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## G = C24⋊4D10order 320 = 26·5

### 3rd semidirect product of C24 and D10 acting via D10/C5=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C24⋊4D10
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C23×D5 — C2×D4×D5 — C24⋊4D10
 Lower central C5 — C2×C10 — C24⋊4D10
 Upper central C1 — C22 — C22≀C2

Generators and relations for C244D10
G = < a,b,c,d,e,f | a2=b2=c2=d2=e10=f2=1, ab=ba, eae-1=faf=ac=ca, ad=da, fbf=bc=cb, ebe-1=bd=db, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef=e-1 >

Subgroups: 1910 in 428 conjugacy classes, 115 normal (27 characteristic)
C1, C2, C2 [×2], C2 [×12], C4 [×9], C22, C22 [×4], C22 [×40], C5, C2×C4, C2×C4 [×2], C2×C4 [×12], D4 [×34], C23 [×2], C23 [×2], C23 [×24], D5 [×6], C10, C10 [×2], C10 [×6], C42, C22⋊C4, C22⋊C4 [×2], C22⋊C4 [×5], C4⋊C4 [×2], C22×C4 [×4], C2×D4, C2×D4 [×2], C2×D4 [×29], C24, C24 [×3], Dic5 [×4], Dic5 [×2], C20 [×3], D10 [×4], D10 [×22], C2×C10, C2×C10 [×4], C2×C10 [×14], C4×D4 [×2], C22≀C2, C22≀C2 [×3], C4⋊D4 [×4], C41D4, C22×D4 [×4], C4×D5 [×4], D20 [×5], C2×Dic5 [×4], C2×Dic5 [×4], C5⋊D4 [×8], C5⋊D4 [×16], C2×C20, C2×C20 [×2], C5×D4 [×5], C22×D5 [×4], C22×D5 [×15], C22×C10 [×2], C22×C10 [×2], C22×C10 [×5], D42, C4×Dic5, C10.D4 [×2], D10⋊C4 [×4], C23.D5, C5×C22⋊C4, C5×C22⋊C4 [×2], C2×C4×D5 [×2], C2×D20, C2×D20 [×2], D4×D5 [×8], C22×Dic5 [×2], C2×C5⋊D4 [×10], C2×C5⋊D4 [×8], D4×C10, D4×C10 [×2], C23×D5, C23×D5 [×2], C23×C10, Dic54D4 [×2], C22⋊D20 [×2], D10⋊D4 [×2], C23⋊D10, Dic5⋊D4 [×2], C20⋊D4, C5×C22≀C2, C2×D4×D5 [×2], C22×C5⋊D4 [×2], C244D10
Quotients: C1, C2 [×15], C22 [×35], D4 [×8], C23 [×15], D5, C2×D4 [×12], C24, D10 [×7], C22×D4 [×2], 2+ 1+4, C22×D5 [×7], D42, D4×D5 [×4], C23×D5, C2×D4×D5 [×2], D46D10, C244D10

Smallest permutation representation of C244D10
On 80 points
Generators in S80
```(1 49)(2 45)(3 41)(4 47)(5 43)(6 57)(7 53)(8 59)(9 55)(10 51)(11 42)(12 48)(13 44)(14 50)(15 46)(16 56)(17 52)(18 58)(19 54)(20 60)(21 73)(22 66)(23 75)(24 68)(25 77)(26 70)(27 79)(28 62)(29 71)(30 64)(31 63)(32 72)(33 65)(34 74)(35 67)(36 76)(37 69)(38 78)(39 61)(40 80)
(1 21)(2 27)(3 23)(4 29)(5 25)(6 30)(7 26)(8 22)(9 28)(10 24)(11 31)(12 37)(13 33)(14 39)(15 35)(16 36)(17 32)(18 38)(19 34)(20 40)(41 75)(42 63)(43 77)(44 65)(45 79)(46 67)(47 71)(48 69)(49 73)(50 61)(51 68)(52 72)(53 70)(54 74)(55 62)(56 76)(57 64)(58 78)(59 66)(60 80)
(1 13)(2 14)(3 15)(4 11)(5 12)(6 17)(7 18)(8 19)(9 20)(10 16)(21 33)(22 34)(23 35)(24 36)(25 37)(26 38)(27 39)(28 40)(29 31)(30 32)(41 46)(42 47)(43 48)(44 49)(45 50)(51 56)(52 57)(53 58)(54 59)(55 60)(61 79)(62 80)(63 71)(64 72)(65 73)(66 74)(67 75)(68 76)(69 77)(70 78)
(1 7)(2 8)(3 9)(4 10)(5 6)(11 16)(12 17)(13 18)(14 19)(15 20)(21 26)(22 27)(23 28)(24 29)(25 30)(31 36)(32 37)(33 38)(34 39)(35 40)(41 55)(42 56)(43 57)(44 58)(45 59)(46 60)(47 51)(48 52)(49 53)(50 54)(61 74)(62 75)(63 76)(64 77)(65 78)(66 79)(67 80)(68 71)(69 72)(70 73)
(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 6)(2 10)(3 9)(4 8)(5 7)(11 19)(12 18)(13 17)(14 16)(15 20)(21 32)(22 31)(23 40)(24 39)(25 38)(26 37)(27 36)(28 35)(29 34)(30 33)(41 60)(42 59)(43 58)(44 57)(45 56)(46 55)(47 54)(48 53)(49 52)(50 51)(61 76)(62 75)(63 74)(64 73)(65 72)(66 71)(67 80)(68 79)(69 78)(70 77)```

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

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

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

53 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 2L 2M 2N 2O 4A 4B 4C 4D 4E 4F 4G 4H 4I 5A 5B 10A ··· 10F 10G ··· 10R 10S 10T 20A ··· 20F order 1 2 2 2 2 2 2 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 10 10 10 10 20 20 4 4 4 10 10 10 10 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 1 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 D4 D5 D10 D10 D10 2+ 1+4 D4×D5 D4⋊6D10 kernel C24⋊4D10 Dic5⋊4D4 C22⋊D20 D10⋊D4 C23⋊D10 Dic5⋊D4 C20⋊D4 C5×C22≀C2 C2×D4×D5 C22×C5⋊D4 C5⋊D4 C22≀C2 C22⋊C4 C2×D4 C24 C10 C22 C2 # reps 1 2 2 2 1 2 1 1 2 2 8 2 6 6 2 1 8 4

Matrix representation of C244D10 in GL6(𝔽41)

 1 23 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 18 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 9 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
,
 40 0 0 0 0 0 9 1 0 0 0 0 0 0 7 7 0 0 0 0 34 40 0 0 0 0 0 0 1 0 0 0 0 0 18 40
,
 1 0 0 0 0 0 32 40 0 0 0 0 0 0 7 7 0 0 0 0 40 34 0 0 0 0 0 0 40 0 0 0 0 0 0 40

`G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,23,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[40,0,0,0,0,0,18,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,9,40],[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],[40,9,0,0,0,0,0,1,0,0,0,0,0,0,7,34,0,0,0,0,7,40,0,0,0,0,0,0,1,18,0,0,0,0,0,40],[1,32,0,0,0,0,0,40,0,0,0,0,0,0,7,40,0,0,0,0,7,34,0,0,0,0,0,0,40,0,0,0,0,0,0,40] >;`

C244D10 in GAP, Magma, Sage, TeX

`C_2^4\rtimes_4D_{10}`
`% in TeX`

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

`G:=SmallGroup(320,1262);`
`// by ID`

`G=gap.SmallGroup(320,1262);`
`# by ID`

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

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

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