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G = D5×C8⋊C22  order 320 = 26·5

Direct product of D5 and C8⋊C22

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

Aliases: D5×C8⋊C22, C40⋊C23, D8⋊3D10, SD16⋊1D10, D40⋊1C22, D20⋊3C23, M4(2)⋊7D10, C20.19C24, Dic10⋊3C23, (D5×D8)⋊1C2, C4○D4⋊9D10, C8⋊1(C22×D5), D8⋊D5⋊1C2, (C2×D4)⋊29D10, C8⋊D10⋊1C2, D40⋊C2⋊1C2, C5⋊2C8⋊3C23, D4⋊D5⋊5C22, (D5×SD16)⋊1C2, (C4×D5).98D4, C4.189(D4×D5), (C8×D5)⋊1C22, D4⋊3(C22×D5), (C5×D8)⋊1C22, (D4×D5)⋊8C22, (C5×D4)⋊3C23, Q8⋊D5⋊4C22, D4⋊D10⋊9C2, (Q8×D5)⋊9C22, (C5×Q8)⋊3C23, Q8⋊3(C22×D5), C20.240(C2×D4), C4○D20⋊7C22, C40⋊C2⋊1C22, C8⋊D5⋊1C22, (D5×M4(2))⋊1C2, D4.D5⋊4C22, C4.19(C23×D5), C22.46(D4×D5), D4.D10⋊9C2, D10.115(C2×D4), D4⋊2D5⋊9C22, (C2×D20)⋊35C22, (D4×C10)⋊21C22, Dic5.99(C2×D4), (C2×Dic5).89D4, Q8⋊2D5⋊9C22, (C5×SD16)⋊1C22, (C4×D5).12C23, (C2×C20).110C23, (C22×D5).138D4, C10.120(C22×D4), (C5×M4(2))⋊1C22, C4.Dic5⋊12C22, (C2×D4×D5)⋊24C2, C5⋊4(C2×C8⋊C22), C2.93(C2×D4×D5), (D5×C4○D4)⋊3C2, (C5×C8⋊C22)⋊1C2, (C2×C10).65(C2×D4), (C5×C4○D4)⋊5C22, (C2×C4×D5).169C22, (C2×C4).94(C22×D5), SmallGroup(320,1444)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C20 — D5×C8⋊C22
Chief series
C1 — C5 — C10 — C20 — C4×D5 — C2×C4×D5 — C2×D4×D5 — D5×C8⋊C22
Lower central
C5 — C10 — C20 — D5×C8⋊C22
Upper central
C1 — C2 — C2×C4 — C8⋊C22

Generators and relations for D5×C8⋊C22
 G = < a,b,c,d,e | a5=b2=c8=d2=e2=1, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd=c3, ece=c5, de=ed >

Subgroups: 1358 in 298 conjugacy classes, 101 normal (51 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, D4, D4, D4, Q8, Q8, C23, D5, D5, C10, C10, C2×C8, M4(2), M4(2), D8, D8, SD16, SD16, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C4○D4, C24, Dic5, Dic5, C20, C20, D10, D10, C2×C10, C2×C10, C2×M4(2), C2×D8, C2×SD16, C8⋊C22, C8⋊C22, C22×D4, C2×C4○D4, C5⋊2C8, C40, Dic10, Dic10, C4×D5, C4×D5, D20, D20, D20, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C5×D4, C5×D4, C5×D4, C5×Q8, C22×D5, C22×D5, C22×C10, C2×C8⋊C22, C8×D5, C8⋊D5, C40⋊C2, D40, C4.Dic5, D4⋊D5, D4.D5, Q8⋊D5, C5×M4(2), C5×D8, C5×SD16, C2×C4×D5, C2×C4×D5, C2×D20, C4○D20, C4○D20, D4×D5, D4×D5, D4×D5, D4⋊2D5, D4⋊2D5, Q8×D5, Q8⋊2D5, C2×C5⋊D4, D4×C10, C5×C4○D4, C23×D5, D5×M4(2), C8⋊D10, D5×D8, D8⋊D5, D5×SD16, D40⋊C2, D4.D10, D4⋊D10, C5×C8⋊C22, C2×D4×D5, D5×C4○D4, D5×C8⋊C22
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C24, D10, C8⋊C22, C22×D4, C22×D5, C2×C8⋊C22, D4×D5, C23×D5, C2×D4×D5, D5×C8⋊C22

Smallest permutation representation of D5×C8⋊C22
►On 40 points
Generators in S40
(1 27 24 38 16)(2 28 17 39 9)(3 29 18 40 10)(4 30 19 33 11)(5 31 20 34 12)(6 32 21 35 13)(7 25 22 36 14)(8 26 23 37 15)

(1 12)(2 13)(3 14)(4 15)(5 16)(6 9)(7 10)(8 11)(17 21)(18 22)(19 23)(20 24)(25 40)(26 33)(27 34)(28 35)(29 36)(30 37)(31 38)(32 39)

(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)

(1 3)(2 6)(5 7)(9 13)(10 16)(12 14)(17 21)(18 24)(20 22)(25 31)(27 29)(28 32)(34 36)(35 39)(38 40)

(1 5)(3 7)(10 14)(12 16)(18 22)(20 24)(25 29)(27 31)(34 38)(36 40)
 
G:=sub<Sym(40)| (1,27,24,38,16)(2,28,17,39,9)(3,29,18,40,10)(4,30,19,33,11)(5,31,20,34,12)(6,32,21,35,13)(7,25,22,36,14)(8,26,23,37,15), (1,12)(2,13)(3,14)(4,15)(5,16)(6,9)(7,10)(8,11)(17,21)(18,22)(19,23)(20,24)(25,40)(26,33)(27,34)(28,35)(29,36)(30,37)(31,38)(32,39), (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), (1,3)(2,6)(5,7)(9,13)(10,16)(12,14)(17,21)(18,24)(20,22)(25,31)(27,29)(28,32)(34,36)(35,39)(38,40), (1,5)(3,7)(10,14)(12,16)(18,22)(20,24)(25,29)(27,31)(34,38)(36,40)>;
 
G:=Group( (1,27,24,38,16)(2,28,17,39,9)(3,29,18,40,10)(4,30,19,33,11)(5,31,20,34,12)(6,32,21,35,13)(7,25,22,36,14)(8,26,23,37,15), (1,12)(2,13)(3,14)(4,15)(5,16)(6,9)(7,10)(8,11)(17,21)(18,22)(19,23)(20,24)(25,40)(26,33)(27,34)(28,35)(29,36)(30,37)(31,38)(32,39), (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), (1,3)(2,6)(5,7)(9,13)(10,16)(12,14)(17,21)(18,24)(20,22)(25,31)(27,29)(28,32)(34,36)(35,39)(38,40), (1,5)(3,7)(10,14)(12,16)(18,22)(20,24)(25,29)(27,31)(34,38)(36,40) );
 
G=PermutationGroup([[(1,27,24,38,16),(2,28,17,39,9),(3,29,18,40,10),(4,30,19,33,11),(5,31,20,34,12),(6,32,21,35,13),(7,25,22,36,14),(8,26,23,37,15)], [(1,12),(2,13),(3,14),(4,15),(5,16),(6,9),(7,10),(8,11),(17,21),(18,22),(19,23),(20,24),(25,40),(26,33),(27,34),(28,35),(29,36),(30,37),(31,38),(32,39)], [(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)], [(1,3),(2,6),(5,7),(9,13),(10,16),(12,14),(17,21),(18,24),(20,22),(25,31),(27,29),(28,32),(34,36),(35,39),(38,40)], [(1,5),(3,7),(10,14),(12,16),(18,22),(20,24),(25,29),(27,31),(34,38),(36,40)]])
 

44 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J2K4A4B4C4D4E4F5A5B8A8B8C8D10A10B10C10D10E···10J20A20B20C20D20E20F40A40B40C40D
order1222222222224444445588881010101010···1020202020202040404040
size11244455102020202241010202244202022448···84444888888

44 irreducible representations

dim1111111111112222222224448
type+++++++++++++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2C2D4D4D4D5D10D10D10D10D10C8⋊C22D4×D5D4×D5D5×C8⋊C22
kernelD5×C8⋊C22D5×M4(2)C8⋊D10D5×D8D8⋊D5D5×SD16D40⋊C2D4.D10D4⋊D10C5×C8⋊C22C2×D4×D5D5×C4○D4C4×D5C2×Dic5C22×D5C8⋊C22M4(2)D8SD16C2×D4C4○D4D5C4C22C1
# reps1112222111112112244222222

Matrix representation of D5×C8⋊C22 ►in GL8(𝔽41)

400100000
040010000
503500000
050350000
00001000
00000100
00000010
00000001
,
10000000
01000000
3604000000
0360400000
000040000
000004000
000000400
000000040
,
040000000
10000000
000400000
00100000
00004040132
000000400
000040000
000002301
,
400000000
01000000
004000000
00010000
00000100
00001000
000000400
000000181
,
400000000
040000000
004000000
000400000
000040000
000004000
00000010
0000232301

G:=sub<GL(8,GF(41))| [40,0,5,0,0,0,0,0,0,40,0,5,0,0,0,0,1,0,35,0,0,0,0,0,0,1,0,35,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,36,0,0,0,0,0,0,1,0,36,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40],[0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,40,0,40,0,0,0,0,0,40,0,0,23,0,0,0,0,1,40,0,0,0,0,0,0,32,0,0,1],[40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,40,18,0,0,0,0,0,0,0,1],[40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,23,0,0,0,0,0,40,0,23,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1] >;
 

D5×C8⋊C22 in GAP, Magma, Sage, TeX 

D_5\times C_8\rtimes C_2^2
 
% in TeX
 
G:=Group("D5xC8:C2^2");
 
// GroupNames label
 
G:=SmallGroup(320,1444);
 
// by ID
 
G=gap.SmallGroup(320,1444);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,570,185,438,235,102,12550]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^5=b^2=c^8=d^2=e^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d=c^3,e*c*e=c^5,d*e=e*d>;
 
// generators/relations
 

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