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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), (D4×D5)⋊8C22, (C5×D4)⋊3C23, (C5×D8)⋊1C22, (C8×D5)⋊1C22, D4⋊3(C22×D5), Q8⋊D5⋊4C22, D4⋊D10⋊9C2, Q8⋊3(C22×D5), (Q8×D5)⋊9C22, (C5×Q8)⋊3C23, C20.240(C2×D4), C4○D20⋊7C22, C40⋊C2⋊1C22, C8⋊D5⋊1C22, (D5×M4(2))⋊1C2, D4.D5⋊4C22, C22.46(D4×D5), C4.19(C23×D5), D4.D10⋊9C2, D10.115(C2×D4), (C2×D20)⋊35C22, D4⋊2D5⋊9C22, (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

Subgroups: 1358 in 298 conjugacy classes, 101 normal (51 characteristic)
C1, C2, C2 [×10], C4 [×2], C4 [×4], C22, C22 [×24], C5, C8 [×2], C8 [×2], C2×C4, C2×C4 [×10], D4, D4 [×2], D4 [×14], Q8, Q8 [×2], C23 [×12], D5 [×2], D5 [×4], C10, C10 [×4], C2×C8 [×2], M4(2), M4(2) [×3], D8 [×2], D8 [×6], SD16 [×2], SD16 [×6], C22×C4 [×2], C2×D4, C2×D4 [×10], C2×Q8, C4○D4, C4○D4 [×5], C24, Dic5 [×2], Dic5, C20 [×2], C20, D10 [×2], D10 [×17], C2×C10, C2×C10 [×5], C2×M4(2), C2×D8 [×2], C2×SD16 [×2], C8⋊C22, C8⋊C22 [×7], C22×D4, C2×C4○D4, C5⋊2C8 [×2], C40 [×2], Dic10, Dic10, C4×D5 [×4], C4×D5 [×3], D20, D20 [×2], D20 [×2], C2×Dic5, C2×Dic5, C5⋊D4 [×7], C2×C20, C2×C20, C5×D4, C5×D4 [×2], C5×D4 [×2], C5×Q8, C22×D5, C22×D5 [×10], C22×C10, C2×C8⋊C22, C8×D5 [×2], C8⋊D5 [×2], C40⋊C2 [×2], D40 [×2], C4.Dic5, D4⋊D5 [×4], D4.D5 [×2], Q8⋊D5 [×2], C5×M4(2), C5×D8 [×2], C5×SD16 [×2], C2×C4×D5, C2×C4×D5, C2×D20, C4○D20, C4○D20, D4×D5, D4×D5 [×4], D4×D5 [×3], 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 [×2], D8⋊D5 [×2], D5×SD16 [×2], D40⋊C2 [×2], D4.D10, D4⋊D10, C5×C8⋊C22, C2×D4×D5, D5×C4○D4, D5×C8⋊C22

Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C24, D10 [×7], C8⋊C22 [×2], C22×D4, C22×D5 [×7], C2×C8⋊C22, D4×D5 [×2], C23×D5, C2×D4×D5, D5×C8⋊C22

Generators and relations
 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 >

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

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

Matrix representation ►G ⊆ 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] >;
 

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

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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