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## G = D20.39C23order 320 = 26·5

### 20th non-split extension by D20 of C23 acting via C23/C22=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — D20.39C23
 Chief series C1 — C5 — C10 — D10 — C22×D5 — C2×C4×D5 — D5×C4○D4 — D20.39C23
 Lower central C5 — C10 — D20.39C23
 Upper central C1 — C2 — 2- 1+4

Generators and relations for D20.39C23
G = < a,b,c,d,e | a20=b2=c2=d2=e2=1, bab=a-1, ac=ca, ad=da, eae=a9, cbc=a10b, bd=db, ebe=a18b, dcd=ece=a10c, de=ed >

Subgroups: 2478 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, D5, C10, C10, C22×C4, C2×D4, C2×Q8, C2×Q8, C4○D4, C4○D4, Dic5, Dic5, C20, D10, D10, 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, C2.C25, C2×C4×D5, C2×D20, C4○D20, D4×D5, D42D5, Q8×D5, Q82D5, Q8×C10, C5×C4○D4, C2×Q82D5, Q8.10D10, D5×C4○D4, D48D10, C5×2- 1+4, D20.39C23
Quotients: C1, C2, C22, C23, D5, C24, D10, C25, C22×D5, C2.C25, C23×D5, D5×C24, D20.39C23

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

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

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

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

68 conjugacy classes

 class 1 2A 2B ··· 2F 2G ··· 2P 4A ··· 4J 4K 4L 4M ··· 4Q 5A 5B 10A 10B 10C ··· 10L 20A ··· 20T 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 D20.39C23 kernel D20.39C23 C2×Q8⋊2D5 Q8.10D10 D5×C4○D4 D4⋊8D10 C5×2- 1+4 2- 1+4 C2×Q8 C4○D4 C5 C1 # reps 1 5 5 10 10 1 2 10 20 2 2

Matrix representation of D20.39C23 in GL6(𝔽41)

 0 40 0 0 0 0 1 35 0 0 0 0 0 0 40 20 4 2 0 0 4 1 0 37 0 0 0 0 40 20 0 0 0 0 4 1
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 21 40 20 0 0 0 40 0 1 0 0 0 0 40 20 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 40 39 0 2 0 0 0 40 0 0 0 0 39 0 1 2 0 0 0 39 0 1
,
 40 0 0 0 0 0 0 40 0 0 0 0 0 0 1 0 40 39 0 0 0 1 0 40 0 0 0 0 40 0 0 0 0 0 0 40
,
 40 0 0 0 0 0 35 1 0 0 0 0 0 0 9 25 27 16 0 0 5 32 36 4 0 0 0 0 32 16 0 0 0 0 36 9

G:=sub<GL(6,GF(41))| [0,1,0,0,0,0,40,35,0,0,0,0,0,0,40,4,0,0,0,0,20,1,0,0,0,0,4,0,40,4,0,0,2,37,20,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,21,40,0,0,0,0,40,0,40,0,0,0,20,1,20,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,39,0,0,0,39,40,0,39,0,0,0,0,1,0,0,0,2,0,2,1],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,40,0,40,0,0,0,39,40,0,40],[40,35,0,0,0,0,0,1,0,0,0,0,0,0,9,5,0,0,0,0,25,32,0,0,0,0,27,36,32,36,0,0,16,4,16,9] >;

D20.39C23 in GAP, Magma, Sage, TeX

D_{20}._{39}C_2^3
% in TeX

G:=Group("D20.39C2^3");
// GroupNames label

G:=SmallGroup(320,1625);
// by ID

G=gap.SmallGroup(320,1625);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,184,570,1684,438,102,12550]);
// Polycyclic

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

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