 0 0 0 0 0 11 8 0 0 0 6 6 0 0 0 0 0 0 16 0 0 0 1 0

`G:=sub<GL(5,GF(17))| [1,0,0,0,0,0,1,10,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,1],[1,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,16],[16,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,16],[1,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,1,0,0,0,0,0,1],[16,0,0,0,0,0,2,4,0,0,0,0,8,0,0,0,0,0,16,0,0,0,0,0,16],[4,0,0,0,0,0,11,6,0,0,0,8,6,0,0,0,0,0,0,1,0,0,0,16,0] >;`

C24.133D4 in GAP, Magma, Sage, TeX

`C_2^4._{133}D_4`
`% in TeX`

`G:=Group("C2^4.133D4");`
`// GroupNames label`

`G:=SmallGroup(128,539);`
`// by ID`

`G=gap.SmallGroup(128,539);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,64,422,2019,248]);`
`// Polycyclic`

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

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