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## G = D20.37C23order 320 = 26·5

### 18th non-split extension by D20 of C23 acting via C23/C22=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — D20.37C23
 Chief series C1 — C5 — C10 — D10 — C22×D5 — C2×C4×D5 — D5×C4○D4 — D20.37C23
 Lower central C5 — C10 — D20.37C23
 Upper central C1 — C2 — 2+ 1+4

Generators and relations for D20.37C23
G = < a,b,c,d,e | a20=b2=c2=d2=1, e2=a10, bab=a-1, ac=ca, ad=da, eae-1=a9, cbc=dbd=a10b, ebe-1=a18b, dcd=ece-1=a10c, ede-1=a10d >

Subgroups: 2350 in 810 conjugacy classes, 443 normal (8 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, D5, C10, C10, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C4○D4, Dic5, Dic5, C20, D10, D10, C2×C10, C2×C10, C2×C4○D4, 2+ 1+4, 2+ 1+4, 2- 1+4, Dic10, C4×D5, D20, C2×Dic5, C5⋊D4, C2×C20, C5×D4, C5×Q8, C22×D5, C22×C10, C2.C25, C2×Dic10, C2×C4×D5, C4○D20, D4×D5, D42D5, Q8×D5, Q82D5, C22×Dic5, C2×C5⋊D4, D4×C10, C5×C4○D4, C2×D42D5, D46D10, D5×C4○D4, D4.10D10, C5×2+ 1+4, D20.37C23
Quotients: C1, C2, C22, C23, D5, C24, D10, C25, C22×D5, C2.C25, C23×D5, D5×C24, D20.37C23

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

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

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

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

68 conjugacy classes

 class 1 2A 2B ··· 2J 2K ··· 2P 4A ··· 4F 4G 4H 4I ··· 4Q 5A 5B 10A 10B 10C ··· 10T 20A ··· 20L order 1 2 2 ··· 2 2 ··· 2 4 ··· 4 4 4 4 ··· 4 5 5 10 10 10 ··· 10 20 ··· 20 size 1 1 2 ··· 2 10 ··· 10 2 ··· 2 5 5 10 ··· 10 2 2 2 2 4 ··· 4 4 ··· 4

68 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 4 8 type + + + + + + + + + - image C1 C2 C2 C2 C2 C2 D5 D10 D10 C2.C25 D20.37C23 kernel D20.37C23 C2×D4⋊2D5 D4⋊6D10 D5×C4○D4 D4.10D10 C5×2+ 1+4 2+ 1+4 C2×D4 C4○D4 C5 C1 # reps 1 9 9 6 6 1 2 18 12 2 2

Matrix representation of D20.37C23 in GL6(𝔽41)

 40 1 0 0 0 0 5 35 0 0 0 0 0 0 1 0 40 40 0 0 0 0 0 1 0 0 2 1 40 40 0 0 0 40 0 0
,
 40 0 0 0 0 0 5 1 0 0 0 0 0 0 40 40 0 1 0 0 0 0 0 40 0 0 39 40 1 1 0 0 0 40 0 0
,
 40 0 0 0 0 0 0 40 0 0 0 0 0 0 32 32 9 0 0 0 0 0 0 9 0 0 23 32 9 9 0 0 0 32 0 0
,
 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 2 1 40 40 0 0 0 0 0 40 0 0 0 0 40 0
,
 6 1 0 0 0 0 6 35 0 0 0 0 0 0 40 40 0 1 0 0 2 1 40 40 0 0 0 0 0 1 0 0 0 0 40 0

`G:=sub<GL(6,GF(41))| [40,5,0,0,0,0,1,35,0,0,0,0,0,0,1,0,2,0,0,0,0,0,1,40,0,0,40,0,40,0,0,0,40,1,40,0],[40,5,0,0,0,0,0,1,0,0,0,0,0,0,40,0,39,0,0,0,40,0,40,40,0,0,0,0,1,0,0,0,1,40,1,0],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,32,0,23,0,0,0,32,0,32,32,0,0,9,0,9,0,0,0,0,9,9,0],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,2,0,0,0,0,0,1,0,0,0,0,0,40,0,40,0,0,0,40,40,0],[6,6,0,0,0,0,1,35,0,0,0,0,0,0,40,2,0,0,0,0,40,1,0,0,0,0,0,40,0,40,0,0,1,40,1,0] >;`

D20.37C23 in GAP, Magma, Sage, TeX

`D_{20}._{37}C_2^3`
`% in TeX`

`G:=Group("D20.37C2^3");`
`// GroupNames label`

`G:=SmallGroup(320,1623);`
`// by ID`

`G=gap.SmallGroup(320,1623);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,184,570,1684,438,12550]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e|a^20=b^2=c^2=d^2=1,e^2=a^10,b*a*b=a^-1,a*c=c*a,a*d=d*a,e*a*e^-1=a^9,c*b*c=d*b*d=a^10*b,e*b*e^-1=a^18*b,d*c*d=e*c*e^-1=a^10*c,e*d*e^-1=a^10*d>;`
`// generators/relations`

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