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G = C23.48D20order 320 = 26·5

19th non-split extension by C23 of D20 acting via D20/D10=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C23.48D20, (C2×C8)⋊21D10, (C2×D20)⋊26C4, (C2×C40)⋊36C22, D20.38(C2×C4), (C2×C4).153D20, C20.417(C2×D4), (C2×C20).173D4, D205C439C2, C2.4(C8⋊D10), (C2×M4(2))⋊11D5, C4⋊Dic548C22, C22.57(C2×D20), C10.20(C8⋊C22), C20.74(C22⋊C4), (C10×M4(2))⋊19C2, C20.174(C22×C4), (C2×C20).773C23, (C22×D20).16C2, (C22×C4).139D10, (C22×C10).101D4, C54(C23.37D4), C4.13(D10⋊C4), (C2×D20).206C22, C23.21D1016C2, (C22×C20).188C22, C22.28(D10⋊C4), C4.73(C2×C4×D5), (C2×C4).53(C4×D5), C4.110(C2×C5⋊D4), (C2×C20).281(C2×C4), (C2×C10).163(C2×D4), (C2×C4).76(C5⋊D4), C10.99(C2×C22⋊C4), C2.30(C2×D10⋊C4), (C2×C4).722(C22×D5), (C2×C10).85(C22⋊C4), SmallGroup(320,758)

Series: Derived Chief Lower central Upper central

C1C20 — C23.48D20
C1C5C10C20C2×C20C2×D20C22×D20 — C23.48D20
C5C10C20 — C23.48D20
C1C22C22×C4C2×M4(2)

Generators and relations for C23.48D20
 G = < a,b,c,d,e | a2=b2=c2=1, d20=c, e2=cb=bc, ab=ba, dad-1=eae-1=ac=ca, bd=db, be=eb, cd=dc, ce=ec, ede-1=bd19 >

Subgroups: 1006 in 190 conjugacy classes, 63 normal (23 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×2], C4 [×2], C4 [×2], C22, C22 [×2], C22 [×18], C5, C8 [×2], C2×C4 [×2], C2×C4 [×4], C2×C4 [×2], D4 [×10], C23, C23 [×10], D5 [×4], C10, C10 [×2], C10 [×2], C42, C22⋊C4, C4⋊C4 [×2], C2×C8 [×2], M4(2) [×2], C22×C4, C2×D4 [×9], C24, Dic5 [×2], C20 [×2], C20 [×2], D10 [×16], C2×C10, C2×C10 [×2], C2×C10 [×2], D4⋊C4 [×4], C42⋊C2, C2×M4(2), C22×D4, C40 [×2], D20 [×4], D20 [×6], C2×Dic5 [×2], C2×C20 [×2], C2×C20 [×4], C22×D5 [×10], C22×C10, C23.37D4, C4×Dic5, C4⋊Dic5 [×2], C23.D5, C2×C40 [×2], C5×M4(2) [×2], C2×D20 [×6], C2×D20 [×3], C22×C20, C23×D5, D205C4 [×4], C23.21D10, C10×M4(2), C22×D20, C23.48D20
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, D5, C22⋊C4 [×4], C22×C4, C2×D4 [×2], D10 [×3], C2×C22⋊C4, C8⋊C22 [×2], C4×D5 [×2], D20 [×2], C5⋊D4 [×2], C22×D5, C23.37D4, D10⋊C4 [×4], C2×C4×D5, C2×D20, C2×C5⋊D4, C8⋊D10 [×2], C2×D10⋊C4, C23.48D20

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

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

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

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

62 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I4A4B4C4D4E4F4G4H5A5B8A8B8C8D10A···10F10G10H10I10J20A···20H20I20J20K20L40A···40P
order12222222224444444455888810···101010101020···202020202040···40
size111122202020202222202020202244442···244442···244444···4

62 irreducible representations

dim11111122222222244
type++++++++++++++
imageC1C2C2C2C2C4D4D4D5D10D10C4×D5D20C5⋊D4D20C8⋊C22C8⋊D10
kernelC23.48D20D205C4C23.21D10C10×M4(2)C22×D20C2×D20C2×C20C22×C10C2×M4(2)C2×C8C22×C4C2×C4C2×C4C2×C4C23C10C2
# reps14111831242848428

Matrix representation of C23.48D20 in GL6(𝔽41)

4000000
0400000
001000
000100
002323400
003838040
,
4000000
0400000
001000
000100
000010
000001
,
100000
010000
0040000
0004000
0000400
0000040
,
40230000
010000
0011633
0000740
00187640
001719134
,
1180000
9400000
00343478
00404021
00222111
00303177

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,23,38,0,0,0,1,23,38,0,0,0,0,40,0,0,0,0,0,0,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[40,0,0,0,0,0,23,1,0,0,0,0,0,0,1,0,18,17,0,0,1,0,7,19,0,0,6,7,6,1,0,0,33,40,40,34],[1,9,0,0,0,0,18,40,0,0,0,0,0,0,34,40,22,30,0,0,34,40,21,31,0,0,7,2,1,7,0,0,8,1,1,7] >;

C23.48D20 in GAP, Magma, Sage, TeX

C_2^3._{48}D_{20}
% in TeX

G:=Group("C2^3.48D20");
// GroupNames label

G:=SmallGroup(320,758);
// by ID

G=gap.SmallGroup(320,758);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,254,387,142,1123,136,12550]);
// Polycyclic

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

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