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G = D20.32C23order 320 = 26·5

13rd non-split extension by D20 of C23 acting via C23/C22=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — D20.32C23
 Chief series C1 — C5 — C10 — C20 — D20 — C2×D20 — D4⋊8D10 — D20.32C23
 Lower central C5 — C10 — C20 — D20.32C23
 Upper central C1 — C2 — C4○D4 — 2+ 1+4

Generators and relations for D20.32C23
G = < a,b,c,d,e | a20=b2=c2=d2=e2=1, bab=dad=a-1, ac=ca, eae=a11, cbc=a10b, dbd=a18b, ebe=a15b, cd=dc, ce=ec, ede=a5d >

Subgroups: 982 in 268 conjugacy classes, 107 normal (20 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, D5, C10, C10, C2×C8, M4(2), D8, SD16, Q16, C2×D4, C2×D4, C4○D4, C4○D4, C4○D4, Dic5, C20, C20, C20, D10, C2×C10, C2×C10, C8○D4, C2×D8, C4○D8, C8⋊C22, 2+ 1+4, 2+ 1+4, C52C8, C52C8, Dic10, C4×D5, D20, D20, C5⋊D4, C2×C20, C2×C20, C5×D4, C5×D4, C5×Q8, C22×D5, C22×C10, D4○D8, C2×C52C8, C4.Dic5, D4⋊D5, D4.D5, Q8⋊D5, C5⋊Q16, C2×D20, C4○D20, D4×D5, Q82D5, D4×C10, D4×C10, C5×C4○D4, C5×C4○D4, C5×C4○D4, C2×D4⋊D5, D4.D10, D4.Dic5, D4⋊D10, D4.8D10, D48D10, C5×2+ 1+4, D20.32C23
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C24, D10, C22×D4, C5⋊D4, C22×D5, D4○D8, C2×C5⋊D4, C23×D5, C22×C5⋊D4, D20.32C23

Smallest permutation representation of D20.32C23
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 20)(2 19)(3 18)(4 17)(5 16)(6 15)(7 14)(8 13)(9 12)(10 11)(21 29)(22 28)(23 27)(24 26)(30 40)(31 39)(32 38)(33 37)(34 36)(41 44)(42 43)(45 60)(46 59)(47 58)(48 57)(49 56)(50 55)(51 54)(52 53)(61 75)(62 74)(63 73)(64 72)(65 71)(66 70)(67 69)(76 80)(77 79)
(1 48)(2 49)(3 50)(4 51)(5 52)(6 53)(7 54)(8 55)(9 56)(10 57)(11 58)(12 59)(13 60)(14 41)(15 42)(16 43)(17 44)(18 45)(19 46)(20 47)(21 79)(22 80)(23 61)(24 62)(25 63)(26 64)(27 65)(28 66)(29 67)(30 68)(31 69)(32 70)(33 71)(34 72)(35 73)(36 74)(37 75)(38 76)(39 77)(40 78)
(1 53)(2 52)(3 51)(4 50)(5 49)(6 48)(7 47)(8 46)(9 45)(10 44)(11 43)(12 42)(13 41)(14 60)(15 59)(16 58)(17 57)(18 56)(19 55)(20 54)(21 63)(22 62)(23 61)(24 80)(25 79)(26 78)(27 77)(28 76)(29 75)(30 74)(31 73)(32 72)(33 71)(34 70)(35 69)(36 68)(37 67)(38 66)(39 65)(40 64)
(1 33)(2 24)(3 35)(4 26)(5 37)(6 28)(7 39)(8 30)(9 21)(10 32)(11 23)(12 34)(13 25)(14 36)(15 27)(16 38)(17 29)(18 40)(19 31)(20 22)(41 74)(42 65)(43 76)(44 67)(45 78)(46 69)(47 80)(48 71)(49 62)(50 73)(51 64)(52 75)(53 66)(54 77)(55 68)(56 79)(57 70)(58 61)(59 72)(60 63)```

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

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

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

56 conjugacy classes

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

56 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 8 type + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 D4 D4 D5 D10 D10 C5⋊D4 C5⋊D4 D4○D8 D20.32C23 kernel D20.32C23 C2×D4⋊D5 D4.D10 D4.Dic5 D4⋊D10 D4.8D10 D4⋊8D10 C5×2+ 1+4 C5×D4 C5×Q8 2+ 1+4 C2×D4 C4○D4 D4 Q8 C5 C1 # reps 1 3 3 1 3 3 1 1 3 1 2 6 8 12 4 2 2

Matrix representation of D20.32C23 in GL6(𝔽41)

 40 1 0 0 0 0 33 7 0 0 0 0 0 0 40 39 0 0 0 0 1 1 0 0 0 0 0 0 1 2 0 0 0 0 40 40
,
 40 0 0 0 0 0 33 1 0 0 0 0 0 0 40 39 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 40 40
,
 40 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 24 24 0 0 0 0 29 17 0 0 24 24 0 0 0 0 29 17 0 0
,
 34 1 0 0 0 0 34 7 0 0 0 0 0 0 0 0 0 17 0 0 0 0 12 24 0 0 24 24 0 0 0 0 29 0 0 0
,
 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))| [40,33,0,0,0,0,1,7,0,0,0,0,0,0,40,1,0,0,0,0,39,1,0,0,0,0,0,0,1,40,0,0,0,0,2,40],[40,33,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,39,1,0,0,0,0,0,0,1,40,0,0,0,0,0,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,24,29,0,0,0,0,24,17,0,0,24,29,0,0,0,0,24,17,0,0],[34,34,0,0,0,0,1,7,0,0,0,0,0,0,0,0,24,29,0,0,0,0,24,0,0,0,0,12,0,0,0,0,17,24,0,0],[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] >;`

D20.32C23 in GAP, Magma, Sage, TeX

`D_{20}._{32}C_2^3`
`% in TeX`

`G:=Group("D20.32C2^3");`
`// GroupNames label`

`G:=SmallGroup(320,1507);`
`// by ID`

`G=gap.SmallGroup(320,1507);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,387,675,1684,235,102,12550]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e|a^20=b^2=c^2=d^2=e^2=1,b*a*b=d*a*d=a^-1,a*c=c*a,e*a*e=a^11,c*b*c=a^10*b,d*b*d=a^18*b,e*b*e=a^15*b,c*d=d*c,c*e=e*c,e*d*e=a^5*d>;`
`// generators/relations`

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