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## G = C24.F5order 320 = 26·5

### 1st non-split extension by C24 of F5 acting via F5/C5=C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C24.F5
 Chief series C1 — C5 — C10 — C2×C10 — C2×Dic5 — C22×Dic5 — C23.2F5 — C24.F5
 Lower central C5 — C10 — C2×C10 — C24.F5
 Upper central C1 — C22 — C23 — C24

Generators and relations for C24.F5
G = < a,b,c,d,e,f | a2=b2=c2=d2=e5=1, f4=c, ab=ba, ac=ca, ad=da, ae=ea, faf-1=abd, bc=cb, fbf-1=bd=db, be=eb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e3 >

Subgroups: 386 in 98 conjugacy classes, 30 normal (22 characteristic)
C1, C2, C2, C4, C22, C22, C5, C8, C2×C4, C23, C23, C23, C10, C10, C22⋊C4, C2×C8, C22×C4, C24, Dic5, C2×C10, C2×C10, C22⋊C8, C2×C22⋊C4, C5⋊C8, C2×Dic5, C2×Dic5, C22×C10, C22×C10, C22×C10, C23⋊C8, C23.D5, C2×C5⋊C8, C22×Dic5, C23×C10, C23.2F5, C2×C23.D5, C24.F5
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, C22⋊C4, C2×C8, M4(2), F5, C22⋊C8, C23⋊C4, C4.D4, C5⋊C8, C2×F5, C23⋊C8, C2×C5⋊C8, C22.F5, C22⋊F5, C23⋊F5, C23.2F5, C23.F5, C24.F5

Smallest permutation representation of C24.F5
On 80 points
Generators in S80
```(2 13)(3 10)(4 8)(6 9)(7 14)(11 15)(17 55)(18 52)(19 23)(21 51)(22 56)(25 29)(27 34)(28 39)(31 38)(32 35)(36 40)(41 63)(42 60)(43 47)(45 59)(46 64)(49 53)(57 61)(65 69)(67 78)(68 75)(71 74)(72 79)(76 80)
(1 5)(2 13)(3 7)(4 15)(6 9)(8 11)(10 14)(12 16)(17 55)(18 22)(19 49)(20 24)(21 51)(23 53)(25 40)(26 30)(27 34)(28 32)(29 36)(31 38)(33 37)(35 39)(41 63)(42 46)(43 57)(44 48)(45 59)(47 61)(50 54)(52 56)(58 62)(60 64)(65 76)(66 70)(67 78)(68 72)(69 80)(71 74)(73 77)(75 79)
(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)(65 69)(66 70)(67 71)(68 72)(73 77)(74 78)(75 79)(76 80)
(1 16)(2 9)(3 10)(4 11)(5 12)(6 13)(7 14)(8 15)(17 51)(18 52)(19 53)(20 54)(21 55)(22 56)(23 49)(24 50)(25 36)(26 37)(27 38)(28 39)(29 40)(30 33)(31 34)(32 35)(41 59)(42 60)(43 61)(44 62)(45 63)(46 64)(47 57)(48 58)(65 80)(66 73)(67 74)(68 75)(69 76)(70 77)(71 78)(72 79)
(1 30 44 54 77)(2 55 31 78 45)(3 79 56 46 32)(4 47 80 25 49)(5 26 48 50 73)(6 51 27 74 41)(7 75 52 42 28)(8 43 76 29 53)(9 21 34 71 63)(10 72 22 64 35)(11 57 65 36 23)(12 37 58 24 66)(13 17 38 67 59)(14 68 18 60 39)(15 61 69 40 19)(16 33 62 20 70)
(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)(65 66 67 68 69 70 71 72)(73 74 75 76 77 78 79 80)```

`G:=sub<Sym(80)| (2,13)(3,10)(4,8)(6,9)(7,14)(11,15)(17,55)(18,52)(19,23)(21,51)(22,56)(25,29)(27,34)(28,39)(31,38)(32,35)(36,40)(41,63)(42,60)(43,47)(45,59)(46,64)(49,53)(57,61)(65,69)(67,78)(68,75)(71,74)(72,79)(76,80), (1,5)(2,13)(3,7)(4,15)(6,9)(8,11)(10,14)(12,16)(17,55)(18,22)(19,49)(20,24)(21,51)(23,53)(25,40)(26,30)(27,34)(28,32)(29,36)(31,38)(33,37)(35,39)(41,63)(42,46)(43,57)(44,48)(45,59)(47,61)(50,54)(52,56)(58,62)(60,64)(65,76)(66,70)(67,78)(68,72)(69,80)(71,74)(73,77)(75,79), (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)(65,69)(66,70)(67,71)(68,72)(73,77)(74,78)(75,79)(76,80), (1,16)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14)(8,15)(17,51)(18,52)(19,53)(20,54)(21,55)(22,56)(23,49)(24,50)(25,36)(26,37)(27,38)(28,39)(29,40)(30,33)(31,34)(32,35)(41,59)(42,60)(43,61)(44,62)(45,63)(46,64)(47,57)(48,58)(65,80)(66,73)(67,74)(68,75)(69,76)(70,77)(71,78)(72,79), (1,30,44,54,77)(2,55,31,78,45)(3,79,56,46,32)(4,47,80,25,49)(5,26,48,50,73)(6,51,27,74,41)(7,75,52,42,28)(8,43,76,29,53)(9,21,34,71,63)(10,72,22,64,35)(11,57,65,36,23)(12,37,58,24,66)(13,17,38,67,59)(14,68,18,60,39)(15,61,69,40,19)(16,33,62,20,70), (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)(65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80)>;`

`G:=Group( (2,13)(3,10)(4,8)(6,9)(7,14)(11,15)(17,55)(18,52)(19,23)(21,51)(22,56)(25,29)(27,34)(28,39)(31,38)(32,35)(36,40)(41,63)(42,60)(43,47)(45,59)(46,64)(49,53)(57,61)(65,69)(67,78)(68,75)(71,74)(72,79)(76,80), (1,5)(2,13)(3,7)(4,15)(6,9)(8,11)(10,14)(12,16)(17,55)(18,22)(19,49)(20,24)(21,51)(23,53)(25,40)(26,30)(27,34)(28,32)(29,36)(31,38)(33,37)(35,39)(41,63)(42,46)(43,57)(44,48)(45,59)(47,61)(50,54)(52,56)(58,62)(60,64)(65,76)(66,70)(67,78)(68,72)(69,80)(71,74)(73,77)(75,79), (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)(65,69)(66,70)(67,71)(68,72)(73,77)(74,78)(75,79)(76,80), (1,16)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14)(8,15)(17,51)(18,52)(19,53)(20,54)(21,55)(22,56)(23,49)(24,50)(25,36)(26,37)(27,38)(28,39)(29,40)(30,33)(31,34)(32,35)(41,59)(42,60)(43,61)(44,62)(45,63)(46,64)(47,57)(48,58)(65,80)(66,73)(67,74)(68,75)(69,76)(70,77)(71,78)(72,79), (1,30,44,54,77)(2,55,31,78,45)(3,79,56,46,32)(4,47,80,25,49)(5,26,48,50,73)(6,51,27,74,41)(7,75,52,42,28)(8,43,76,29,53)(9,21,34,71,63)(10,72,22,64,35)(11,57,65,36,23)(12,37,58,24,66)(13,17,38,67,59)(14,68,18,60,39)(15,61,69,40,19)(16,33,62,20,70), (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)(65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80) );`

`G=PermutationGroup([[(2,13),(3,10),(4,8),(6,9),(7,14),(11,15),(17,55),(18,52),(19,23),(21,51),(22,56),(25,29),(27,34),(28,39),(31,38),(32,35),(36,40),(41,63),(42,60),(43,47),(45,59),(46,64),(49,53),(57,61),(65,69),(67,78),(68,75),(71,74),(72,79),(76,80)], [(1,5),(2,13),(3,7),(4,15),(6,9),(8,11),(10,14),(12,16),(17,55),(18,22),(19,49),(20,24),(21,51),(23,53),(25,40),(26,30),(27,34),(28,32),(29,36),(31,38),(33,37),(35,39),(41,63),(42,46),(43,57),(44,48),(45,59),(47,61),(50,54),(52,56),(58,62),(60,64),(65,76),(66,70),(67,78),(68,72),(69,80),(71,74),(73,77),(75,79)], [(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),(65,69),(66,70),(67,71),(68,72),(73,77),(74,78),(75,79),(76,80)], [(1,16),(2,9),(3,10),(4,11),(5,12),(6,13),(7,14),(8,15),(17,51),(18,52),(19,53),(20,54),(21,55),(22,56),(23,49),(24,50),(25,36),(26,37),(27,38),(28,39),(29,40),(30,33),(31,34),(32,35),(41,59),(42,60),(43,61),(44,62),(45,63),(46,64),(47,57),(48,58),(65,80),(66,73),(67,74),(68,75),(69,76),(70,77),(71,78),(72,79)], [(1,30,44,54,77),(2,55,31,78,45),(3,79,56,46,32),(4,47,80,25,49),(5,26,48,50,73),(6,51,27,74,41),(7,75,52,42,28),(8,43,76,29,53),(9,21,34,71,63),(10,72,22,64,35),(11,57,65,36,23),(12,37,58,24,66),(13,17,38,67,59),(14,68,18,60,39),(15,61,69,40,19),(16,33,62,20,70)], [(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),(65,66,67,68,69,70,71,72),(73,74,75,76,77,78,79,80)]])`

38 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E 4F 5 8A ··· 8H 10A ··· 10O order 1 2 2 2 2 2 2 2 4 4 4 4 4 4 5 8 ··· 8 10 ··· 10 size 1 1 1 1 2 2 4 4 10 10 10 10 20 20 4 20 ··· 20 4 ··· 4

38 irreducible representations

 dim 1 1 1 1 1 1 2 2 4 4 4 4 4 4 4 4 4 type + + + + + + + - + - + image C1 C2 C2 C4 C4 C8 D4 M4(2) F5 C23⋊C4 C4.D4 C5⋊C8 C2×F5 C22.F5 C22⋊F5 C23⋊F5 C23.F5 kernel C24.F5 C23.2F5 C2×C23.D5 C22×Dic5 C23×C10 C22×C10 C2×Dic5 C2×C10 C24 C10 C10 C23 C23 C22 C22 C2 C2 # reps 1 2 1 2 2 8 2 2 1 1 1 2 1 2 2 4 4

Matrix representation of C24.F5 in GL6(𝔽41)

 1 0 0 0 0 0 0 40 0 0 0 0 0 0 1 0 0 0 0 0 20 40 0 0 0 0 0 0 40 0 0 0 0 0 21 1
,
 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 10 0 0 0 0 0 17 37 0 0 0 0 0 0 18 0 0 0 0 0 20 16
,
 0 1 0 0 0 0 9 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 9 36 0 0 0 0 0 32 0 0

`G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,40,0,0,0,0,0,0,1,20,0,0,0,0,0,40,0,0,0,0,0,0,40,21,0,0,0,0,0,1],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,10,17,0,0,0,0,0,37,0,0,0,0,0,0,18,20,0,0,0,0,0,16],[0,9,0,0,0,0,1,0,0,0,0,0,0,0,0,0,9,0,0,0,0,0,36,32,0,0,1,0,0,0,0,0,0,1,0,0] >;`

C24.F5 in GAP, Magma, Sage, TeX

`C_2^4.F_5`
`% in TeX`

`G:=Group("C2^4.F5");`
`// GroupNames label`

`G:=SmallGroup(320,271);`
`// by ID`

`G=gap.SmallGroup(320,271);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,387,100,1123,6278,3156]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=e^5=1,f^4=c,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,f*a*f^-1=a*b*d,b*c=c*b,f*b*f^-1=b*d=d*b,b*e=e*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=e^3>;`
`// generators/relations`

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