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## G = C20.C24order 320 = 26·5

### 35th non-split extension by C20 of C24 acting via C24/C22=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — C20.C24
 Chief series C1 — C5 — C10 — C20 — D20 — C2×D20 — C2×C4○D20 — C20.C24
 Lower central C5 — C10 — C20 — C20.C24
 Upper central C1 — C4 — C22×C4 — C2×C4○D4

Generators and relations for C20.C24
G = < a,b,c,d,e | a20=b2=c2=e2=1, d2=a10, bab=a-1, ac=ca, ad=da, eae=a11, bc=cb, bd=db, ebe=a5b, cd=dc, ece=a10c, de=ed >

Subgroups: 830 in 262 conjugacy classes, 107 normal (45 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, D5, C10, C10, C2×C8, M4(2), D8, SD16, Q16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4, C4○D4, Dic5, C20, C20, D10, C2×C10, C2×C10, C2×M4(2), C4○D8, C8⋊C22, C8.C22, C2×C4○D4, C2×C4○D4, C52C8, Dic10, Dic10, C4×D5, D20, D20, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C5×D4, C5×D4, C5×Q8, C5×Q8, C22×D5, C22×C10, C22×C10, D8⋊C22, C2×C52C8, C4.Dic5, D4⋊D5, D4.D5, Q8⋊D5, C5⋊Q16, C2×Dic10, C2×C4×D5, C2×D20, C4○D20, C4○D20, C2×C5⋊D4, C22×C20, C22×C20, D4×C10, D4×C10, Q8×C10, C5×C4○D4, C5×C4○D4, C2×C4.Dic5, D4.D10, C20.C23, D4⋊D10, D4.8D10, D4.9D10, C2×C4○D20, C10×C4○D4, C20.C24
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C24, D10, C22×D4, C5⋊D4, C22×D5, D8⋊C22, C2×C5⋊D4, C23×D5, C22×C5⋊D4, C20.C24

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

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

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

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

62 conjugacy classes

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

62 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 D4 D4 D5 D10 D10 D10 D10 C5⋊D4 C5⋊D4 D8⋊C22 C20.C24 kernel C20.C24 C2×C4.Dic5 D4.D10 C20.C23 D4⋊D10 D4.8D10 D4.9D10 C2×C4○D20 C10×C4○D4 C2×C20 C22×C10 C2×C4○D4 C22×C4 C2×D4 C2×Q8 C4○D4 C2×C4 C23 C5 C1 # reps 1 1 2 2 2 4 2 1 1 3 1 2 2 2 2 8 12 4 2 8

Matrix representation of C20.C24 in GL6(𝔽41)

 0 40 0 0 0 0 1 6 0 0 0 0 0 0 0 1 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 1 0
,
 35 6 0 0 0 0 1 6 0 0 0 0 0 0 1 0 0 0 0 0 0 40 0 0 0 0 0 0 0 1 0 0 0 0 1 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 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 32 0 0 0 0 0 0 32 0 0 0 0 0 0 32 0 0 0 0 0 0 32
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0

`G:=sub<GL(6,GF(41))| [0,1,0,0,0,0,40,6,0,0,0,0,0,0,0,40,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,40,0],[35,1,0,0,0,0,6,6,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,32,0,0,0,0,0,0,32,0,0,0,0,0,0,32,0,0,0,0,0,0,32],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0] >;`

C20.C24 in GAP, Magma, Sage, TeX

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

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

`G:=SmallGroup(320,1494);`
`// by ID`

`G=gap.SmallGroup(320,1494);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,184,675,570,1684,235,102,12550]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e|a^20=b^2=c^2=e^2=1,d^2=a^10,b*a*b=a^-1,a*c=c*a,a*d=d*a,e*a*e=a^11,b*c=c*b,b*d=d*b,e*b*e=a^5*b,c*d=d*c,e*c*e=a^10*c,d*e=e*d>;`
`// generators/relations`

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