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## G = C24.2D10order 320 = 26·5

### 2nd non-split extension by C24 of D10 acting via D10/C5=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C24.2D10
 Chief series C1 — C5 — C10 — C2×C10 — C22×C10 — C23×C10 — C2×C23.D5 — C24.2D10
 Lower central C5 — C10 — C2×C10 — C24.2D10
 Upper central C1 — C22 — C24 — C2×C22⋊C4

Generators and relations for C24.2D10
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e10=b, f2=abcd, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, fbf-1=bd=db, be=eb, ece-1=fcf-1=cd=dc, de=ed, df=fd, fef-1=cde9 >

Subgroups: 518 in 142 conjugacy classes, 51 normal (39 characteristic)
C1, C2, C2, C2, C4, C22, C22, C5, C2×C4, C23, C23, C10, C10, C10, C22⋊C4, C22⋊C4, C22×C4, C24, Dic5, C20, C2×C10, C2×C10, C2×C22⋊C4, C2×C22⋊C4, C2×Dic5, C2×C20, C22×C10, C22×C10, C23.9D4, C23.D5, C23.D5, C5×C22⋊C4, C5×C22⋊C4, C22×Dic5, C22×Dic5, C22×C20, C23×C10, C2×C23.D5, C10×C22⋊C4, C24.2D10
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, D5, C42, C22⋊C4, C4⋊C4, Dic5, D10, C2.C42, C23⋊C4, Dic10, C4×D5, D20, C2×Dic5, C5⋊D4, C23.9D4, C4×Dic5, C10.D4, C4⋊Dic5, D10⋊C4, C23.D5, C23.1D10, C10.10C42, C24.2D10

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

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

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

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

62 conjugacy classes

 class 1 2A 2B 2C 2D ··· 2I 4A 4B 4C 4D 4E ··· 4L 5A 5B 10A ··· 10N 10O ··· 10V 20A ··· 20P order 1 2 2 2 2 ··· 2 4 4 4 4 4 ··· 4 5 5 10 ··· 10 10 ··· 10 20 ··· 20 size 1 1 1 1 2 ··· 2 4 4 4 4 20 ··· 20 2 2 2 ··· 2 4 ··· 4 4 ··· 4

62 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 type + + + + - + - + - + + image C1 C2 C2 C4 C4 C4 D4 Q8 D5 Dic5 D10 Dic10 C4×D5 D20 C5⋊D4 C23⋊C4 C23.1D10 kernel C24.2D10 C2×C23.D5 C10×C22⋊C4 C23.D5 C5×C22⋊C4 C22×Dic5 C22×C10 C22×C10 C2×C22⋊C4 C22⋊C4 C24 C23 C23 C23 C23 C10 C2 # reps 1 2 1 4 4 4 3 1 2 4 2 4 8 4 8 2 8

Matrix representation of C24.2D10 in GL6(𝔽41)

 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 1 0 0 0 0 0 0 1
,
 40 0 0 0 0 0 0 40 0 0 0 0 0 0 17 7 6 35 0 0 35 24 0 35 0 0 0 0 18 6 0 0 0 0 35 23
,
 40 0 0 0 0 0 0 40 0 0 0 0 0 0 24 34 2 39 0 0 6 17 2 0 0 0 0 0 18 6 0 0 0 0 35 23
,
 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
,
 0 40 0 0 0 0 1 0 0 0 0 0 0 0 0 36 26 13 0 0 33 9 12 14 0 0 14 39 40 6 0 0 27 28 8 33
,
 32 0 0 0 0 0 0 9 0 0 0 0 0 0 7 34 31 7 0 0 1 34 4 18 0 0 0 0 20 18 0 0 0 0 21 21

`G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,17,35,0,0,0,0,7,24,0,0,0,0,6,0,18,35,0,0,35,35,6,23],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,24,6,0,0,0,0,34,17,0,0,0,0,2,2,18,35,0,0,39,0,6,23],[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],[0,1,0,0,0,0,40,0,0,0,0,0,0,0,0,33,14,27,0,0,36,9,39,28,0,0,26,12,40,8,0,0,13,14,6,33],[32,0,0,0,0,0,0,9,0,0,0,0,0,0,7,1,0,0,0,0,34,34,0,0,0,0,31,4,20,21,0,0,7,18,18,21] >;`

C24.2D10 in GAP, Magma, Sage, TeX

`C_2^4._2D_{10}`
`% in TeX`

`G:=Group("C2^4.2D10");`
`// GroupNames label`

`G:=SmallGroup(320,85);`
`// by ID`

`G=gap.SmallGroup(320,85);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,253,64,1123,851,12550]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^10=b,f^2=a*b*c*d,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,f*b*f^-1=b*d=d*b,b*e=e*b,e*c*e^-1=f*c*f^-1=c*d=d*c,d*e=e*d,d*f=f*d,f*e*f^-1=c*d*e^9>;`
`// generators/relations`

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