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G = C23.9D20order 320 = 26·5

2nd non-split extension by C23 of D20 acting via D20/C10=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C23.9D20, C20.50C42, (C2×C40)⋊12C4, (C2×C8)⋊2Dic5, (C4×Dic5)⋊4C4, C20.43(C4⋊C4), (C2×C20).12Q8, C54(C4.9C42), (C2×C20).110D4, C4.25(C4×Dic5), (C2×C4).6Dic10, (C22×C10).47D4, (C22×C4).63D10, (C2×M4(2)).8D5, C4.18(C23.D5), C20.134(C22⋊C4), C4.12(C10.D4), (C10×M4(2)).12C2, C22.10(C4⋊Dic5), (C22×C20).126C22, C22.20(D10⋊C4), C10.35(C2.C42), C23.21D10.10C2, C2.16(C10.10C42), (C2×C4).140(C4×D5), (C2×C10).68(C4⋊C4), (C2×C20).236(C2×C4), (C2×C4).22(C5⋊D4), (C2×C4).77(C2×Dic5), (C2×C10).75(C22⋊C4), SmallGroup(320,115)

Series: Derived Chief Lower central Upper central

C1C20 — C23.9D20
C1C5C10C2×C10C2×C20C22×C20C23.21D10 — C23.9D20
C5C20 — C23.9D20
C1C4C2×M4(2)

Generators and relations for C23.9D20
 G = < a,b,c,d,e | a2=b2=c2=1, d20=c, e2=abc, ab=ba, dad-1=eae-1=ac=ca, ebe-1=bc=cb, bd=db, cd=dc, ce=ec, ede-1=bd19 >

Subgroups: 310 in 94 conjugacy classes, 47 normal (21 characteristic)
C1, C2, C2 [×3], C4 [×2], C4 [×2], C4 [×4], C22, C22 [×2], C22, C5, C8 [×2], C2×C4 [×2], C2×C4 [×4], C2×C4 [×4], C23, C10, C10 [×3], C42 [×4], C22⋊C4 [×2], C4⋊C4 [×2], C2×C8 [×2], M4(2) [×2], C22×C4, Dic5 [×4], C20 [×2], C20 [×2], C2×C10, C2×C10 [×2], C2×C10, C42⋊C2 [×2], C2×M4(2), C40 [×2], C2×Dic5 [×4], C2×C20 [×2], C2×C20 [×4], C22×C10, C4.9C42, C4×Dic5 [×4], C4⋊Dic5 [×2], C23.D5 [×2], C2×C40 [×2], C5×M4(2) [×2], C22×C20, C23.21D10 [×2], C10×M4(2), C23.9D20
Quotients: C1, C2 [×3], C4 [×6], C22, C2×C4 [×3], D4 [×3], Q8, D5, C42, C22⋊C4 [×3], C4⋊C4 [×3], Dic5 [×2], D10, C2.C42, Dic10, C4×D5 [×2], D20, C2×Dic5, C5⋊D4 [×2], C4.9C42, C4×Dic5, C10.D4 [×2], C4⋊Dic5, D10⋊C4 [×2], C23.D5, C10.10C42, C23.9D20

Smallest permutation representation of C23.9D20
On 80 points
Generators in S80
(1 65)(2 46)(3 67)(4 48)(5 69)(6 50)(7 71)(8 52)(9 73)(10 54)(11 75)(12 56)(13 77)(14 58)(15 79)(16 60)(17 41)(18 62)(19 43)(20 64)(21 45)(22 66)(23 47)(24 68)(25 49)(26 70)(27 51)(28 72)(29 53)(30 74)(31 55)(32 76)(33 57)(34 78)(35 59)(36 80)(37 61)(38 42)(39 63)(40 44)
(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 41)(18 42)(19 43)(20 44)(21 45)(22 46)(23 47)(24 48)(25 49)(26 50)(27 51)(28 52)(29 53)(30 54)(31 55)(32 56)(33 57)(34 58)(35 59)(36 60)(37 61)(38 62)(39 63)(40 64)
(1 21)(2 22)(3 23)(4 24)(5 25)(6 26)(7 27)(8 28)(9 29)(10 30)(11 31)(12 32)(13 33)(14 34)(15 35)(16 36)(17 37)(18 38)(19 39)(20 40)(41 61)(42 62)(43 63)(44 64)(45 65)(46 66)(47 67)(48 68)(49 69)(50 70)(51 71)(52 72)(53 73)(54 74)(55 75)(56 76)(57 77)(58 78)(59 79)(60 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 69 21 49)(2 4)(3 67 23 47)(5 65 25 45)(6 40)(7 63 27 43)(8 38)(9 61 29 41)(10 36)(11 59 31 79)(12 34)(13 57 33 77)(14 32)(15 55 35 75)(16 30)(17 53 37 73)(18 28)(19 51 39 71)(20 26)(22 24)(42 72)(44 70)(46 68)(48 66)(50 64)(52 62)(54 60)(56 58)(74 80)(76 78)

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

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

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

62 conjugacy classes

class 1 2A2B2C2D4A4B4C4D4E4F···4M5A5B8A8B8C8D10A···10F10G10H10I10J20A···20H20I20J20K20L40A···40P
order12222444444···455888810···101010101020···202020202040···40
size112221122220···202244442···244442···244444···4

62 irreducible representations

dim11111222222222244
type++++-++-+-+
imageC1C2C2C4C4D4Q8D4D5Dic5D10Dic10C4×D5C5⋊D4D20C4.9C42C23.9D20
kernelC23.9D20C23.21D10C10×M4(2)C4×Dic5C2×C40C2×C20C2×C20C22×C10C2×M4(2)C2×C8C22×C4C2×C4C2×C4C2×C4C23C5C1
# reps12184211242488428

Matrix representation of C23.9D20 in GL4(𝔽41) generated by

2440261
1174033
00171
004024
,
244000
11700
002440
00117
,
40000
04000
00400
00040
,
34291721
1232200
1319712
2223299
,
117431
24401837
00930
001132
G:=sub<GL(4,GF(41))| [24,1,0,0,40,17,0,0,26,40,17,40,1,33,1,24],[24,1,0,0,40,17,0,0,0,0,24,1,0,0,40,17],[40,0,0,0,0,40,0,0,0,0,40,0,0,0,0,40],[34,12,13,22,29,32,19,23,17,20,7,29,21,0,12,9],[1,24,0,0,17,40,0,0,4,18,9,11,31,37,30,32] >;

C23.9D20 in GAP, Magma, Sage, TeX

C_2^3._9D_{20}
% in TeX

G:=Group("C2^3.9D20");
// GroupNames label

G:=SmallGroup(320,115);
// by ID

G=gap.SmallGroup(320,115);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,253,64,387,1123,136,102,12550]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^2=1,d^20=c,e^2=a*b*c,a*b=b*a,d*a*d^-1=e*a*e^-1=a*c=c*a,e*b*e^-1=b*c=c*b,b*d=d*b,c*d=d*c,c*e=e*c,e*d*e^-1=b*d^19>;
// generators/relations

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