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## G = C24.56D10order 320 = 26·5

### 14th non-split extension by C24 of D10 acting via D10/D5=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C24.56D10
 Chief series C1 — C5 — C10 — C2×C10 — C2×Dic5 — C22×Dic5 — C23×Dic5 — C24.56D10
 Lower central C5 — C2×C10 — C24.56D10
 Upper central C1 — C22 — C22≀C2

Generators and relations for C24.56D10
G = < a,b,c,d,e,f | a2=b2=c2=d2=e10=1, f2=d, ab=ba, ac=ca, eae-1=faf-1=ad=da, ebe-1=fbf-1=bc=cb, bd=db, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e-1 >

Subgroups: 1070 in 330 conjugacy classes, 111 normal (39 characteristic)
C1, C2, C2 [×2], C2 [×8], C4 [×12], C22, C22 [×6], C22 [×20], C5, C2×C4, C2×C4 [×2], C2×C4 [×25], D4 [×14], Q8 [×2], C23 [×2], C23 [×2], C23 [×7], D5, C10, C10 [×2], C10 [×7], C42 [×2], C22⋊C4, C22⋊C4 [×2], C22⋊C4 [×7], C4⋊C4 [×6], C22×C4 [×12], C2×D4, C2×D4 [×2], C2×D4 [×4], C2×Q8, C4○D4 [×4], C24, Dic5 [×4], Dic5 [×5], C20 [×3], D10 [×3], C2×C10, C2×C10 [×6], C2×C10 [×17], C42⋊C2, C4×D4 [×4], C22≀C2, C22≀C2, C4⋊D4 [×2], C22⋊Q8 [×2], C22.D4 [×2], C23×C4, C2×C4○D4, Dic10 [×2], C4×D5 [×2], C2×Dic5 [×3], C2×Dic5 [×8], C2×Dic5 [×12], C5⋊D4 [×8], C2×C20, C2×C20 [×2], C5×D4 [×6], C22×D5, C22×C10 [×2], C22×C10 [×2], C22×C10 [×6], C22.19C24, C4×Dic5 [×2], C10.D4 [×4], C4⋊Dic5 [×2], D10⋊C4 [×2], C23.D5, C23.D5 [×4], C5×C22⋊C4, C5×C22⋊C4 [×2], C2×Dic10, C2×C4×D5, D42D5 [×4], C22×Dic5 [×3], C22×Dic5 [×4], C22×Dic5 [×4], C2×C5⋊D4 [×2], C2×C5⋊D4 [×2], D4×C10, D4×C10 [×2], C23×C10, C23.11D10, Dic5.14D4 [×2], Dic54D4 [×2], C22.D20, D4×Dic5 [×2], C23.18D10, Dic5⋊D4 [×2], C242D5, C5×C22≀C2, C2×D42D5, C23×Dic5, C24.56D10
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C4○D4 [×4], C24, D10 [×7], C22×D4, C2×C4○D4 [×2], C22×D5 [×7], C22.19C24, D4×D5 [×2], D42D5 [×4], C23×D5, C2×D4×D5, C2×D42D5 [×2], C24.56D10

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

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

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

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

56 conjugacy classes

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

56 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 D4 D5 C4○D4 D10 D10 D10 D4×D5 D4⋊2D5 kernel C24.56D10 C23.11D10 Dic5.14D4 Dic5⋊4D4 C22.D20 D4×Dic5 C23.18D10 Dic5⋊D4 C24⋊2D5 C5×C22≀C2 C2×D4⋊2D5 C23×Dic5 C2×Dic5 C22≀C2 C2×C10 C22⋊C4 C2×D4 C24 C22 C22 # reps 1 1 2 2 1 2 1 2 1 1 1 1 4 2 8 6 6 2 4 8

Matrix representation of C24.56D10 in GL6(𝔽41)

 40 23 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 23 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 1 0 0 0 0 0 0 1
,
 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
,
 1 0 0 0 0 0 9 40 0 0 0 0 0 0 0 34 0 0 0 0 6 35 0 0 0 0 0 0 0 1 0 0 0 0 1 0
,
 9 0 0 0 0 0 40 32 0 0 0 0 0 0 13 14 0 0 0 0 29 28 0 0 0 0 0 0 0 9 0 0 0 0 9 0

`G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,23,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40],[40,0,0,0,0,0,23,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[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],[1,9,0,0,0,0,0,40,0,0,0,0,0,0,0,6,0,0,0,0,34,35,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[9,40,0,0,0,0,0,32,0,0,0,0,0,0,13,29,0,0,0,0,14,28,0,0,0,0,0,0,0,9,0,0,0,0,9,0] >;`

C24.56D10 in GAP, Magma, Sage, TeX

`C_2^4._{56}D_{10}`
`% in TeX`

`G:=Group("C2^4.56D10");`
`// GroupNames label`

`G:=SmallGroup(320,1258);`
`// by ID`

`G=gap.SmallGroup(320,1258);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,570,185,12550]);`
`// 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,a*c=c*a,e*a*e^-1=f*a*f^-1=a*d=d*a,e*b*e^-1=f*b*f^-1=b*c=c*b,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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