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## G = C23⋊F5⋊5C2order 320 = 26·5

### The semidirect product of C23⋊F5 and C2 acting through Inn(C23⋊F5)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C23⋊F5⋊5C2
 Chief series C1 — C5 — C10 — D10 — C22×D5 — C22⋊F5 — D10.C23 — C23⋊F5⋊5C2
 Lower central C5 — C10 — C2×C10 — C23⋊F5⋊5C2
 Upper central C1 — C4 — C2×C4 — C22×C4

Generators and relations for C23⋊F55C2
G = < a,b,c,d,e,f | a2=b2=c2=d5=e4=f2=1, ab=ba, faf=ac=ca, ad=da, eae-1=abc, ebe-1=bc=cb, bd=db, bf=fb, cd=dc, ce=ec, cf=fc, ede-1=d3, fdf=d-1, fef=bce >

Subgroups: 714 in 158 conjugacy classes, 48 normal (30 characteristic)
C1, C2, C2 [×5], C4 [×2], C4 [×8], C22, C22 [×7], C5, C2×C4 [×2], C2×C4 [×14], D4 [×6], Q8 [×2], C23, C23 [×2], D5 [×3], C10, C10 [×2], C42 [×2], C22⋊C4 [×4], C4⋊C4 [×2], C22×C4, C22×C4 [×2], C2×D4 [×3], C2×Q8, C4○D4 [×4], Dic5 [×2], Dic5, C20 [×2], C20, F5 [×4], D10 [×2], D10 [×3], C2×C10, C2×C10 [×2], C23⋊C4 [×4], C42⋊C2 [×2], C2×C4○D4, Dic10 [×2], C4×D5 [×4], C4×D5 [×2], D20 [×2], C2×Dic5 [×2], C5⋊D4 [×4], C2×C20 [×2], C2×C20 [×2], C2×F5 [×4], C22×D5 [×2], C22×C10, C23.C23, C4×F5 [×2], C4⋊F5 [×2], C22⋊F5 [×4], C2×Dic10, C2×C4×D5 [×2], C2×D20, C4○D20 [×4], C2×C5⋊D4 [×2], C22×C20, D10.D4 [×2], C23⋊F5 [×2], D10.C23 [×2], C2×C4○D20, C23⋊F55C2
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×2], F5, C2×C22⋊C4, C2×F5 [×3], C23.C23, C22⋊F5 [×2], C22×F5, C2×C22⋊F5, C23⋊F55C2

Smallest permutation representation of C23⋊F55C2
On 80 points
Generators in S80
```(1 24)(2 25)(3 21)(4 22)(5 23)(6 26)(7 27)(8 28)(9 29)(10 30)(11 31)(12 32)(13 33)(14 34)(15 35)(16 36)(17 37)(18 38)(19 39)(20 40)(41 61)(42 62)(43 63)(44 64)(45 65)(46 66)(47 67)(48 68)(49 69)(50 70)(51 71)(52 72)(53 73)(54 74)(55 75)(56 76)(57 77)(58 78)(59 79)(60 80)
(1 14)(2 15)(3 11)(4 12)(5 13)(6 16)(7 17)(8 18)(9 19)(10 20)(21 31)(22 32)(23 33)(24 34)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)(41 51)(42 52)(43 53)(44 54)(45 55)(46 56)(47 57)(48 58)(49 59)(50 60)(61 71)(62 72)(63 73)(64 74)(65 75)(66 76)(67 77)(68 78)(69 79)(70 80)
(1 9)(2 10)(3 6)(4 7)(5 8)(11 16)(12 17)(13 18)(14 19)(15 20)(21 26)(22 27)(23 28)(24 29)(25 30)(31 36)(32 37)(33 38)(34 39)(35 40)(41 46)(42 47)(43 48)(44 49)(45 50)(51 56)(52 57)(53 58)(54 59)(55 60)(61 66)(62 67)(63 68)(64 69)(65 70)(71 76)(72 77)(73 78)(74 79)(75 80)
(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)
(1 64 19 74)(2 61 18 72)(3 63 17 75)(4 65 16 73)(5 62 20 71)(6 68 12 80)(7 70 11 78)(8 67 15 76)(9 69 14 79)(10 66 13 77)(21 53 32 50)(22 55 31 48)(23 52 35 46)(24 54 34 49)(25 51 33 47)(26 58 37 45)(27 60 36 43)(28 57 40 41)(29 59 39 44)(30 56 38 42)
(1 79)(2 78)(3 77)(4 76)(5 80)(6 72)(7 71)(8 75)(9 74)(10 73)(11 67)(12 66)(13 70)(14 69)(15 68)(16 62)(17 61)(18 65)(19 64)(20 63)(21 52)(22 51)(23 55)(24 54)(25 53)(26 57)(27 56)(28 60)(29 59)(30 58)(31 42)(32 41)(33 45)(34 44)(35 43)(36 47)(37 46)(38 50)(39 49)(40 48)```

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

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

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

38 conjugacy classes

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

38 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 4 4 4 4 4 4 type + + + + + + + + + + image C1 C2 C2 C2 C2 C4 C4 C4 C4 D4 F5 C2×F5 C2×F5 C23.C23 C22⋊F5 C23⋊F5⋊5C2 kernel C23⋊F5⋊5C2 D10.D4 C23⋊F5 D10.C23 C2×C4○D20 C2×Dic10 C2×C4×D5 C2×C5⋊D4 C22×C20 C4×D5 C22×C4 C2×C4 C23 C5 C4 C1 # reps 1 2 2 2 1 2 2 2 2 4 1 2 1 2 4 8

Matrix representation of C23⋊F55C2 in GL4(𝔽41) generated by

 5 10 32 19 22 27 32 13 28 9 14 19 22 9 31 36
,
 22 0 3 3 38 19 38 0 0 38 19 38 3 3 0 22
,
 40 0 0 0 0 40 0 0 0 0 40 0 0 0 0 40
,
 0 1 0 0 0 0 1 0 0 0 0 1 40 40 40 40
,
 3 6 7 40 1 34 35 38 1 4 7 8 3 4 37 38
,
 38 35 34 1 38 37 4 3 40 7 6 3 8 7 4 1
`G:=sub<GL(4,GF(41))| [5,22,28,22,10,27,9,9,32,32,14,31,19,13,19,36],[22,38,0,3,0,19,38,3,3,38,19,0,3,0,38,22],[40,0,0,0,0,40,0,0,0,0,40,0,0,0,0,40],[0,0,0,40,1,0,0,40,0,1,0,40,0,0,1,40],[3,1,1,3,6,34,4,4,7,35,7,37,40,38,8,38],[38,38,40,8,35,37,7,7,34,4,6,4,1,3,3,1] >;`

C23⋊F55C2 in GAP, Magma, Sage, TeX

`C_2^3\rtimes F_5\rtimes_5C_2`
`% in TeX`

`G:=Group("C2^3:F5:5C2");`
`// GroupNames label`

`G:=SmallGroup(320,1083);`
`// by ID`

`G=gap.SmallGroup(320,1083);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,232,422,297,1684,6278,1595]);`
`// Polycyclic`

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

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