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## G = C24.34D10order 320 = 26·5

### 34th non-split extension by C24 of D10 acting via D10/C5=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C24.34D10
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C23×D5 — C23⋊D10 — C24.34D10
 Lower central C5 — C2×C10 — C24.34D10
 Upper central C1 — C22 — C22≀C2

Generators and relations for C24.34D10
G = < a,b,c,d,e,f | a2=b2=c2=d2=e10=1, f2=d, ab=ba, eae-1=faf-1=ac=ca, ad=da, bc=cb, ebe-1=bd=db, fbf-1=bcd, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=de-1 >

Subgroups: 1406 in 334 conjugacy classes, 103 normal (39 characteristic)
C1, C2, C2 [×2], C2 [×8], C4 [×10], C22, C22 [×2], C22 [×28], C5, C2×C4, C2×C4 [×2], C2×C4 [×13], D4 [×22], Q8 [×2], C23 [×2], C23 [×2], C23 [×11], D5 [×3], C10, C10 [×2], C10 [×5], C42 [×2], C22⋊C4, C22⋊C4 [×2], C22⋊C4 [×7], C4⋊C4 [×2], C22×C4 [×3], C2×D4, C2×D4 [×2], C2×D4 [×16], C2×Q8, C4○D4 [×4], C24, C24, Dic5 [×4], Dic5 [×3], C20 [×3], D10 [×13], C2×C10, C2×C10 [×2], C2×C10 [×15], C42⋊C2, C22≀C2, C22≀C2 [×3], C4⋊D4 [×4], C4.4D4 [×2], C41D4 [×2], C22×D4, C2×C4○D4, Dic10 [×2], C4×D5 [×2], D20 [×2], C2×Dic5 [×3], C2×Dic5 [×6], C2×Dic5 [×2], C5⋊D4 [×16], C2×C20, C2×C20 [×2], C5×D4 [×4], C22×D5, C22×D5 [×2], C22×D5 [×4], C22×C10 [×2], C22×C10 [×2], C22×C10 [×4], C22.29C24, C4×Dic5 [×2], C10.D4 [×2], D10⋊C4 [×4], C23.D5, C23.D5 [×2], C5×C22⋊C4, C5×C22⋊C4 [×2], C2×Dic10, C2×C4×D5, C2×D20 [×2], D42D5 [×4], C22×Dic5 [×2], C2×C5⋊D4 [×2], C2×C5⋊D4 [×8], C2×C5⋊D4 [×4], D4×C10, D4×C10 [×2], C23×D5, C23×C10, C23.11D10, C22⋊D20, D10⋊D4 [×2], Dic5.5D4 [×2], C23⋊D10, Dic5⋊D4 [×2], C20⋊D4 [×2], C242D5, C5×C22≀C2, C2×D42D5, C22×C5⋊D4, C24.34D10
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], C22.29C24, D4×D5 [×2], C23×D5, C2×D4×D5, D46D10 [×2], C24.34D10

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

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

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

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

50 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 5A 5B 10A ··· 10F 10G ··· 10R 10S 10T 20A ··· 20F order 1 2 2 2 2 2 2 2 2 2 2 2 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 4 4 20 20 20 4 4 4 10 10 10 10 20 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 C24.34D10 C23.11D10 C22⋊D20 D10⋊D4 Dic5.5D4 C23⋊D10 Dic5⋊D4 C20⋊D4 C24⋊2D5 C5×C22≀C2 C2×D4⋊2D5 C22×C5⋊D4 C2×Dic5 C22≀C2 C22⋊C4 C2×D4 C24 C10 C22 C2 # reps 1 1 1 2 2 1 2 2 1 1 1 1 4 2 6 6 2 2 4 8

Matrix representation of C24.34D10 in GL6(𝔽41)

 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 1 0 0 0 0 0 0 1
,
 40 0 0 0 0 0 0 1 0 0 0 0 0 0 17 40 0 0 0 0 1 24 0 0 0 0 0 0 24 1 0 0 0 0 40 17
,
 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 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 7 7 0 0 0 0 34 40 0 0 7 7 0 0 0 0 34 40 0 0
,
 0 40 0 0 0 0 1 0 0 0 0 0 0 0 0 0 40 34 0 0 0 0 0 1 0 0 1 7 0 0 0 0 0 40 0 0

`G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[40,0,0,0,0,0,0,1,0,0,0,0,0,0,17,1,0,0,0,0,40,24,0,0,0,0,0,0,24,40,0,0,0,0,1,17],[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],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,7,34,0,0,0,0,7,40,0,0,7,34,0,0,0,0,7,40,0,0],[0,1,0,0,0,0,40,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,7,40,0,0,40,0,0,0,0,0,34,1,0,0] >;`

C24.34D10 in GAP, Magma, Sage, TeX

`C_2^4._{34}D_{10}`
`% in TeX`

`G:=Group("C2^4.34D10");`
`// GroupNames label`

`G:=SmallGroup(320,1264);`
`// by ID`

`G=gap.SmallGroup(320,1264);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,758,219,675,297,12550]);`
`// 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,e*a*e^-1=f*a*f^-1=a*c=c*a,a*d=d*a,b*c=c*b,e*b*e^-1=b*d=d*b,f*b*f^-1=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^-1=d*e^-1>;`
`// generators/relations`

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