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G = C2×C4○D20order 160 = 25·5

Direct product of C2 and C4○D20

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C2×C4○D20, C10.4C24, D2012C22, C20.43C23, D10.1C23, C23.26D10, Dic5.2C23, Dic1011C22, (C2×C4)⋊10D10, (C22×C4)⋊6D5, (C2×D20)⋊14C2, C101(C4○D4), (C22×C20)⋊8C2, (C4×D5)⋊6C22, C5⋊D46C22, C2.5(C23×D5), (C2×C20)⋊13C22, C4.43(C22×D5), (C2×Dic10)⋊15C2, (C2×C10).65C23, C22.5(C22×D5), (C22×C10).46C22, (C2×Dic5).46C22, (C22×D5).31C22, C51(C2×C4○D4), (C2×C4×D5)⋊15C2, (C2×C5⋊D4)⋊12C2, SmallGroup(160,216)

Series: Derived Chief Lower central Upper central

C1C10 — C2×C4○D20
C1C5C10D10C22×D5C2×C4×D5 — C2×C4○D20
C5C10 — C2×C4○D20
C1C2×C4C22×C4

Generators and relations for C2×C4○D20
 G = < a,b,c,d | a2=b4=d2=1, c10=b2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b2c9 >

Subgroups: 456 in 164 conjugacy classes, 89 normal (17 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×4], C4 [×4], C22, C22 [×2], C22 [×10], C5, C2×C4 [×2], C2×C4 [×4], C2×C4 [×10], D4 [×12], Q8 [×4], C23, C23 [×2], D5 [×4], C10, C10 [×2], C10 [×2], C22×C4, C22×C4 [×2], C2×D4 [×3], C2×Q8, C4○D4 [×8], Dic5 [×4], C20 [×4], D10 [×4], D10 [×4], C2×C10, C2×C10 [×2], C2×C10 [×2], C2×C4○D4, Dic10 [×4], C4×D5 [×8], D20 [×4], C2×Dic5 [×2], C5⋊D4 [×8], C2×C20 [×2], C2×C20 [×4], C22×D5 [×2], C22×C10, C2×Dic10, C2×C4×D5 [×2], C2×D20, C4○D20 [×8], C2×C5⋊D4 [×2], C22×C20, C2×C4○D20
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], D5, C4○D4 [×2], C24, D10 [×7], C2×C4○D4, C22×D5 [×7], C4○D20 [×2], C23×D5, C2×C4○D20

Smallest permutation representation of C2×C4○D20
On 80 points
Generators in S80
(1 38)(2 39)(3 40)(4 21)(5 22)(6 23)(7 24)(8 25)(9 26)(10 27)(11 28)(12 29)(13 30)(14 31)(15 32)(16 33)(17 34)(18 35)(19 36)(20 37)(41 62)(42 63)(43 64)(44 65)(45 66)(46 67)(47 68)(48 69)(49 70)(50 71)(51 72)(52 73)(53 74)(54 75)(55 76)(56 77)(57 78)(58 79)(59 80)(60 61)
(1 77 11 67)(2 78 12 68)(3 79 13 69)(4 80 14 70)(5 61 15 71)(6 62 16 72)(7 63 17 73)(8 64 18 74)(9 65 19 75)(10 66 20 76)(21 59 31 49)(22 60 32 50)(23 41 33 51)(24 42 34 52)(25 43 35 53)(26 44 36 54)(27 45 37 55)(28 46 38 56)(29 47 39 57)(30 48 40 58)
(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 37)(2 36)(3 35)(4 34)(5 33)(6 32)(7 31)(8 30)(9 29)(10 28)(11 27)(12 26)(13 25)(14 24)(15 23)(16 22)(17 21)(18 40)(19 39)(20 38)(41 71)(42 70)(43 69)(44 68)(45 67)(46 66)(47 65)(48 64)(49 63)(50 62)(51 61)(52 80)(53 79)(54 78)(55 77)(56 76)(57 75)(58 74)(59 73)(60 72)

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

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

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

C2×C4○D20 is a maximal subgroup of
C22⋊C8⋊D5  D20.32D4  D2014D4  C4○D209C4  C4○D2010C4  C424D10  (C2×D20)⋊25C4  D2017D4  D20.37D4  (C22×C8)⋊D5  C23.23D20  C4.89(C2×D20)  M4(2).31D10  C23.49D20  C23.20D20  C23⋊F55C2  (C4×D5).D4  C42.276D10  C24.27D10  C10.82+ 1+4  C10.2- 1+4  C10.2+ 1+4  C42.188D10  C42.91D10  C428D10  C429D10  C42.92D10  C4212D10  C42.228D10  D2023D4  D2024D4  Dic1023D4  Dic1024D4  Dic1020D4  C10.382+ 1+4  C10.392+ 1+4  D2020D4  C10.162- 1+4  C10.172- 1+4  D2022D4  Dic1022D4  C10.1212+ 1+4  C10.822- 1+4  C40.47C23  C40.9C23  C24.72D10  C24.41D10  C10.442- 1+4  C20.C24  (C2×C20)⋊17D4  C10.1472+ 1+4  C2×D5×C4○D4  C10.C25
C2×C4○D20 is a maximal quotient of
C2×C4×Dic10  C42.274D10  C2×C4×D20  C42.276D10  C42.277D10  C24.27D10  C24.30D10  C24.31D10  C10.2- 1+4  C10.102+ 1+4  C10.52- 1+4  C10.112+ 1+4  C10.62- 1+4  C42.89D10  C4210D10  C42.93D10  C42.94D10  C42.95D10  C42.96D10  C42.97D10  C42.98D10  C42.99D10  C42.100D10  C42.102D10  C42.104D10  C42.105D10  C42.106D10  C4212D10  C42.228D10  D2023D4  D2024D4  Dic1023D4  Dic1024D4  C4216D10  C42.229D10  C42.113D10  C42.114D10  C4217D10  C42.115D10  C42.116D10  C42.117D10  C42.118D10  C42.119D10  Dic1010Q8  C42.122D10  C42.232D10  D2010Q8  C42.131D10  C42.132D10  C42.133D10  C42.134D10  C42.135D10  C42.136D10  C2×C4×C5⋊D4  C24.72D10

52 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I4A4B4C4D4E4F4G4H4I4J5A5B10A···10N20A···20P
order122222222244444444445510···1020···20
size1111221010101011112210101010222···22···2

52 irreducible representations

dim111111122222
type++++++++++
imageC1C2C2C2C2C2C2D5C4○D4D10D10C4○D20
kernelC2×C4○D20C2×Dic10C2×C4×D5C2×D20C4○D20C2×C5⋊D4C22×C20C22×C4C10C2×C4C23C2
# reps11218212412216

Matrix representation of C2×C4○D20 in GL3(𝔽41) generated by

4000
0400
0040
,
100
0320
0032
,
4000
03927
03025
,
4000
0394
0302
G:=sub<GL(3,GF(41))| [40,0,0,0,40,0,0,0,40],[1,0,0,0,32,0,0,0,32],[40,0,0,0,39,30,0,27,25],[40,0,0,0,39,30,0,4,2] >;

C2×C4○D20 in GAP, Magma, Sage, TeX

C_2\times C_4\circ D_{20}
% in TeX

G:=Group("C2xC4oD20");
// GroupNames label

G:=SmallGroup(160,216);
// by ID

G=gap.SmallGroup(160,216);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,86,579,4613]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^4=d^2=1,c^10=b^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=b^2*c^9>;
// generators/relations

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