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## G = C24.132D4order 128 = 27

### 1st 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 — C4 — C24.132D4
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C23×C4 — C23×C8 — C24.132D4
 Lower central C1 — C2 — C4 — C24.132D4
 Upper central C1 — C22×C4 — C23×C4 — C24.132D4
 Jennings C1 — C2 — C2 — C22×C4 — C24.132D4

Generators and relations for C24.132D4
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e4=d, f2=cb=bc, ab=ba, ac=ca, faf-1=ad=da, ae=ea, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=bde3 >

Subgroups: 340 in 212 conjugacy classes, 116 normal (14 characteristic)
C1, C2, C2 [×6], C2 [×4], C4 [×2], C4 [×6], C4 [×8], C22, C22 [×10], C22 [×12], C8 [×4], C2×C4 [×2], C2×C4 [×26], C2×C4 [×16], C23, C23 [×6], C23 [×4], C42 [×8], C22⋊C4 [×8], C4⋊C4 [×8], C4⋊C4 [×4], C2×C8 [×4], C2×C8 [×12], C22×C4 [×2], C22×C4 [×12], C22×C4 [×4], C24, C2×C42 [×2], C2×C22⋊C4 [×2], C2×C4⋊C4 [×4], C42⋊C2 [×8], C42⋊C2 [×4], C22×C8 [×6], C22×C8 [×4], C23×C4, C22.4Q16 [×4], C2×C42⋊C2 [×2], C23×C8, C24.132D4
Quotients: C1, C2 [×7], C4 [×12], C22 [×7], C2×C4 [×18], D4 [×6], Q8 [×2], C23, C42 [×4], C22⋊C4 [×12], C4⋊C4 [×12], C22×C4 [×3], C2×D4 [×3], C2×Q8, C2.C42 [×8], C2×C42, C2×C22⋊C4 [×3], C2×C4⋊C4 [×3], C4○D8 [×4], C2×C2.C42, C23.24D4 [×4], C23.25D4 [×2], C24.132D4

Smallest permutation representation of C24.132D4
On 64 points
Generators in S64
```(17 21)(18 22)(19 23)(20 24)(33 37)(34 38)(35 39)(36 40)(49 53)(50 54)(51 55)(52 56)(57 61)(58 62)(59 63)(60 64)
(1 12)(2 13)(3 14)(4 15)(5 16)(6 9)(7 10)(8 11)(17 59)(18 60)(19 61)(20 62)(21 63)(22 64)(23 57)(24 58)(25 46)(26 47)(27 48)(28 41)(29 42)(30 43)(31 44)(32 45)(33 49)(34 50)(35 51)(36 52)(37 53)(38 54)(39 55)(40 56)
(1 31)(2 32)(3 25)(4 26)(5 27)(6 28)(7 29)(8 30)(9 41)(10 42)(11 43)(12 44)(13 45)(14 46)(15 47)(16 48)(17 56)(18 49)(19 50)(20 51)(21 52)(22 53)(23 54)(24 55)(33 60)(34 61)(35 62)(36 63)(37 64)(38 57)(39 58)(40 59)
(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 64 44 53)(2 21 45 36)(3 62 46 51)(4 19 47 34)(5 60 48 49)(6 17 41 40)(7 58 42 55)(8 23 43 38)(9 59 28 56)(10 24 29 39)(11 57 30 54)(12 22 31 37)(13 63 32 52)(14 20 25 35)(15 61 26 50)(16 18 27 33)```

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

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

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

56 conjugacy classes

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

56 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 type + + + + + - + image C1 C2 C2 C2 C4 C4 D4 Q8 D4 C4○D8 kernel C24.132D4 C22.4Q16 C2×C42⋊C2 C23×C8 C42⋊C2 C22×C8 C22×C4 C22×C4 C24 C22 # reps 1 4 2 1 16 8 5 2 1 16

Matrix representation of C24.132D4 in GL4(𝔽17) generated by

 16 0 0 0 0 16 0 0 0 0 1 0 0 0 7 16
,
 16 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1
,
 16 0 0 0 0 16 0 0 0 0 16 0 0 0 0 16
,
 1 0 0 0 0 1 0 0 0 0 16 0 0 0 0 16
,
 13 0 0 0 0 1 0 0 0 0 8 0 0 0 1 15
,
 1 0 0 0 0 13 0 0 0 0 3 4 0 0 6 14
`G:=sub<GL(4,GF(17))| [16,0,0,0,0,16,0,0,0,0,1,7,0,0,0,16],[16,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[16,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[1,0,0,0,0,1,0,0,0,0,16,0,0,0,0,16],[13,0,0,0,0,1,0,0,0,0,8,1,0,0,0,15],[1,0,0,0,0,13,0,0,0,0,3,6,0,0,4,14] >;`

C24.132D4 in GAP, Magma, Sage, TeX

`C_2^4._{132}D_4`
`% in TeX`

`G:=Group("C2^4.132D4");`
`// GroupNames label`

`G:=SmallGroup(128,467);`
`// by ID`

`G=gap.SmallGroup(128,467);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,2,-2,112,141,232,352,2019,248,4037,124]);`
`// Polycyclic`

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

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