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## G = C24⋊8D10order 320 = 26·5

### 7th semidirect product of C24 and D10 acting via D10/C5=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C24⋊8D10
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C23×D5 — C22×C5⋊D4 — C24⋊8D10
 Lower central C5 — C2×C10 — C24⋊8D10
 Upper central C1 — C22 — C22×D4

Generators and relations for C248D10
G = < a,b,c,d,e,f | a2=b2=c2=d2=e10=f2=1, ab=ba, ac=ca, faf=ad=da, ae=ea, bc=cb, ebe-1=bd=db, fbf=bcd, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef=e-1 >

Subgroups: 1294 in 346 conjugacy classes, 111 normal (25 characteristic)
C1, C2, C2 [×2], C2 [×10], C4 [×8], C22, C22 [×6], C22 [×30], C5, C2×C4 [×2], C2×C4 [×12], D4 [×20], C23 [×3], C23 [×6], C23 [×12], D5 [×2], C10, C10 [×2], C10 [×8], C22⋊C4 [×12], C4⋊C4 [×4], C22×C4, C22×C4 [×3], C2×D4 [×4], C2×D4 [×16], C24 [×2], C24, Dic5 [×6], C20 [×2], D10 [×10], C2×C10, C2×C10 [×6], C2×C10 [×20], C2×C22⋊C4, C22≀C2 [×4], C4⋊D4 [×4], C22.D4 [×4], C22×D4, C22×D4, C2×Dic5 [×6], C2×Dic5 [×4], C5⋊D4 [×12], C2×C20 [×2], C2×C20 [×2], C5×D4 [×8], C22×D5 [×2], C22×D5 [×4], C22×C10 [×3], C22×C10 [×6], C22×C10 [×6], C233D4, C10.D4 [×4], D10⋊C4 [×4], C23.D5 [×8], C22×Dic5, C22×Dic5 [×2], C2×C5⋊D4 [×8], C2×C5⋊D4 [×4], C22×C20, D4×C10 [×4], D4×C10 [×4], C23×D5, C23×C10 [×2], C23.23D10 [×2], C23.18D10 [×2], C23⋊D10 [×2], Dic5⋊D4 [×4], C2×C23.D5, C242D5 [×2], C22×C5⋊D4, D4×C2×C10, C248D10
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C24, D10 [×7], C22×D4, 2+ 1+4 [×2], C5⋊D4 [×4], C22×D5 [×7], C233D4, C2×C5⋊D4 [×6], C23×D5, D46D10 [×2], C22×C5⋊D4, C248D10

Smallest permutation representation of C248D10
On 80 points
Generators in S80
```(1 25)(2 21)(3 22)(4 23)(5 24)(6 36)(7 37)(8 38)(9 39)(10 40)(11 34)(12 35)(13 31)(14 32)(15 33)(16 26)(17 27)(18 28)(19 29)(20 30)(41 80)(42 71)(43 72)(44 73)(45 74)(46 75)(47 76)(48 77)(49 78)(50 79)(51 67)(52 68)(53 69)(54 70)(55 61)(56 62)(57 63)(58 64)(59 65)(60 66)
(1 74)(2 80)(3 76)(4 72)(5 78)(6 77)(7 73)(8 79)(9 75)(10 71)(11 55)(12 51)(13 57)(14 53)(15 59)(16 52)(17 58)(18 54)(19 60)(20 56)(21 41)(22 47)(23 43)(24 49)(25 45)(26 68)(27 64)(28 70)(29 66)(30 62)(31 63)(32 69)(33 65)(34 61)(35 67)(36 48)(37 44)(38 50)(39 46)(40 42)
(1 13)(2 14)(3 15)(4 11)(5 12)(6 19)(7 20)(8 16)(9 17)(10 18)(21 32)(22 33)(23 34)(24 35)(25 31)(26 38)(27 39)(28 40)(29 36)(30 37)(41 69)(42 70)(43 61)(44 62)(45 63)(46 64)(47 65)(48 66)(49 67)(50 68)(51 78)(52 79)(53 80)(54 71)(55 72)(56 73)(57 74)(58 75)(59 76)(60 77)
(1 8)(2 9)(3 10)(4 6)(5 7)(11 19)(12 20)(13 16)(14 17)(15 18)(21 39)(22 40)(23 36)(24 37)(25 38)(26 31)(27 32)(28 33)(29 34)(30 35)(41 46)(42 47)(43 48)(44 49)(45 50)(51 56)(52 57)(53 58)(54 59)(55 60)(61 66)(62 67)(63 68)(64 69)(65 70)(71 76)(72 77)(73 78)(74 79)(75 80)
(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 4)(6 9)(7 8)(11 14)(12 13)(16 20)(17 19)(21 36)(22 40)(23 39)(24 38)(25 37)(26 35)(27 34)(28 33)(29 32)(30 31)(41 61)(42 70)(43 69)(44 68)(45 67)(46 66)(47 65)(48 64)(49 63)(50 62)(51 79)(52 78)(53 77)(54 76)(55 75)(56 74)(57 73)(58 72)(59 71)(60 80)```

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

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

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

62 conjugacy classes

 class 1 2A 2B 2C 2D ··· 2I 2J 2K 2L 2M 4A 4B 4C ··· 4H 5A 5B 10A ··· 10N 10O ··· 10AD 20A ··· 20H order 1 2 2 2 2 ··· 2 2 2 2 2 4 4 4 ··· 4 5 5 10 ··· 10 10 ··· 10 20 ··· 20 size 1 1 1 1 2 ··· 2 4 4 20 20 4 4 20 ··· 20 2 2 2 ··· 2 4 ··· 4 4 ··· 4

62 irreducible representations

 dim 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 D4 D5 D10 D10 D10 C5⋊D4 2+ 1+4 D4⋊6D10 kernel C24⋊8D10 C23.23D10 C23.18D10 C23⋊D10 Dic5⋊D4 C2×C23.D5 C24⋊2D5 C22×C5⋊D4 D4×C2×C10 C22×C10 C22×D4 C22×C4 C2×D4 C24 C23 C10 C2 # reps 1 2 2 2 4 1 2 1 1 4 2 2 8 4 16 2 8

Matrix representation of C248D10 in GL6(𝔽41)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 24 1 0 0 0 0 40 17 0 0 0 0 0 0 23 1 0 0 0 0 5 18
,
 16 32 0 0 0 0 1 25 0 0 0 0 0 0 24 1 38 23 0 0 40 17 36 23 0 0 0 0 18 40 0 0 0 0 36 23
,
 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
,
 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
,
 40 0 0 0 0 0 0 40 0 0 0 0 0 0 7 7 0 0 0 0 34 40 0 0 0 0 0 14 0 34 0 0 27 39 6 35
,
 1 0 0 0 0 0 40 40 0 0 0 0 0 0 7 7 0 0 0 0 40 34 0 0 0 0 0 0 6 34 0 0 0 0 5 35

`G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,24,40,0,0,0,0,1,17,0,0,0,0,0,0,23,5,0,0,0,0,1,18],[16,1,0,0,0,0,32,25,0,0,0,0,0,0,24,40,0,0,0,0,1,17,0,0,0,0,38,36,18,36,0,0,23,23,40,23],[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],[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],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,7,34,0,27,0,0,7,40,14,39,0,0,0,0,0,6,0,0,0,0,34,35],[1,40,0,0,0,0,0,40,0,0,0,0,0,0,7,40,0,0,0,0,7,34,0,0,0,0,0,0,6,5,0,0,0,0,34,35] >;`

C248D10 in GAP, Magma, Sage, TeX

`C_2^4\rtimes_8D_{10}`
`% in TeX`

`G:=Group("C2^4:8D10");`
`// GroupNames label`

`G:=SmallGroup(320,1476);`
`// by ID`

`G=gap.SmallGroup(320,1476);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,758,675,570,12550]);`
`// 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,a*c=c*a,f*a*f=a*d=d*a,a*e=e*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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