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## G = C24.36D10order 320 = 26·5

### 36th non-split extension by C24 of D10 acting via D10/C5=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C24.36D10
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C2×C4×D5 — D10.12D4 — C24.36D10
 Lower central C5 — C2×C10 — C24.36D10
 Upper central C1 — C22 — C22≀C2

Generators and relations for C24.36D10
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e10=f2=d, ab=ba, eae-1=ac=ca, ad=da, faf-1=acd, fbf-1=bc=cb, ebe-1=bd=db, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e9 >

Subgroups: 998 in 252 conjugacy classes, 91 normal (27 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×9], C22, C22 [×22], C5, C2×C4, C2×C4 [×2], C2×C4 [×9], D4 [×12], C23 [×2], C23 [×2], C23 [×5], D5 [×2], C10, C10 [×2], C10 [×4], C42, C22⋊C4, C22⋊C4 [×2], C22⋊C4 [×9], C4⋊C4 [×6], C22×C4 [×3], C2×D4, C2×D4 [×2], C2×D4 [×9], C24, Dic5 [×6], C20 [×3], D10 [×6], C2×C10, C2×C10 [×16], C22≀C2, C22≀C2 [×2], C4⋊D4 [×6], C22.D4 [×3], C422C2 [×2], C41D4, C4×D5 [×2], D20, C2×Dic5 [×6], C2×Dic5, C5⋊D4 [×8], C2×C20, C2×C20 [×2], C5×D4 [×3], C22×D5 [×2], C22×C10 [×2], C22×C10 [×2], C22×C10 [×3], C22.54C24, C4×Dic5, C10.D4 [×4], C4⋊Dic5 [×2], D10⋊C4 [×2], C23.D5, C23.D5 [×6], C5×C22⋊C4, C5×C22⋊C4 [×2], C2×C4×D5 [×2], C2×D20, C22×Dic5, C2×C5⋊D4 [×8], D4×C10, D4×C10 [×2], C23×C10, C23.D10 [×2], D10.12D4 [×2], D10⋊D4 [×2], C23.18D10, C202D4 [×2], Dic5⋊D4 [×2], C20⋊D4, C242D5 [×2], C5×C22≀C2, C24.36D10
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], D5, C24, D10 [×7], 2+ 1+4 [×3], C22×D5 [×7], C22.54C24, C23×D5, D46D10 [×3], C24.36D10

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

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

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

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

47 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 4A 4B 4C 4D ··· 4I 5A 5B 10A ··· 10F 10G ··· 10R 10S 10T 20A ··· 20F order 1 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 4 4 4 4 20 20 4 4 4 20 ··· 20 2 2 2 ··· 2 4 ··· 4 8 8 8 ··· 8

47 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 4 4 type + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 D5 D10 D10 D10 2+ 1+4 D4⋊6D10 kernel C24.36D10 C23.D10 D10.12D4 D10⋊D4 C23.18D10 C20⋊2D4 Dic5⋊D4 C20⋊D4 C24⋊2D5 C5×C22≀C2 C22≀C2 C22⋊C4 C2×D4 C24 C10 C2 # reps 1 2 2 2 1 2 2 1 2 1 2 6 6 2 3 12

Matrix representation of C24.36D10 in GL8(𝔽41)

 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 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
,
 40 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 1 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 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 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
,
 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
,
 0 18 0 0 0 0 0 0 23 0 0 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 25 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 40 0
,
 0 0 0 16 0 0 0 0 0 0 25 0 0 0 0 0 0 18 0 0 0 0 0 0 23 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 40 0 0 0 0 0 0 1 0 0 0 0 0 0 40 0 0 0

`G:=sub<GL(8,GF(41))| [1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40],[40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40],[40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[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],[0,23,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,0,25,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0],[0,0,0,23,0,0,0,0,0,0,18,0,0,0,0,0,0,25,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0] >;`

C24.36D10 in GAP, Magma, Sage, TeX

`C_2^4._{36}D_{10}`
`% in TeX`

`G:=Group("C2^4.36D10");`
`// GroupNames label`

`G:=SmallGroup(320,1267);`
`// by ID`

`G=gap.SmallGroup(320,1267);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,758,219,1571,570,12550]);`
`// 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,e*a*e^-1=a*c=c*a,a*d=d*a,f*a*f^-1=a*c*d,f*b*f^-1=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^-1=e^9>;`
`// generators/relations`

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