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## G = C20.72C24order 320 = 26·5

### 19th non-split extension by C20 of C24 acting via C24/C23=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C20.72C24
 Chief series C1 — C5 — C10 — C20 — C4×D5 — C2×C4×D5 — D5×C4○D4 — C20.72C24
 Lower central C5 — C10 — C20.72C24
 Upper central C1 — C4 — C8○D4

Generators and relations for C20.72C24
G = < a,b,c,d | a40=b2=c2=d2=1, bab=a29, cac=a21, ad=da, bc=cb, bd=db, dcd=a20c >

Subgroups: 734 in 258 conjugacy classes, 147 normal (24 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C5, C8, C8, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, D5, C10, C10, C2×C8, C2×C8, M4(2), M4(2), C22×C4, C2×D4, C2×Q8, C4○D4, C4○D4, Dic5, Dic5, C20, C20, D10, D10, D10, C2×C10, C2×M4(2), C8○D4, C8○D4, C2×C4○D4, C52C8, C52C8, C40, C40, Dic10, C4×D5, C4×D5, D20, C2×Dic5, C5⋊D4, C2×C20, C5×D4, C5×Q8, C22×D5, Q8○M4(2), C8×D5, C8⋊D5, C8⋊D5, C2×C52C8, C4.Dic5, C2×C40, C5×M4(2), C2×C4×D5, C4○D20, D4×D5, D42D5, Q8×D5, Q82D5, C5×C4○D4, C2×C8⋊D5, D20.3C4, D5×M4(2), D20.2C4, D4.Dic5, C5×C8○D4, D5×C4○D4, C20.72C24
Quotients: C1, C2, C4, C22, C2×C4, C23, D5, C22×C4, C24, D10, C23×C4, C4×D5, C22×D5, Q8○M4(2), C2×C4×D5, C23×D5, D5×C22×C4, C20.72C24

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

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

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

74 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 4A 4B 4C 4D 4E 4F 4G 4H 4I 5A 5B 8A ··· 8H 8I ··· 8P 10A 10B 10C ··· 10H 20A 20B 20C 20D 20E ··· 20J 40A ··· 40H 40I ··· 40T 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 20 ··· 20 40 ··· 40 40 ··· 40 size 1 1 2 2 2 10 10 10 10 1 1 2 2 2 10 10 10 10 2 2 2 ··· 2 10 ··· 10 2 2 4 ··· 4 2 2 2 2 4 ··· 4 2 ··· 2 4 ··· 4

74 irreducible representations

 dim 1 1 1 1 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 C4 C4 C4 C4 D5 D10 D10 D10 C4×D5 C4×D5 Q8○M4(2) C20.72C24 kernel C20.72C24 C2×C8⋊D5 D20.3C4 D5×M4(2) D20.2C4 D4.Dic5 C5×C8○D4 D5×C4○D4 D4×D5 D4⋊2D5 Q8×D5 Q8⋊2D5 C8○D4 C2×C8 M4(2) C4○D4 D4 Q8 C5 C1 # reps 1 3 3 3 3 1 1 1 6 6 2 2 2 6 6 2 12 4 2 8

Matrix representation of C20.72C24 in GL6(𝔽41)

 6 35 0 0 0 0 6 1 0 0 0 0 0 0 0 39 0 0 0 0 25 0 0 0 0 0 40 0 40 2 0 0 20 40 36 1
,
 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 1 0 0 0 1 0 1 40
,
 40 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 1 0 0 0 20 0 20 1 0 0 1 0 0 0 0 0 21 1 21 0
,
 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 0 0 40 0 0 0 1 0 0 40

`G:=sub<GL(6,GF(41))| [6,6,0,0,0,0,35,1,0,0,0,0,0,0,0,25,40,20,0,0,39,0,0,40,0,0,0,0,40,36,0,0,0,0,2,1],[35,1,0,0,0,0,6,6,0,0,0,0,0,0,1,0,0,1,0,0,0,40,0,0,0,0,0,0,1,1,0,0,0,0,0,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,20,1,21,0,0,0,0,0,1,0,0,1,20,0,21,0,0,0,1,0,0],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,1,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40] >;`

C20.72C24 in GAP, Magma, Sage, TeX

`C_{20}._{72}C_2^4`
`% in TeX`

`G:=Group("C20.72C2^4");`
`// GroupNames label`

`G:=SmallGroup(320,1422);`
`// by ID`

`G=gap.SmallGroup(320,1422);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,387,1123,80,102,12550]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^40=b^2=c^2=d^2=1,b*a*b=a^29,c*a*c=a^21,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=a^20*c>;`
`// generators/relations`

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