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## G = C20.76C24order 320 = 26·5

### 23rd non-split extension by C20 of C24 acting via C24/C23=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C20.76C24
 Chief series C1 — C5 — C10 — C20 — C5⋊2C8 — C2×C5⋊2C8 — D4.Dic5 — C20.76C24
 Lower central C5 — C10 — C20.76C24
 Upper central C1 — C4 — C2×C4○D4

Generators and relations for C20.76C24
G = < a,b,c,d,e | a20=c2=d2=e2=1, b2=a5, bab-1=a9, ac=ca, ad=da, ae=ea, bc=cb, bd=db, ebe=a10b, dcd=a10c, ce=ec, de=ed >

Subgroups: 494 in 258 conjugacy classes, 187 normal (17 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C5, C8, C2×C4, C2×C4, D4, Q8, C23, C10, C10, C2×C8, M4(2), C22×C4, C2×D4, C2×Q8, C4○D4, C20, C20, C2×C10, C2×C10, C2×C10, C2×M4(2), C8○D4, C2×C4○D4, C52C8, C2×C20, C2×C20, C5×D4, C5×Q8, C22×C10, Q8○M4(2), C2×C52C8, C4.Dic5, C22×C20, D4×C10, Q8×C10, C5×C4○D4, C2×C4.Dic5, D4.Dic5, C10×C4○D4, C20.76C24
Quotients: C1, C2, C4, C22, C2×C4, C23, D5, C22×C4, C24, Dic5, D10, C23×C4, C2×Dic5, C22×D5, Q8○M4(2), C22×Dic5, C23×D5, C23×Dic5, C20.76C24

Smallest permutation representation of C20.76C24
On 80 points
Generators in S80
```(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 55 6 60 11 45 16 50)(2 44 7 49 12 54 17 59)(3 53 8 58 13 43 18 48)(4 42 9 47 14 52 19 57)(5 51 10 56 15 41 20 46)(21 66 26 71 31 76 36 61)(22 75 27 80 32 65 37 70)(23 64 28 69 33 74 38 79)(24 73 29 78 34 63 39 68)(25 62 30 67 35 72 40 77)
(1 26)(2 27)(3 28)(4 29)(5 30)(6 31)(7 32)(8 33)(9 34)(10 35)(11 36)(12 37)(13 38)(14 39)(15 40)(16 21)(17 22)(18 23)(19 24)(20 25)(41 77)(42 78)(43 79)(44 80)(45 61)(46 62)(47 63)(48 64)(49 65)(50 66)(51 67)(52 68)(53 69)(54 70)(55 71)(56 72)(57 73)(58 74)(59 75)(60 76)
(1 31)(2 32)(3 33)(4 34)(5 35)(6 36)(7 37)(8 38)(9 39)(10 40)(11 21)(12 22)(13 23)(14 24)(15 25)(16 26)(17 27)(18 28)(19 29)(20 30)(41 62)(42 63)(43 64)(44 65)(45 66)(46 67)(47 68)(48 69)(49 70)(50 71)(51 72)(52 73)(53 74)(54 75)(55 76)(56 77)(57 78)(58 79)(59 80)(60 61)
(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)```

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

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

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

74 conjugacy classes

 class 1 2A 2B ··· 2H 4A 4B 4C ··· 4I 5A 5B 8A ··· 8P 10A ··· 10F 10G ··· 10R 20A ··· 20H 20I ··· 20T order 1 2 2 ··· 2 4 4 4 ··· 4 5 5 8 ··· 8 10 ··· 10 10 ··· 10 20 ··· 20 20 ··· 20 size 1 1 2 ··· 2 1 1 2 ··· 2 2 2 10 ··· 10 2 ··· 2 4 ··· 4 2 ··· 2 4 ··· 4

74 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 4 4 type + + + + + + - - - + image C1 C2 C2 C2 C4 C4 C4 D5 D10 Dic5 Dic5 Dic5 D10 Q8○M4(2) C20.76C24 kernel C20.76C24 C2×C4.Dic5 D4.Dic5 C10×C4○D4 D4×C10 Q8×C10 C5×C4○D4 C2×C4○D4 C22×C4 C2×D4 C2×Q8 C4○D4 C4○D4 C5 C1 # reps 1 6 8 1 6 2 8 2 6 6 2 8 8 2 8

Matrix representation of C20.76C24 in GL6(𝔽41)

 23 0 0 0 0 0 38 25 0 0 0 0 0 0 9 0 0 0 0 0 0 9 0 0 0 0 0 0 9 0 0 0 0 0 0 9
,
 5 24 0 0 0 0 16 36 0 0 0 0 0 0 0 15 1 23 0 0 0 17 0 26 0 0 9 40 0 1 0 0 0 5 0 24
,
 40 0 0 0 0 0 0 40 0 0 0 0 0 0 0 9 26 32 0 0 32 0 24 40 0 0 0 0 1 0 0 0 0 0 36 40
,
 40 0 0 0 0 0 0 40 0 0 0 0 0 0 0 1 29 27 0 0 1 0 11 32 0 0 0 0 9 20 0 0 0 0 37 32
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 35 0 0 0 1 0 26 0 0 0 0 40 0 0 0 0 0 0 40

`G:=sub<GL(6,GF(41))| [23,38,0,0,0,0,0,25,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,9],[5,16,0,0,0,0,24,36,0,0,0,0,0,0,0,0,9,0,0,0,15,17,40,5,0,0,1,0,0,0,0,0,23,26,1,24],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,32,0,0,0,0,9,0,0,0,0,0,26,24,1,36,0,0,32,40,0,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,29,11,9,37,0,0,27,32,20,32],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,35,26,0,40] >;`

C20.76C24 in GAP, Magma, Sage, TeX

`C_{20}._{76}C_2^4`
`% in TeX`

`G:=Group("C20.76C2^4");`
`// GroupNames label`

`G:=SmallGroup(320,1491);`
`// by ID`

`G=gap.SmallGroup(320,1491);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,387,1123,102,12550]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e|a^20=c^2=d^2=e^2=1,b^2=a^5,b*a*b^-1=a^9,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,e*b*e=a^10*b,d*c*d=a^10*c,c*e=e*c,d*e=e*d>;`
`// generators/relations`

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