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## G = C10.C25order 320 = 26·5

### 14th non-split extension by C10 of C25 acting via C25/C24=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C10.C25
 Chief series C1 — C5 — C10 — D10 — C22×D5 — C2×C4×D5 — D5×C4○D4 — C10.C25
 Lower central C5 — C10 — C10.C25
 Upper central C1 — C4 — C2×C4○D4

Generators and relations for C10.C25
G = < a,b,c,d,e,f | a10=b2=c2=e2=f2=1, d2=a5, bab=a-1, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, fbf=a5b, cd=dc, ece=a5c, cf=fc, de=ed, df=fd, ef=fe >

Subgroups: 2414 in 810 conjugacy classes, 443 normal (17 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C5, C2×C4, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, D5, C10, C10, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4, C4○D4, Dic5, C20, C20, D10, D10, C2×C10, C2×C10, C2×C10, C2×C4○D4, C2×C4○D4, 2+ 1+4, 2- 1+4, Dic10, C4×D5, D20, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C5×D4, C5×Q8, C22×D5, C22×C10, C2.C25, C2×Dic10, C2×C4×D5, C2×D20, C4○D20, D4×D5, D42D5, Q8×D5, Q82D5, C2×C5⋊D4, C22×C20, D4×C10, Q8×C10, C5×C4○D4, C2×C4○D20, D46D10, Q8.10D10, D5×C4○D4, D48D10, D4.10D10, C10×C4○D4, C10.C25
Quotients: C1, C2, C22, C23, D5, C24, D10, C25, C22×D5, C2.C25, C23×D5, D5×C24, C10.C25

Smallest permutation representation of C10.C25
On 80 points
Generators in S80
```(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 23)(2 22)(3 21)(4 30)(5 29)(6 28)(7 27)(8 26)(9 25)(10 24)(11 40)(12 39)(13 38)(14 37)(15 36)(16 35)(17 34)(18 33)(19 32)(20 31)(41 65)(42 64)(43 63)(44 62)(45 61)(46 70)(47 69)(48 68)(49 67)(50 66)(51 80)(52 79)(53 78)(54 77)(55 76)(56 75)(57 74)(58 73)(59 72)(60 71)
(1 13)(2 14)(3 15)(4 16)(5 17)(6 18)(7 19)(8 20)(9 11)(10 12)(21 36)(22 37)(23 38)(24 39)(25 40)(26 31)(27 32)(28 33)(29 34)(30 35)(41 51)(42 52)(43 53)(44 54)(45 55)(46 56)(47 57)(48 58)(49 59)(50 60)(61 76)(62 77)(63 78)(64 79)(65 80)(66 71)(67 72)(68 73)(69 74)(70 75)
(1 48 6 43)(2 49 7 44)(3 50 8 45)(4 41 9 46)(5 42 10 47)(11 56 16 51)(12 57 17 52)(13 58 18 53)(14 59 19 54)(15 60 20 55)(21 66 26 61)(22 67 27 62)(23 68 28 63)(24 69 29 64)(25 70 30 65)(31 76 36 71)(32 77 37 72)(33 78 38 73)(34 79 39 74)(35 80 40 75)
(1 23)(2 24)(3 25)(4 26)(5 27)(6 28)(7 29)(8 30)(9 21)(10 22)(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 13)(2 14)(3 15)(4 16)(5 17)(6 18)(7 19)(8 20)(9 11)(10 12)(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)```

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

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

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

74 conjugacy classes

 class 1 2A 2B ··· 2H 2I ··· 2P 4A 4B 4C ··· 4I 4J ··· 4Q 5A 5B 10A ··· 10F 10G ··· 10R 20A ··· 20H 20I ··· 20T order 1 2 2 ··· 2 2 ··· 2 4 4 4 ··· 4 4 ··· 4 5 5 10 ··· 10 10 ··· 10 20 ··· 20 20 ··· 20 size 1 1 2 ··· 2 10 ··· 10 1 1 2 ··· 2 10 ··· 10 2 2 2 ··· 2 4 ··· 4 2 ··· 2 4 ··· 4

74 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 4 4 type + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 D5 D10 D10 D10 D10 C2.C25 C10.C25 kernel C10.C25 C2×C4○D20 D4⋊6D10 Q8.10D10 D5×C4○D4 D4⋊8D10 D4.10D10 C10×C4○D4 C2×C4○D4 C22×C4 C2×D4 C2×Q8 C4○D4 C5 C1 # reps 1 6 6 2 8 4 4 1 2 6 6 2 16 2 8

Matrix representation of C10.C25 in GL6(𝔽41)

 40 7 0 0 0 0 34 7 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
,
 7 34 0 0 0 0 1 34 0 0 0 0 0 0 40 0 37 0 0 0 20 0 40 40 0 0 0 0 1 0 0 0 21 40 1 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 37 0 0 0 0 0 40 0 0 0 0 20 1 0 40 0 0 20 1 40 0
,
 40 0 0 0 0 0 0 40 0 0 0 0 0 0 32 0 0 0 0 0 0 32 0 0 0 0 0 0 32 0 0 0 0 0 0 32
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 40 0 37 0 0 0 0 0 40 1 0 0 0 0 1 0 0 0 0 1 1 0
,
 40 0 0 0 0 0 0 40 0 0 0 0 0 0 1 37 0 0 0 0 0 40 0 0 0 0 0 1 0 1 0 0 0 1 1 0

`G:=sub<GL(6,GF(41))| [40,34,0,0,0,0,7,7,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[7,1,0,0,0,0,34,34,0,0,0,0,0,0,40,20,0,21,0,0,0,0,0,40,0,0,37,40,1,1,0,0,0,40,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,20,20,0,0,37,40,1,1,0,0,0,0,0,40,0,0,0,0,40,0],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,32,0,0,0,0,0,0,32,0,0,0,0,0,0,32,0,0,0,0,0,0,32],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,37,40,1,1,0,0,0,1,0,0],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,37,40,1,1,0,0,0,0,0,1,0,0,0,0,1,0] >;`

C10.C25 in GAP, Magma, Sage, TeX

`C_{10}.C_2^5`
`% in TeX`

`G:=Group("C10.C2^5");`
`// GroupNames label`

`G:=SmallGroup(320,1621);`
`// by ID`

`G=gap.SmallGroup(320,1621);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,184,570,1684,12550]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e,f|a^10=b^2=c^2=e^2=f^2=1,d^2=a^5,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,b*d=d*b,b*e=e*b,f*b*f=a^5*b,c*d=d*c,e*c*e=a^5*c,c*f=f*c,d*e=e*d,d*f=f*d,e*f=f*e>;`
`// generators/relations`

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