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## G = C24.4F5order 320 = 26·5

### 2nd non-split extension by C24 of F5 acting via F5/D5=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C24.4F5
 Chief series C1 — C5 — C10 — Dic5 — C2×Dic5 — C2×C5⋊C8 — C23.2F5 — C24.4F5
 Lower central C5 — C2×C10 — C24.4F5
 Upper central C1 — C22 — C24

Generators and relations for C24.4F5
G = < a,b,c,d,e,f | a2=b2=c2=d2=e5=1, f4=d, ab=ba, faf-1=ac=ca, ad=da, ae=ea, bc=cb, fbf-1=bd=db, be=eb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e3 >

Subgroups: 602 in 190 conjugacy classes, 60 normal (16 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C5, C8, C2×C4, C23, C23, C23, C10, C10, C10, C2×C8, M4(2), C22×C4, C24, Dic5, Dic5, C2×C10, C2×C10, C2×C10, C22⋊C8, C2×M4(2), C23×C4, C5⋊C8, C2×Dic5, C2×Dic5, C2×Dic5, C22×C10, C22×C10, C22×C10, C24.4C4, C2×C5⋊C8, C22.F5, C22×Dic5, C22×Dic5, C22×Dic5, C23×C10, C23.2F5, C2×C22.F5, C23×Dic5, C24.4F5
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, M4(2), C22×C4, C2×D4, F5, C2×C22⋊C4, C2×M4(2), C2×F5, C24.4C4, C22.F5, C22⋊F5, C22×F5, C2×C22.F5, C2×C22⋊F5, C24.4F5

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

44 conjugacy classes

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

44 irreducible representations

 dim 1 1 1 1 1 1 2 2 4 4 4 4 type + + + + + + + - + image C1 C2 C2 C2 C4 C4 D4 M4(2) F5 C2×F5 C22.F5 C22⋊F5 kernel C24.4F5 C23.2F5 C2×C22.F5 C23×Dic5 C22×Dic5 C23×C10 C2×Dic5 C2×C10 C24 C23 C22 C22 # reps 1 4 2 1 6 2 4 8 1 3 8 4

Matrix representation of C24.4F5 in GL6(𝔽41)

 1 0 0 0 0 0 0 40 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 40 0 0 0 0 0 0 40
,
 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 34 40 0 0 0 0 1 0 0 0 0 0 0 0 7 7 0 0 0 0 34 40
,
 0 1 0 0 0 0 40 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 38 3 0 0 0 0 24 3 0 0

`G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,34,1,0,0,0,0,40,0,0,0,0,0,0,0,7,34,0,0,0,0,7,40],[0,40,0,0,0,0,1,0,0,0,0,0,0,0,0,0,38,24,0,0,0,0,3,3,0,0,1,0,0,0,0,0,0,1,0,0] >;`

C24.4F5 in GAP, Magma, Sage, TeX

`C_2^4._4F_5`
`% in TeX`

`G:=Group("C2^4.4F5");`
`// GroupNames label`

`G:=SmallGroup(320,1136);`
`// by ID`

`G=gap.SmallGroup(320,1136);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,253,758,136,6278,1595]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=e^5=1,f^4=d,a*b=b*a,f*a*f^-1=a*c=c*a,a*d=d*a,a*e=e*a,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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