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## G = D20.34C23order 320 = 26·5

### 15th non-split extension by D20 of C23 acting via C23/C22=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — D20.34C23
 Chief series C1 — C5 — C10 — C20 — D20 — C2×D20 — D4⋊8D10 — D20.34C23
 Lower central C5 — C10 — C20 — D20.34C23
 Upper central C1 — C2 — C4○D4 — 2- 1+4

Generators and relations for D20.34C23
G = < a,b,c,d,e | a20=b2=c2=d2=1, e2=a10, bab=dad=a-1, ac=ca, eae-1=a11, cbc=a10b, dbd=a18b, ebe-1=a15b, cd=dc, ce=ec, ede-1=a5d >

Subgroups: 918 in 258 conjugacy classes, 107 normal (20 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, D4, D4, Q8, Q8, Q8, C23, D5, C10, C10, C2×C8, M4(2), D8, SD16, Q16, C2×D4, C2×Q8, C2×Q8, C4○D4, C4○D4, C4○D4, Dic5, C20, C20, C20, D10, C2×C10, C2×C10, C8○D4, C2×SD16, C4○D8, C8⋊C22, 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×D4, C5×Q8, C5×Q8, C5×Q8, C22×D5, D4○SD16, C2×C52C8, C4.Dic5, D4⋊D5, D4.D5, Q8⋊D5, C5⋊Q16, C2×D20, C4○D20, D4×D5, Q82D5, Q8×C10, Q8×C10, C5×C4○D4, C5×C4○D4, C5×C4○D4, C2×Q8⋊D5, C20.C23, D4.Dic5, D4⋊D10, D4.8D10, D48D10, C5×2- 1+4, D20.34C23
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C24, D10, C22×D4, C5⋊D4, C22×D5, D4○SD16, C2×C5⋊D4, C23×D5, C22×C5⋊D4, D20.34C23

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

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

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

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

56 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 4A 4B 4C 4D 4E 4F 4G 4H 5A 5B 8A 8B 8C 8D 8E 10A 10B 10C ··· 10L 20A ··· 20T order 1 2 2 2 2 2 2 2 2 4 4 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 20 20 20 2 2 2 2 4 4 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○SD16 D20.34C23 kernel D20.34C23 C2×Q8⋊D5 C20.C23 D4.Dic5 D4⋊D10 D4.8D10 D4⋊8D10 C5×2- 1+4 C5×D4 C5×Q8 2- 1+4 C2×Q8 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 D20.34C23 in GL6(𝔽41)

 0 1 0 0 0 0 40 6 0 0 0 0 0 0 1 16 0 0 0 0 5 40 0 0 0 0 0 0 40 25 0 0 0 0 36 1
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 16 0 0 0 0 0 40 0 0 0 0 0 0 1 0 0 0 0 0 5 40
,
 40 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 0 6 0 0 0 0 34 0 0 0 0 35 0 0 0 0 7 0 0 0
,
 1 0 0 0 0 0 6 40 0 0 0 0 0 0 0 0 30 35 0 0 0 0 34 0 0 0 0 35 0 0 0 0 34 11 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 40 0 0 0 0 0 0 40 0 0

`G:=sub<GL(6,GF(41))| [0,40,0,0,0,0,1,6,0,0,0,0,0,0,1,5,0,0,0,0,16,40,0,0,0,0,0,0,40,36,0,0,0,0,25,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,16,40,0,0,0,0,0,0,1,5,0,0,0,0,0,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,7,0,0,0,0,35,0,0,0,0,34,0,0,0,0,6,0,0,0],[1,6,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,34,0,0,0,0,35,11,0,0,30,34,0,0,0,0,35,0,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,1,0,0,0,0,0,0,1,0,0] >;`

D20.34C23 in GAP, Magma, Sage, TeX

`D_{20}._{34}C_2^3`
`% in TeX`

`G:=Group("D20.34C2^3");`
`// GroupNames label`

`G:=SmallGroup(320,1509);`
`// by ID`

`G=gap.SmallGroup(320,1509);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,387,675,136,1684,235,102,12550]);`
`// Polycyclic`

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

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