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## G = C24⋊3D10order 320 = 26·5

### 2nd semidirect product of C24 and D10 acting via D10/C5=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C24⋊3D10
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C23×D5 — C2×D4×D5 — C24⋊3D10
 Lower central C5 — C2×C10 — C24⋊3D10
 Upper central C1 — C22 — C22≀C2

Generators and relations for C243D10
G = < a,b,c,d,e,f | a2=b2=c2=d2=e10=f2=1, ab=ba, eae-1=faf=ac=ca, ad=da, bc=cb, ebe-1=bd=db, fbf=bcd, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef=e-1 >

Subgroups: 1502 in 346 conjugacy classes, 103 normal (39 characteristic)
C1, C2, C2 [×2], C2 [×10], C4 [×8], C22, C22 [×2], C22 [×34], C5, C2×C4, C2×C4 [×2], C2×C4 [×11], D4 [×20], C23 [×2], C23 [×2], C23 [×17], D5 [×5], C10, C10 [×2], C10 [×5], C22⋊C4, C22⋊C4 [×2], C22⋊C4 [×9], C4⋊C4 [×4], C22×C4 [×4], C2×D4, C2×D4 [×2], C2×D4 [×17], C24, C24 [×2], Dic5 [×5], C20 [×3], D10 [×4], D10 [×15], C2×C10, C2×C10 [×2], C2×C10 [×15], C2×C22⋊C4, C22≀C2, C22≀C2 [×3], C4⋊D4 [×4], C22.D4 [×4], C22×D4 [×2], C4×D5 [×4], D20 [×2], C2×Dic5, C2×Dic5 [×4], C2×Dic5 [×2], C5⋊D4 [×14], C2×C20, C2×C20 [×2], C5×D4 [×4], C22×D5 [×3], C22×D5 [×4], C22×D5 [×6], C22×C10 [×2], C22×C10 [×2], C22×C10 [×4], C233D4, C10.D4 [×2], C4⋊Dic5 [×2], D10⋊C4 [×4], C23.D5, C23.D5 [×4], C5×C22⋊C4, C5×C22⋊C4 [×2], C2×C4×D5, C2×C4×D5 [×2], C2×D20, D4×D5 [×4], C22×Dic5, C2×C5⋊D4 [×2], C2×C5⋊D4 [×6], C2×C5⋊D4 [×4], D4×C10, D4×C10 [×2], C23×D5 [×2], C23×C10, D5×C22⋊C4, D10.12D4 [×2], D10⋊D4 [×2], C22.D20, C23.18D10, C23⋊D10 [×2], C202D4 [×2], C242D5, C5×C22≀C2, C2×D4×D5, C22×C5⋊D4, C243D10
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C24, D10 [×7], C22×D4, 2+ 1+4 [×2], C22×D5 [×7], C233D4, D4×D5 [×2], C23×D5, C2×D4×D5, D46D10 [×2], C243D10

Smallest permutation representation of C243D10
On 80 points
Generators in S80
```(1 63)(2 31)(3 65)(4 33)(5 67)(6 35)(7 69)(8 37)(9 61)(10 39)(11 74)(12 48)(13 76)(14 50)(15 78)(16 42)(17 80)(18 44)(19 72)(20 46)(21 43)(22 71)(23 45)(24 73)(25 47)(26 75)(27 49)(28 77)(29 41)(30 79)(32 59)(34 51)(36 53)(38 55)(40 57)(52 68)(54 70)(56 62)(58 64)(60 66)
(1 24)(2 74)(3 26)(4 76)(5 28)(6 78)(7 30)(8 80)(9 22)(10 72)(11 31)(12 59)(13 33)(14 51)(15 35)(16 53)(17 37)(18 55)(19 39)(20 57)(21 70)(23 62)(25 64)(27 66)(29 68)(32 48)(34 50)(36 42)(38 44)(40 46)(41 52)(43 54)(45 56)(47 58)(49 60)(61 71)(63 73)(65 75)(67 77)(69 79)
(1 57)(2 58)(3 59)(4 60)(5 51)(6 52)(7 53)(8 54)(9 55)(10 56)(11 25)(12 26)(13 27)(14 28)(15 29)(16 30)(17 21)(18 22)(19 23)(20 24)(31 64)(32 65)(33 66)(34 67)(35 68)(36 69)(37 70)(38 61)(39 62)(40 63)(41 78)(42 79)(43 80)(44 71)(45 72)(46 73)(47 74)(48 75)(49 76)(50 77)
(1 63)(2 64)(3 65)(4 66)(5 67)(6 68)(7 69)(8 70)(9 61)(10 62)(11 47)(12 48)(13 49)(14 50)(15 41)(16 42)(17 43)(18 44)(19 45)(20 46)(21 80)(22 71)(23 72)(24 73)(25 74)(26 75)(27 76)(28 77)(29 78)(30 79)(31 58)(32 59)(33 60)(34 51)(35 52)(36 53)(37 54)(38 55)(39 56)(40 57)
(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 10)(2 9)(3 8)(4 7)(5 6)(11 22)(12 21)(13 30)(14 29)(15 28)(16 27)(17 26)(18 25)(19 24)(20 23)(31 38)(32 37)(33 36)(34 35)(39 40)(41 77)(42 76)(43 75)(44 74)(45 73)(46 72)(47 71)(48 80)(49 79)(50 78)(51 52)(53 60)(54 59)(55 58)(56 57)(61 64)(62 63)(65 70)(66 69)(67 68)```

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

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

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

50 conjugacy classes

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

50 irreducible representations

 dim 1 1 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 C2 C2 D4 D5 D10 D10 D10 2+ 1+4 D4×D5 D4⋊6D10 kernel C24⋊3D10 D5×C22⋊C4 D10.12D4 D10⋊D4 C22.D20 C23.18D10 C23⋊D10 C20⋊2D4 C24⋊2D5 C5×C22≀C2 C2×D4×D5 C22×C5⋊D4 C22×D5 C22≀C2 C22⋊C4 C2×D4 C24 C10 C22 C2 # reps 1 1 2 2 1 1 2 2 1 1 1 1 4 2 6 6 2 2 4 8

Matrix representation of C243D10 in GL8(𝔽41)

 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 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 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 0 0 0 0 0 0 0 0 23 35 0 0 0 0 0 0 6 18 0 0 0 0 0 0 0 0 18 6 0 0 0 0 0 0 35 23
,
 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 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
,
 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 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
,
 9 13 11 25 0 0 0 0 28 13 16 25 0 0 0 0 30 16 32 28 0 0 0 0 25 16 13 28 0 0 0 0 0 0 0 0 0 0 40 35 0 0 0 0 0 0 6 35 0 0 0 0 40 35 0 0 0 0 0 0 6 35 0 0
,
 9 0 11 0 0 0 0 0 28 32 16 30 0 0 0 0 30 0 32 0 0 0 0 0 25 11 13 9 0 0 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 6 1 0 0 0 0 40 0 0 0 0 0 0 0 6 1 0 0

`G:=sub<GL(8,GF(41))| [40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,23,6,0,0,0,0,0,0,35,18,0,0,0,0,0,0,0,0,18,35,0,0,0,0,0,0,6,23],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40],[40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40],[9,28,30,25,0,0,0,0,13,13,16,16,0,0,0,0,11,16,32,13,0,0,0,0,25,25,28,28,0,0,0,0,0,0,0,0,0,0,40,6,0,0,0,0,0,0,35,35,0,0,0,0,40,6,0,0,0,0,0,0,35,35,0,0],[9,28,30,25,0,0,0,0,0,32,0,11,0,0,0,0,11,16,32,13,0,0,0,0,0,30,0,9,0,0,0,0,0,0,0,0,0,0,40,6,0,0,0,0,0,0,0,1,0,0,0,0,40,6,0,0,0,0,0,0,0,1,0,0] >;`

C243D10 in GAP, Magma, Sage, TeX

`C_2^4\rtimes_3D_{10}`
`% in TeX`

`G:=Group("C2^4:3D10");`
`// GroupNames label`

`G:=SmallGroup(320,1261);`
`// by ID`

`G=gap.SmallGroup(320,1261);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,219,675,297,12550]);`
`// 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,b*c=c*b,e*b*e^-1=b*d=d*b,f*b*f=b*c*d,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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