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G = D20.39C23order 320 = 26·5

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

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C10.18C25, C20.53C24, D20.39C23, D10.10C24, 2- 1+45D5, Dic5.13C24, Dic10.40C23, C4○D413D10, (C2×Q8)⋊25D10, (D4×D5)⋊15C22, (C2×C10).9C24, D48D1012C2, (Q8×D5)⋊18C22, C2.19(D5×C24), C4.50(C23×D5), C5⋊D4.5C23, C4○D2015C22, (C2×D20)⋊41C22, C53(C2.C25), (Q8×C10)⋊25C22, D4.33(C22×D5), (C5×D4).33C23, (C4×D5).22C23, (C5×Q8).34C23, Q8.34(C22×D5), D42D519C22, C22.6(C23×D5), (C2×C20).124C23, Q8.10D108C2, Q82D517C22, (C5×2- 1+4)⋊5C2, (C2×Dic5).310C23, (C22×D5).144C23, (D5×C4○D4)⋊10C2, (C2×C4×D5)⋊38C22, (C2×Q82D5)⋊22C2, (C5×C4○D4)⋊13C22, (C2×C4).108(C22×D5), SmallGroup(320,1625)

Series: Derived Chief Lower central Upper central

C1C10 — D20.39C23
C1C5C10D10C22×D5C2×C4×D5D5×C4○D4 — D20.39C23
C5C10 — D20.39C23
C1C22- 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 [×15], C4 [×10], C4 [×6], C22 [×5], C22 [×25], C5, C2×C4 [×15], C2×C4 [×45], D4 [×10], D4 [×50], Q8 [×10], Q8 [×10], C23 [×15], D5 [×10], C10, C10 [×5], C22×C4 [×15], C2×D4 [×45], C2×Q8 [×5], C2×Q8 [×10], C4○D4 [×10], C4○D4 [×70], Dic5, Dic5 [×5], C20 [×10], D10 [×10], D10 [×15], C2×C10 [×5], C2×C4○D4 [×15], 2+ 1+4 [×10], 2- 1+4, 2- 1+4 [×5], Dic10 [×10], C4×D5 [×40], D20 [×30], C2×Dic5 [×5], C5⋊D4 [×20], C2×C20 [×15], C5×D4 [×10], C5×Q8 [×10], C22×D5 [×15], C2.C25, C2×C4×D5 [×15], C2×D20 [×15], C4○D20 [×30], D4×D5 [×30], D42D5 [×10], Q8×D5 [×10], Q82D5 [×30], Q8×C10 [×5], C5×C4○D4 [×10], C2×Q82D5 [×5], Q8.10D10 [×5], D5×C4○D4 [×10], D48D10 [×10], C5×2- 1+4, D20.39C23
Quotients: C1, C2 [×31], C22 [×155], C23 [×155], D5, C24 [×31], D10 [×15], C25, C22×D5 [×35], C2.C25, C23×D5 [×15], 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 25)(22 24)(26 40)(27 39)(28 38)(29 37)(30 36)(31 35)(32 34)(41 55)(42 54)(43 53)(44 52)(45 51)(46 50)(47 49)(56 60)(57 59)(61 73)(62 72)(63 71)(64 70)(65 69)(66 68)(74 80)(75 79)(76 78)
(1 26)(2 27)(3 28)(4 29)(5 30)(6 31)(7 32)(8 33)(9 34)(10 35)(11 36)(12 37)(13 38)(14 39)(15 40)(16 21)(17 22)(18 23)(19 24)(20 25)(41 65)(42 66)(43 67)(44 68)(45 69)(46 70)(47 71)(48 72)(49 73)(50 74)(51 75)(52 76)(53 77)(54 78)(55 79)(56 80)(57 61)(58 62)(59 63)(60 64)
(21 31)(22 32)(23 33)(24 34)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)(41 51)(42 52)(43 53)(44 54)(45 55)(46 56)(47 57)(48 58)(49 59)(50 60)
(1 70)(2 79)(3 68)(4 77)(5 66)(6 75)(7 64)(8 73)(9 62)(10 71)(11 80)(12 69)(13 78)(14 67)(15 76)(16 65)(17 74)(18 63)(19 72)(20 61)(21 51)(22 60)(23 49)(24 58)(25 47)(26 56)(27 45)(28 54)(29 43)(30 52)(31 41)(32 50)(33 59)(34 48)(35 57)(36 46)(37 55)(38 44)(39 53)(40 42)

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

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

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

68 conjugacy classes

class 1 2A2B···2F2G···2P4A···4J4K4L4M···4Q5A5B10A10B10C···10L20A···20T
order122···22···24···4444···455101010···1020···20
size112···210···102···25510···1022224···44···4

68 irreducible representations

dim11111122248
type++++++++++
imageC1C2C2C2C2C2D5D10D10C2.C25D20.39C23
kernelD20.39C23C2×Q82D5Q8.10D10D5×C4○D4D48D10C5×2- 1+42- 1+4C2×Q8C4○D4C5C1
# reps155101012102022

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

0400000
1350000
00402042
0041037
00004020
000041
,
010000
100000
001214020
0004001
00004020
000001
,
100000
010000
00403902
0004000
0039012
0003901
,
4000000
0400000
00104039
0001040
0000400
0000040
,
4000000
3510000
009252716
00532364
00003216
0000369

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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