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## G = C24.152D4order 128 = 27

### 7th non-split extension by C24 of D4 acting via D4/C22=C2

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C4 — C24.152D4
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C23×C4 — C22×M4(2) — C24.152D4
 Lower central C1 — C2 — C4 — C24.152D4
 Upper central C1 — C23 — C23×C4 — C24.152D4
 Jennings C1 — C2 — C2 — C22×C4 — C24.152D4

Generators and relations for C24.152D4
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e4=d, f2=cb=bc, ab=ba, ac=ca, eae-1=faf-1=ad=da, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=bde3 >

Subgroups: 388 in 224 conjugacy classes, 116 normal (26 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×4], C4 [×4], C4 [×8], C22 [×3], C22 [×8], C22 [×12], C8 [×4], C2×C4 [×6], C2×C4 [×22], C2×C4 [×24], C23, C23 [×6], C23 [×4], C42 [×4], C22⋊C4 [×4], C4⋊C4 [×8], C4⋊C4 [×8], C2×C8 [×4], C2×C8 [×4], M4(2) [×8], C22×C4 [×2], C22×C4 [×12], C22×C4 [×12], C24, C2×C42, C2×C22⋊C4, C2×C4⋊C4 [×8], C2×C4⋊C4 [×3], C42⋊C2 [×4], C42⋊C2 [×2], C22×C8 [×2], C2×M4(2) [×4], C2×M4(2) [×4], C23×C4, C23×C4, C22.4Q16 [×4], C22×C4⋊C4, C2×C42⋊C2, C22×M4(2), C24.152D4
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], C8⋊C22 [×2], C8.C22 [×2], C2×C2.C42, C23.36D4 [×2], C23.37D4, C23.38D4, M4(2)⋊C4 [×2], C24.152D4

Smallest permutation representation of C24.152D4
On 64 points
Generators in S64
```(1 12)(2 9)(3 14)(4 11)(5 16)(6 13)(7 10)(8 15)(17 59)(18 64)(19 61)(20 58)(21 63)(22 60)(23 57)(24 62)(25 46)(26 43)(27 48)(28 45)(29 42)(30 47)(31 44)(32 41)(33 55)(34 52)(35 49)(36 54)(37 51)(38 56)(39 53)(40 50)
(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 51)(34 52)(35 53)(36 54)(37 55)(38 56)(39 49)(40 50)
(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 52)(18 53)(19 54)(20 55)(21 56)(22 49)(23 50)(24 51)(33 58)(34 59)(35 60)(36 61)(37 62)(38 63)(39 64)(40 57)
(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 49)(2 21 45 38)(3 62 46 55)(4 19 47 36)(5 60 48 53)(6 17 41 34)(7 58 42 51)(8 23 43 40)(9 59 28 52)(10 24 29 33)(11 57 30 50)(12 22 31 39)(13 63 32 56)(14 20 25 37)(15 61 26 54)(16 18 27 35)```

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

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

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

44 conjugacy classes

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

44 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 4 4 type + + + + + + - + + - image C1 C2 C2 C2 C2 C4 C4 C4 D4 Q8 D4 C8⋊C22 C8.C22 kernel C24.152D4 C22.4Q16 C22×C4⋊C4 C2×C42⋊C2 C22×M4(2) C2×C4⋊C4 C42⋊C2 C2×M4(2) C22×C4 C22×C4 C24 C22 C22 # reps 1 4 1 1 1 8 8 8 5 2 1 2 2

Matrix representation of C24.152D4 in GL8(𝔽17)

 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 1 0 1 0 0 0 0 0 1 0 0 1
,
 16 0 0 0 0 0 0 0 0 16 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 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16
,
 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 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16
,
 1 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 13 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 16 0 15 0 0 0 0 0 0 0 16 1 0 0 0 0 0 1 1 0 0 0 0 0 1 0 1 0
,
 0 4 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 13 0 0 0 0 0 0 13 0 0 0 0 0 0 0 0 0 9 0 15 3 0 0 0 0 16 0 6 9 0 0 0 0 8 10 1 7 0 0 0 0 7 1 1 7

`G:=sub<GL(8,GF(17))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,1,1,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16],[1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,16,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,15,16,1,1,0,0,0,0,0,1,0,0],[0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,0,9,16,8,7,0,0,0,0,0,0,10,1,0,0,0,0,15,6,1,1,0,0,0,0,3,9,7,7] >;`

C24.152D4 in GAP, Magma, Sage, TeX

`C_2^4._{152}D_4`
`% in TeX`

`G:=Group("C2^4.152D4");`
`// GroupNames label`

`G:=SmallGroup(128,468);`
`// by ID`

`G=gap.SmallGroup(128,468);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,2,-2,112,141,232,723,2019,248,4037,124]);`
`// 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,e*a*e^-1=f*a*f^-1=a*d=d*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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