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## G = D10.C24order 320 = 26·5

### 15th non-split extension by D10 of C24 acting via C24/C23=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — D10.C24
 Chief series C1 — C5 — D5 — D10 — C2×F5 — C22×F5 — D4×F5 — D10.C24
 Lower central C5 — C10 — D10.C24
 Upper central C1 — C2 — C2×D4

Generators and relations for D10.C24
G = < a,b,c,d,e,f | a10=b2=d2=e2=f2=1, c2=a-1b, bab=a-1, cac-1=a3, ad=da, ae=ea, af=fa, cbc-1=a2b, bd=db, be=eb, bf=fb, cd=dc, ce=ec, fcf=a5c, ede=a5d, df=fd, ef=fe >

Subgroups: 1402 in 338 conjugacy classes, 138 normal (18 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C5, C2×C4, C2×C4, D4, D4, C23, C23, D5, D5, C10, C10, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×D4, C24, Dic5, C20, F5, D10, D10, D10, C2×C10, C2×C10, C2×C10, C2×C22⋊C4, C42⋊C2, C4×D4, C22×D4, C4×D5, D20, C2×Dic5, C5⋊D4, C2×C20, C5×D4, C2×F5, C2×F5, C22×D5, C22×D5, C22×D5, C22×C10, C22.11C24, C4×F5, C4⋊F5, C22⋊F5, C2×C4×D5, C2×D20, D4×D5, C2×C5⋊D4, D4×C10, C22×F5, C23×D5, D10.C23, D4×F5, C2×C22⋊F5, C2×D4×D5, D10.C24
Quotients: C1, C2, C4, C22, C2×C4, C23, C22×C4, C24, F5, C23×C4, 2+ 1+4, C2×F5, C22.11C24, C22×F5, C23×F5, D10.C24

Smallest permutation representation of D10.C24
On 40 points
Generators in S40
```(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)
(1 5)(2 4)(6 10)(7 9)(11 19)(12 18)(13 17)(14 16)(21 29)(22 28)(23 27)(24 26)(31 39)(32 38)(33 37)(34 36)
(1 13 6 18)(2 20 5 11)(3 17 4 14)(7 15 10 16)(8 12 9 19)(21 39 30 32)(22 36 29 35)(23 33 28 38)(24 40 27 31)(25 37 26 34)
(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)
(21 26)(22 27)(23 28)(24 29)(25 30)(31 36)(32 37)(33 38)(34 39)(35 40)
(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)```

`G:=sub<Sym(40)| (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), (1,5)(2,4)(6,10)(7,9)(11,19)(12,18)(13,17)(14,16)(21,29)(22,28)(23,27)(24,26)(31,39)(32,38)(33,37)(34,36), (1,13,6,18)(2,20,5,11)(3,17,4,14)(7,15,10,16)(8,12,9,19)(21,39,30,32)(22,36,29,35)(23,33,28,38)(24,40,27,31)(25,37,26,34), (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), (21,26)(22,27)(23,28)(24,29)(25,30)(31,36)(32,37)(33,38)(34,39)(35,40), (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)>;`

`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), (1,5)(2,4)(6,10)(7,9)(11,19)(12,18)(13,17)(14,16)(21,29)(22,28)(23,27)(24,26)(31,39)(32,38)(33,37)(34,36), (1,13,6,18)(2,20,5,11)(3,17,4,14)(7,15,10,16)(8,12,9,19)(21,39,30,32)(22,36,29,35)(23,33,28,38)(24,40,27,31)(25,37,26,34), (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), (21,26)(22,27)(23,28)(24,29)(25,30)(31,36)(32,37)(33,38)(34,39)(35,40), (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) );`

`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)], [(1,5),(2,4),(6,10),(7,9),(11,19),(12,18),(13,17),(14,16),(21,29),(22,28),(23,27),(24,26),(31,39),(32,38),(33,37),(34,36)], [(1,13,6,18),(2,20,5,11),(3,17,4,14),(7,15,10,16),(8,12,9,19),(21,39,30,32),(22,36,29,35),(23,33,28,38),(24,40,27,31),(25,37,26,34)], [(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)], [(21,26),(22,27),(23,28),(24,29),(25,30),(31,36),(32,37),(33,38),(34,39),(35,40)], [(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)]])`

44 conjugacy classes

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

44 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 4 4 4 4 4 8 type + + + + + + + + + + + image C1 C2 C2 C2 C2 C4 C4 C4 C4 F5 2+ 1+4 C2×F5 C2×F5 C2×F5 D10.C24 kernel D10.C24 D10.C23 D4×F5 C2×C22⋊F5 C2×D4×D5 C2×D20 D4×D5 C2×C5⋊D4 D4×C10 C2×D4 D5 C2×C4 D4 C23 C1 # reps 1 2 8 4 1 2 8 4 2 1 2 1 4 2 2

Matrix representation of D10.C24 in GL8(𝔽41)

 0 0 0 1 0 0 0 0 40 40 40 40 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40
,
 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 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 1
,
 40 0 0 0 0 0 0 0 0 0 0 40 0 0 0 0 0 40 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 28 36 0 0 0 0 0 0 34 13 0 0 0 0 0 0 0 0 28 36 0 0 0 0 0 0 34 13
,
 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0
,
 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40 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 40 0 0 0 0 0 0 0 0 40
,
 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 13 5 0 0 0 0 0 0 32 28 0 0 0 0 0 0 0 0 13 5 0 0 0 0 0 0 32 28

`G:=sub<GL(8,GF(41))| [0,40,1,0,0,0,0,0,0,40,0,1,0,0,0,0,0,40,0,0,0,0,0,0,1,40,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40],[0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[40,0,0,1,0,0,0,0,0,0,40,1,0,0,0,0,0,0,0,1,0,0,0,0,0,40,0,1,0,0,0,0,0,0,0,0,28,34,0,0,0,0,0,0,36,13,0,0,0,0,0,0,0,0,28,34,0,0,0,0,0,0,36,13],[40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40],[40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,13,32,0,0,0,0,0,0,5,28,0,0,0,0,0,0,0,0,13,32,0,0,0,0,0,0,5,28] >;`

D10.C24 in GAP, Magma, Sage, TeX

`D_{10}.C_2^4`
`% in TeX`

`G:=Group("D10.C2^4");`
`// GroupNames label`

`G:=SmallGroup(320,1596);`
`// by ID`

`G=gap.SmallGroup(320,1596);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,387,1123,6278,818]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e,f|a^10=b^2=d^2=e^2=f^2=1,c^2=a^-1*b,b*a*b=a^-1,c*a*c^-1=a^3,a*d=d*a,a*e=e*a,a*f=f*a,c*b*c^-1=a^2*b,b*d=d*b,b*e=e*b,b*f=f*b,c*d=d*c,c*e=e*c,f*c*f=a^5*c,e*d*e=a^5*d,d*f=f*d,e*f=f*e>;`
`// generators/relations`

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