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## G = C24.41D10order 320 = 26·5

### 41st non-split extension by C24 of D10 acting via D10/C5=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C24.41D10
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C2×C4×D5 — C2×C4○D20 — C24.41D10
 Lower central C5 — C2×C10 — C24.41D10
 Upper central C1 — C22 — C22×D4

Generators and relations for C24.41D10
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e10=f2=d, ab=ba, ac=ca, eae-1=ad=da, faf-1=acd, bc=cb, fbf-1=bd=db, be=eb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e9 >

Subgroups: 1198 in 334 conjugacy classes, 111 normal (21 characteristic)
C1, C2, C2 [×2], C2 [×8], C4 [×4], C4 [×6], C22, C22 [×2], C22 [×28], C5, C2×C4 [×2], C2×C4 [×4], C2×C4 [×10], D4 [×22], Q8 [×2], C23, C23 [×4], C23 [×10], D5 [×2], C10, C10 [×2], C10 [×6], C42 [×2], C22⋊C4 [×10], C4⋊C4 [×2], C22×C4, C22×C4 [×2], C2×D4 [×4], C2×D4 [×15], C2×Q8, C4○D4 [×4], C24 [×2], Dic5 [×6], C20 [×4], D10 [×6], C2×C10, C2×C10 [×2], C2×C10 [×22], C42⋊C2, C22≀C2 [×4], C4⋊D4 [×4], C4.4D4 [×2], C41D4 [×2], C22×D4, C2×C4○D4, Dic10 [×2], C4×D5 [×4], D20 [×2], C2×Dic5 [×6], C5⋊D4 [×12], C2×C20 [×2], C2×C20 [×4], C5×D4 [×8], C22×D5 [×2], C22×C10, C22×C10 [×4], C22×C10 [×8], C22.29C24, C4×Dic5 [×2], C4⋊Dic5 [×2], C23.D5 [×10], C2×Dic10, C2×C4×D5 [×2], C2×D20, C4○D20 [×4], C2×C5⋊D4 [×10], C22×C20, D4×C10 [×4], D4×C10 [×4], C23×C10 [×2], C23.21D10, C20.17D4 [×2], C202D4 [×4], C20⋊D4 [×2], C242D5 [×4], C2×C4○D20, D4×C2×C10, C24.41D10
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], C22.29C24, C2×C5⋊D4 [×6], C23×D5, D46D10 [×2], C22×C5⋊D4, C24.41D10

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

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

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

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

62 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 4A 4B 4C 4D 4E ··· 4J 5A 5B 10A ··· 10N 10O ··· 10AD 20A ··· 20H order 1 2 2 2 2 2 2 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 20 ··· 20 2 2 2 ··· 2 4 ··· 4 4 ··· 4

62 irreducible representations

 dim 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 D4 D5 D10 D10 D10 C5⋊D4 2+ 1+4 D4⋊6D10 kernel C24.41D10 C23.21D10 C20.17D4 C20⋊2D4 C20⋊D4 C24⋊2D5 C2×C4○D20 D4×C2×C10 C2×C20 C22×D4 C22×C4 C2×D4 C24 C2×C4 C10 C2 # reps 1 1 2 4 2 4 1 1 4 2 2 8 4 16 2 8

Matrix representation of C24.41D10 in GL6(𝔽41)

 40 0 0 0 0 0 17 1 0 0 0 0 0 0 1 0 0 0 0 0 21 40 0 0 0 0 26 0 40 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 26 0 40 0 0 0 26 0 0 40
,
 40 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 0 0 0 0 0 40
,
 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
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 4 25 0 0 0 0 2 37 0 0 0 0 18 38 0 31 0 0 4 38 10 0
,
 1 17 0 0 0 0 0 40 0 0 0 0 0 0 13 0 0 40 0 0 27 0 10 10 0 0 4 4 0 28 0 0 6 0 0 28

`G:=sub<GL(6,GF(41))| [40,17,0,0,0,0,0,1,0,0,0,0,0,0,1,21,26,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,26,26,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,2,18,4,0,0,25,37,38,38,0,0,0,0,0,10,0,0,0,0,31,0],[1,0,0,0,0,0,17,40,0,0,0,0,0,0,13,27,4,6,0,0,0,0,4,0,0,0,0,10,0,0,0,0,40,10,28,28] >;`

C24.41D10 in GAP, Magma, Sage, TeX

`C_2^4._{41}D_{10}`
`% in TeX`

`G:=Group("C2^4.41D10");`
`// GroupNames label`

`G:=SmallGroup(320,1477);`
`// by ID`

`G=gap.SmallGroup(320,1477);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,758,675,570,12550]);`
`// Polycyclic`

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

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