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## G = C24.32D10order 320 = 26·5

### 32nd non-split extension by C24 of D10 acting via D10/C5=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C24.32D10
 Chief series C1 — C5 — C10 — C2×C10 — C2×Dic5 — C22×Dic5 — D4×Dic5 — C24.32D10
 Lower central C5 — C2×C10 — C24.32D10
 Upper central C1 — C22 — C22≀C2

Generators and relations for C24.32D10
G = < a,b,c,d,e,f | a2=b2=c2=d2=e10=1, f2=d, ab=ba, eae-1=faf-1=ac=ca, ad=da, fbf-1=bc=cb, ebe-1=bd=db, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e-1 >

Subgroups: 758 in 248 conjugacy classes, 99 normal (27 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C5, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C23, C10, C10, C10, C42, C22⋊C4, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×D4, C2×Q8, C24, Dic5, C20, C2×C10, C2×C10, C2×C10, C2×C22⋊C4, C42⋊C2, C4×D4, C22≀C2, C22⋊Q8, C22.D4, C4.4D4, C422C2, Dic10, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C5×D4, C22×C10, C22×C10, C22×C10, C22.45C24, C4×Dic5, C4×Dic5, C10.D4, C4⋊Dic5, C23.D5, C23.D5, C5×C22⋊C4, C5×C22⋊C4, C2×Dic10, C22×Dic5, C22×Dic5, D4×C10, D4×C10, C23×C10, C23.11D10, Dic5.14D4, C23.D10, D4×Dic5, C23.18D10, C23.18D10, C20.17D4, C2×C23.D5, C5×C22≀C2, C24.32D10
Quotients: C1, C2, C22, C23, D5, C4○D4, C24, D10, C2×C4○D4, 2+ 1+4, C22×D5, C22.45C24, D42D5, C23×D5, C2×D42D5, D46D10, C24.32D10

Smallest permutation representation of C24.32D10
On 80 points
Generators in S80
```(2 13)(4 15)(6 17)(8 19)(10 11)(21 78)(23 80)(25 72)(27 74)(29 76)(32 50)(34 42)(36 44)(38 46)(40 48)(52 69)(54 61)(56 63)(58 65)(60 67)
(1 33)(3 35)(5 37)(7 39)(9 31)(12 41)(14 43)(16 45)(18 47)(20 49)(21 60)(22 79)(23 52)(24 71)(25 54)(26 73)(27 56)(28 75)(29 58)(30 77)(51 68)(53 70)(55 62)(57 64)(59 66)(61 72)(63 74)(65 76)(67 78)(69 80)
(1 12)(2 13)(3 14)(4 15)(5 16)(6 17)(7 18)(8 19)(9 20)(10 11)(21 78)(22 79)(23 80)(24 71)(25 72)(26 73)(27 74)(28 75)(29 76)(30 77)(31 49)(32 50)(33 41)(34 42)(35 43)(36 44)(37 45)(38 46)(39 47)(40 48)(51 68)(52 69)(53 70)(54 61)(55 62)(56 63)(57 64)(58 65)(59 66)(60 67)
(1 33)(2 34)(3 35)(4 36)(5 37)(6 38)(7 39)(8 40)(9 31)(10 32)(11 50)(12 41)(13 42)(14 43)(15 44)(16 45)(17 46)(18 47)(19 48)(20 49)(21 67)(22 68)(23 69)(24 70)(25 61)(26 62)(27 63)(28 64)(29 65)(30 66)(51 79)(52 80)(53 71)(54 72)(55 73)(56 74)(57 75)(58 76)(59 77)(60 78)
(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 76 33 58)(2 75 34 57)(3 74 35 56)(4 73 36 55)(5 72 37 54)(6 71 38 53)(7 80 39 52)(8 79 40 51)(9 78 31 60)(10 77 32 59)(11 30 50 66)(12 29 41 65)(13 28 42 64)(14 27 43 63)(15 26 44 62)(16 25 45 61)(17 24 46 70)(18 23 47 69)(19 22 48 68)(20 21 49 67)```

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

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

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

53 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 4A 4B 4C 4D ··· 4K 4L 4M 4N 4O 5A 5B 10A ··· 10F 10G ··· 10R 10S 10T 20A ··· 20F order 1 2 2 2 2 2 2 2 2 2 4 4 4 4 ··· 4 4 4 4 4 5 5 10 ··· 10 10 ··· 10 10 10 20 ··· 20 size 1 1 1 1 2 2 2 2 4 4 4 4 4 10 ··· 10 20 20 20 20 2 2 2 ··· 2 4 ··· 4 8 8 8 ··· 8

53 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 C2 C2 D5 C4○D4 D10 D10 D10 2+ 1+4 D4⋊2D5 D4⋊6D10 kernel C24.32D10 C23.11D10 Dic5.14D4 C23.D10 D4×Dic5 C23.18D10 C20.17D4 C2×C23.D5 C5×C22≀C2 C22≀C2 C2×C10 C22⋊C4 C2×D4 C24 C10 C22 C2 # reps 1 2 2 2 2 3 1 2 1 2 8 6 6 2 1 8 4

Matrix representation of C24.32D10 in GL6(𝔽41)

 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 1 0 0 0 0 0 0 40
,
 40 0 0 0 0 0 18 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 1
,
 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 40 0 0 0 0 0 0 40
,
 40 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 9 0 0 0 0 0 1 0 0 0 0 0 0 7 7 0 0 0 0 34 40 0 0 0 0 0 0 0 1 0 0 0 0 1 0
,
 32 0 0 0 0 0 0 32 0 0 0 0 0 0 27 11 0 0 0 0 27 14 0 0 0 0 0 0 0 9 0 0 0 0 9 0

`G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40],[40,18,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[40,0,0,0,0,0,9,1,0,0,0,0,0,0,7,34,0,0,0,0,7,40,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[32,0,0,0,0,0,0,32,0,0,0,0,0,0,27,27,0,0,0,0,11,14,0,0,0,0,0,0,0,9,0,0,0,0,9,0] >;`

C24.32D10 in GAP, Magma, Sage, TeX

`C_2^4._{32}D_{10}`
`% in TeX`

`G:=Group("C2^4.32D10");`
`// GroupNames label`

`G:=SmallGroup(320,1259);`
`// by ID`

`G=gap.SmallGroup(320,1259);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,219,1571,297,12550]);`
`// Polycyclic`

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

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