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G = C2×Dic10order 80 = 24·5

Direct product of C2 and Dic10

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

Aliases: C2×Dic10, C10⋊Q8, C4.11D10, C10.1C23, C22.8D10, C20.11C22, Dic5.1C22, C51(C2×Q8), (C2×C4).4D5, (C2×C20).4C2, C2.3(C22×D5), (C2×C10).8C22, (C2×Dic5).3C2, SmallGroup(80,35)

Series: Derived Chief Lower central Upper central

C1C10 — C2×Dic10
C1C5C10Dic5C2×Dic5 — C2×Dic10
C5C10 — C2×Dic10
C1C22C2×C4

Generators and relations for C2×Dic10
 G = < a,b,c | a2=b20=1, c2=b10, ab=ba, ac=ca, cbc-1=b-1 >

5C4
5C4
5C4
5C4
5C2×C4
5Q8
5C2×C4
5Q8
5Q8
5Q8
5C2×Q8

Character table of C2×Dic10

 class 12A2B2C4A4B4C4D4E4F5A5B10A10B10C10D10E10F20A20B20C20D20E20F20G20H
 size 111122101010102222222222222222
ρ111111111111111111111111111    trivial
ρ21-11-11-1-11-11111-11-1-1-11-1-1-1-1111    linear of order 2
ρ31-11-1-11-111-1111-11-1-1-1-11111-1-1-1    linear of order 2
ρ41111-1-111-1-111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ51111-1-1-1-11111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ61-11-1-111-1-11111-11-1-1-1-11111-1-1-1    linear of order 2
ρ71-11-11-11-11-1111-11-1-1-11-1-1-1-1111    linear of order 2
ρ8111111-1-1-1-11111111111111111    linear of order 2
ρ92222220000-1+5/2-1-5/2-1+5/2-1+5/2-1-5/2-1-5/2-1+5/2-1-5/2-1-5/2-1+5/2-1+5/2-1-5/2-1-5/2-1-5/2-1+5/2-1+5/2    orthogonal lifted from D5
ρ102222-2-20000-1-5/2-1+5/2-1-5/2-1-5/2-1+5/2-1+5/2-1-5/2-1+5/21-5/21+5/21+5/21-5/21-5/21-5/21+5/21+5/2    orthogonal lifted from D10
ρ112222220000-1-5/2-1+5/2-1-5/2-1-5/2-1+5/2-1+5/2-1-5/2-1+5/2-1+5/2-1-5/2-1-5/2-1+5/2-1+5/2-1+5/2-1-5/2-1-5/2    orthogonal lifted from D5
ρ122-22-2-220000-1+5/2-1-5/2-1+5/21-5/2-1-5/21+5/21-5/21+5/21+5/2-1+5/2-1+5/2-1-5/2-1-5/21+5/21-5/21-5/2    orthogonal lifted from D10
ρ132-22-22-20000-1-5/2-1+5/2-1-5/21+5/2-1+5/21-5/21+5/21-5/2-1+5/21+5/21+5/21-5/21-5/2-1+5/2-1-5/2-1-5/2    orthogonal lifted from D10
ρ142222-2-20000-1+5/2-1-5/2-1+5/2-1+5/2-1-5/2-1-5/2-1+5/2-1-5/21+5/21-5/21-5/21+5/21+5/21+5/21-5/21-5/2    orthogonal lifted from D10
ρ152-22-22-20000-1+5/2-1-5/2-1+5/21-5/2-1-5/21+5/21-5/21+5/2-1-5/21-5/21-5/21+5/21+5/2-1-5/2-1+5/2-1+5/2    orthogonal lifted from D10
ρ162-22-2-220000-1-5/2-1+5/2-1-5/21+5/2-1+5/21-5/21+5/21-5/21-5/2-1-5/2-1-5/2-1+5/2-1+5/21-5/21+5/21+5/2    orthogonal lifted from D10
ρ172-2-2200000022-2-2-222-200000000    symplectic lifted from Q8, Schur index 2
ρ1822-2-200000022-22-2-2-2200000000    symplectic lifted from Q8, Schur index 2
ρ192-2-22000000-1+5/2-1-5/21-5/21-5/21+5/2-1-5/2-1+5/21+5/24ζ534ζ5243ζ5443ζ5ζ43ζ5443ζ5ζ4ζ534ζ524ζ534ζ52ζ4ζ534ζ52ζ43ζ5443ζ543ζ5443ζ5    symplectic lifted from Dic10, Schur index 2
ρ2022-2-2000000-1+5/2-1-5/21-5/2-1+5/21+5/21+5/21-5/2-1-5/24ζ534ζ52ζ43ζ5443ζ543ζ5443ζ54ζ534ζ52ζ4ζ534ζ52ζ4ζ534ζ52ζ43ζ5443ζ543ζ5443ζ5    symplectic lifted from Dic10, Schur index 2
ρ212-2-22000000-1+5/2-1-5/21-5/21-5/21+5/2-1-5/2-1+5/21+5/2ζ4ζ534ζ52ζ43ζ5443ζ543ζ5443ζ54ζ534ζ52ζ4ζ534ζ524ζ534ζ5243ζ5443ζ5ζ43ζ5443ζ5    symplectic lifted from Dic10, Schur index 2
ρ2222-2-2000000-1-5/2-1+5/21+5/2-1-5/21-5/21-5/21+5/2-1+5/243ζ5443ζ54ζ534ζ52ζ4ζ534ζ5243ζ5443ζ5ζ43ζ5443ζ5ζ43ζ5443ζ54ζ534ζ52ζ4ζ534ζ52    symplectic lifted from Dic10, Schur index 2
ρ232-2-22000000-1-5/2-1+5/21+5/21+5/21-5/2-1+5/2-1-5/21-5/2ζ43ζ5443ζ54ζ534ζ52ζ4ζ534ζ5243ζ5443ζ5ζ43ζ5443ζ543ζ5443ζ5ζ4ζ534ζ524ζ534ζ52    symplectic lifted from Dic10, Schur index 2
ρ242-2-22000000-1-5/2-1+5/21+5/21+5/21-5/2-1+5/2-1-5/21-5/243ζ5443ζ5ζ4ζ534ζ524ζ534ζ52ζ43ζ5443ζ543ζ5443ζ5ζ43ζ5443ζ54ζ534ζ52ζ4ζ534ζ52    symplectic lifted from Dic10, Schur index 2
ρ2522-2-2000000-1-5/2-1+5/21+5/2-1-5/21-5/21-5/21+5/2-1+5/2ζ43ζ5443ζ5ζ4ζ534ζ524ζ534ζ52ζ43ζ5443ζ543ζ5443ζ543ζ5443ζ5ζ4ζ534ζ524ζ534ζ52    symplectic lifted from Dic10, Schur index 2
ρ2622-2-2000000-1+5/2-1-5/21-5/2-1+5/21+5/21+5/21-5/2-1-5/2ζ4ζ534ζ5243ζ5443ζ5ζ43ζ5443ζ5ζ4ζ534ζ524ζ534ζ524ζ534ζ5243ζ5443ζ5ζ43ζ5443ζ5    symplectic lifted from Dic10, Schur index 2

Smallest permutation representation of C2×Dic10
Regular action on 80 points
Generators in S80
(1 31)(2 32)(3 33)(4 34)(5 35)(6 36)(7 37)(8 38)(9 39)(10 40)(11 21)(12 22)(13 23)(14 24)(15 25)(16 26)(17 27)(18 28)(19 29)(20 30)(41 66)(42 67)(43 68)(44 69)(45 70)(46 71)(47 72)(48 73)(49 74)(50 75)(51 76)(52 77)(53 78)(54 79)(55 80)(56 61)(57 62)(58 63)(59 64)(60 65)
(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 51 11 41)(2 50 12 60)(3 49 13 59)(4 48 14 58)(5 47 15 57)(6 46 16 56)(7 45 17 55)(8 44 18 54)(9 43 19 53)(10 42 20 52)(21 66 31 76)(22 65 32 75)(23 64 33 74)(24 63 34 73)(25 62 35 72)(26 61 36 71)(27 80 37 70)(28 79 38 69)(29 78 39 68)(30 77 40 67)

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

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

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

C2×Dic10 is a maximal subgroup of
C10.Q16  C20.44D4  C4.12D20  Dic5.D4  C202Q8  C4.D20  Dic5.14D4  Dic5.5D4  Dic53Q8  C20⋊Q8  D10⋊Q8  D102Q8  C8.D10  C20.48D4  C20.17D4  Dic5⋊Q8  D4.9D10  C2×Q8×D5  D4.10D10
C2×Dic10 is a maximal quotient of
C202Q8  C20.6Q8  Dic5.14D4  C20⋊Q8  C4.Dic10  C20.48D4

Matrix representation of C2×Dic10 in GL4(𝔽41) generated by

1000
0100
00400
00040
,
0100
40000
00140
00834
,
401100
11100
00388
00403
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,40,0,0,0,0,40],[0,40,0,0,1,0,0,0,0,0,1,8,0,0,40,34],[40,11,0,0,11,1,0,0,0,0,38,40,0,0,8,3] >;

C2×Dic10 in GAP, Magma, Sage, TeX

C_2\times {\rm Dic}_{10}
% in TeX

G:=Group("C2xDic10");
// GroupNames label

G:=SmallGroup(80,35);
// by ID

G=gap.SmallGroup(80,35);
# by ID

G:=PCGroup([5,-2,-2,-2,-2,-5,40,182,42,1604]);
// Polycyclic

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

Export

Subgroup lattice of C2×Dic10 in TeX
Character table of C2×Dic10 in TeX

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