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G = C10.D4order 80 = 24·5

1st non-split extension by C10 of D4 acting via D4/C22=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C10.5D4, C10.1Q8, Dic51C4, C2.1Dic10, C22.4D10, C52(C4⋊C4), C2.4(C4×D5), (C2×C4).1D5, (C2×C20).1C2, C10.11(C2×C4), C2.1(C5⋊D4), (C2×C10).4C22, (C2×Dic5).1C2, SmallGroup(80,12)

Series: Derived Chief Lower central Upper central

C1C10 — C10.D4
C1C5C10C2×C10C2×Dic5 — C10.D4
C5C10 — C10.D4
C1C22C2×C4

Generators and relations for C10.D4
 G = < a,b,c | a10=b4=1, c2=a5, bab-1=cac-1=a-1, cbc-1=b-1 >

2C4
5C4
5C4
10C4
5C2×C4
5C2×C4
2C20
2Dic5
5C4⋊C4

Character table of C10.D4

 class 12A2B2C4A4B4C4D4E4F5A5B10A10B10C10D10E10F20A20B20C20D20E20F20G20H
 size 111122101010102222222222222222
ρ111111111111111111111111111    trivial
ρ21111-1-1-11-1111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ31111-1-11-11-111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ4111111-1-1-1-11111111111111111    linear of order 2
ρ51-1-11-iii-1-i111-11-1-1-11iiii-i-i-i-i    linear of order 4
ρ61-1-11i-i-i-1i111-11-1-1-11-i-i-i-iiiii    linear of order 4
ρ71-1-11i-ii1-i-111-11-1-1-11-i-i-i-iiiii    linear of order 4
ρ81-1-11-ii-i1i-111-11-1-1-11iiii-i-i-i-i    linear of order 4
ρ92-22-200000022-2-2-222-200000000    orthogonal lifted from D4
ρ102222220000-1+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
ρ112222-2-20000-1+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
ρ122222-2-20000-1-5/2-1+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
ρ132222220000-1-5/2-1+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
ρ1422-2-2000000222-22-2-2-200000000    symplectic lifted from Q8, Schur index 2
ρ1522-2-2000000-1-5/2-1+5/2-1-5/21-5/2-1+5/21+5/21-5/21+5/24ζ534ζ5243ζ5443ζ5ζ43ζ5443ζ5ζ4ζ534ζ52ζ43ζ5443ζ543ζ5443ζ54ζ534ζ52ζ4ζ534ζ52    symplectic lifted from Dic10, Schur index 2
ρ1622-2-2000000-1-5/2-1+5/2-1-5/21-5/2-1+5/21+5/21-5/21+5/2ζ4ζ534ζ52ζ43ζ5443ζ543ζ5443ζ54ζ534ζ5243ζ5443ζ5ζ43ζ5443ζ5ζ4ζ534ζ524ζ534ζ52    symplectic lifted from Dic10, Schur index 2
ρ1722-2-2000000-1+5/2-1-5/2-1+5/21+5/2-1-5/21-5/21+5/21-5/2ζ43ζ5443ζ54ζ534ζ52ζ4ζ534ζ5243ζ5443ζ5ζ4ζ534ζ524ζ534ζ52ζ43ζ5443ζ543ζ5443ζ5    symplectic lifted from Dic10, Schur index 2
ρ1822-2-2000000-1+5/2-1-5/2-1+5/21+5/2-1-5/21-5/21+5/21-5/243ζ5443ζ5ζ4ζ534ζ524ζ534ζ52ζ43ζ5443ζ54ζ534ζ52ζ4ζ534ζ5243ζ5443ζ5ζ43ζ5443ζ5    symplectic lifted from Dic10, Schur index 2
ρ192-22-2000000-1-5/2-1+5/21+5/21-5/21-5/2-1-5/2-1+5/21+5/2ζ5352545ζ5455352545ζ5455352ζ5352    complex lifted from C5⋊D4
ρ202-22-2000000-1+5/2-1-5/21-5/21+5/21+5/2-1+5/2-1-5/21-5/25455352ζ5352ζ5455352ζ5352ζ545545    complex lifted from C5⋊D4
ρ212-22-2000000-1-5/2-1+5/21+5/21-5/21-5/2-1-5/2-1+5/21+5/25352ζ545545ζ5352ζ545545ζ53525352    complex lifted from C5⋊D4
ρ222-22-2000000-1+5/2-1-5/21-5/21+5/21+5/2-1+5/2-1-5/21-5/2ζ545ζ53525352545ζ53525352545ζ545    complex lifted from C5⋊D4
ρ232-2-222i-2i0000-1+5/2-1-5/21-5/2-1-5/21+5/21-5/21+5/2-1+5/2ζ43ζ5443ζ5ζ43ζ5343ζ52ζ43ζ5343ζ52ζ43ζ5443ζ5ζ4ζ534ζ52ζ4ζ534ζ52ζ4ζ544ζ5ζ4ζ544ζ5    complex lifted from C4×D5
ρ242-2-22-2i2i0000-1-5/2-1+5/21+5/2-1+5/21-5/21+5/21-5/2-1-5/2ζ4ζ534ζ52ζ4ζ544ζ5ζ4ζ544ζ5ζ4ζ534ζ52ζ43ζ5443ζ5ζ43ζ5443ζ5ζ43ζ5343ζ52ζ43ζ5343ζ52    complex lifted from C4×D5
ρ252-2-22-2i2i0000-1+5/2-1-5/21-5/2-1-5/21+5/21-5/21+5/2-1+5/2ζ4ζ544ζ5ζ4ζ534ζ52ζ4ζ534ζ52ζ4ζ544ζ5ζ43ζ5343ζ52ζ43ζ5343ζ52ζ43ζ5443ζ5ζ43ζ5443ζ5    complex lifted from C4×D5
ρ262-2-222i-2i0000-1-5/2-1+5/21+5/2-1+5/21-5/21+5/21-5/2-1-5/2ζ43ζ5343ζ52ζ43ζ5443ζ5ζ43ζ5443ζ5ζ43ζ5343ζ52ζ4ζ544ζ5ζ4ζ544ζ5ζ4ζ534ζ52ζ4ζ534ζ52    complex lifted from C4×D5

Smallest permutation representation of C10.D4
Regular action 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 65 30 58)(2 64 21 57)(3 63 22 56)(4 62 23 55)(5 61 24 54)(6 70 25 53)(7 69 26 52)(8 68 27 51)(9 67 28 60)(10 66 29 59)(11 50 79 39)(12 49 80 38)(13 48 71 37)(14 47 72 36)(15 46 73 35)(16 45 74 34)(17 44 75 33)(18 43 76 32)(19 42 77 31)(20 41 78 40)
(1 73 6 78)(2 72 7 77)(3 71 8 76)(4 80 9 75)(5 79 10 74)(11 29 16 24)(12 28 17 23)(13 27 18 22)(14 26 19 21)(15 25 20 30)(31 57 36 52)(32 56 37 51)(33 55 38 60)(34 54 39 59)(35 53 40 58)(41 65 46 70)(42 64 47 69)(43 63 48 68)(44 62 49 67)(45 61 50 66)

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

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

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

C10.D4 is a maximal subgroup of
C4×Dic10  C20.6Q8  C42⋊D5  C422D5  C23.11D10  Dic5.14D4  C23.D10  Dic54D4  D10.12D4  D10⋊D4  Dic53Q8  C20⋊Q8  Dic5.Q8  C4.Dic10  D5×C4⋊C4  D10.13D4  D10⋊Q8  C4⋊C4⋊D5  C20.48D4  C4×C5⋊D4  C23.23D10  C23.18D10  Dic5⋊D4  Dic5⋊Q8  D103Q8  C30.Q8  Dic155C4  C30.4Q8  C50.D4  Dic5⋊Dic5  C10.Dic10  C102.22C22  D5.Dic10
C10.D4 is a maximal quotient of
C10.D8  C20.Q8  C20.8Q8  C20.53D4  C10.10C42  C30.Q8  Dic155C4  C30.4Q8  C50.D4  Dic5⋊Dic5  C10.Dic10  C102.22C22  D5.Dic10

Matrix representation of C10.D4 in GL4(𝔽41) generated by

13400
73400
0010
0001
,
1000
74000
0001
00400
,
32000
19900
00400
0001
G:=sub<GL(4,GF(41))| [1,7,0,0,34,34,0,0,0,0,1,0,0,0,0,1],[1,7,0,0,0,40,0,0,0,0,0,40,0,0,1,0],[32,19,0,0,0,9,0,0,0,0,40,0,0,0,0,1] >;

C10.D4 in GAP, Magma, Sage, TeX

C_{10}.D_4
% in TeX

G:=Group("C10.D4");
// GroupNames label

G:=SmallGroup(80,12);
// by ID

G=gap.SmallGroup(80,12);
# by ID

G:=PCGroup([5,-2,-2,-2,-2,-5,40,101,26,1604]);
// Polycyclic

G:=Group<a,b,c|a^10=b^4=1,c^2=a^5,b*a*b^-1=c*a*c^-1=a^-1,c*b*c^-1=b^-1>;
// generators/relations

Export

Subgroup lattice of C10.D4 in TeX
Character table of C10.D4 in TeX

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