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G = C2×D4×D5order 160 = 25·5

Direct product of C2, D4 and D5

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

Aliases: C2×D4×D5, C20⋊C23, C233D10, D207C22, D102C23, C10.5C24, Dic51C23, (C2×C4)⋊6D10, C102(C2×D4), (C2×C10)⋊C23, C52(C22×D4), (D4×C10)⋊5C2, C41(C22×D5), (C2×D20)⋊11C2, (C2×C20)⋊2C22, (C23×D5)⋊4C2, (C4×D5)⋊3C22, (C5×D4)⋊5C22, C5⋊D41C22, C2.6(C23×D5), C221(C22×D5), (C22×C10)⋊4C22, (C2×Dic5)⋊8C22, (C22×D5)⋊6C22, (C2×C4×D5)⋊3C2, (C2×C5⋊D4)⋊9C2, SmallGroup(160,217)

Series: Derived Chief Lower central Upper central

C1C10 — C2×D4×D5
C1C5C10D10C22×D5C23×D5 — C2×D4×D5
C5C10 — C2×D4×D5
C1C22C2×D4

Generators and relations for C2×D4×D5
 G = < a,b,c,d,e | a2=b4=c2=d5=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 808 in 236 conjugacy classes, 97 normal (15 characteristic)
C1, C2, C2 [×2], C2 [×12], C4 [×2], C4 [×2], C22, C22 [×4], C22 [×34], C5, C2×C4, C2×C4 [×5], D4 [×4], D4 [×12], C23 [×2], C23 [×19], D5 [×4], D5 [×4], C10, C10 [×2], C10 [×4], C22×C4, C2×D4, C2×D4 [×11], C24 [×2], Dic5 [×2], C20 [×2], D10 [×10], D10 [×20], C2×C10, C2×C10 [×4], C2×C10 [×4], C22×D4, C4×D5 [×4], D20 [×4], C2×Dic5, C5⋊D4 [×8], C2×C20, C5×D4 [×4], C22×D5, C22×D5 [×10], C22×D5 [×8], C22×C10 [×2], C2×C4×D5, C2×D20, D4×D5 [×8], C2×C5⋊D4 [×2], D4×C10, C23×D5 [×2], C2×D4×D5
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C24, D10 [×7], C22×D4, C22×D5 [×7], D4×D5 [×2], C23×D5, C2×D4×D5

Smallest permutation representation of C2×D4×D5
On 40 points
Generators in S40
(1 14)(2 15)(3 11)(4 12)(5 13)(6 16)(7 17)(8 18)(9 19)(10 20)(21 31)(22 32)(23 33)(24 34)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)
(1 24 9 29)(2 25 10 30)(3 21 6 26)(4 22 7 27)(5 23 8 28)(11 31 16 36)(12 32 17 37)(13 33 18 38)(14 34 19 39)(15 35 20 40)
(21 26)(22 27)(23 28)(24 29)(25 30)(31 36)(32 37)(33 38)(34 39)(35 40)
(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 18)(2 17)(3 16)(4 20)(5 19)(6 11)(7 15)(8 14)(9 13)(10 12)(21 36)(22 40)(23 39)(24 38)(25 37)(26 31)(27 35)(28 34)(29 33)(30 32)

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

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

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

C2×D4×D5 is a maximal subgroup of
(D4×D5)⋊C4  D4⋊D20  D20.8D4  D20⋊D4  D106SD16  (D4×C10)⋊C4  (C2×D4)⋊7F5  D5⋊(C4.D4)  (C2×F5)⋊D4  C2.(D4×F5)  C4211D10  D45D20  C243D10  C244D10  C10.372+ 1+4  C10.382+ 1+4  D2019D4  C10.402+ 1+4  D2020D4  C10.1202+ 1+4  C10.1212+ 1+4  C4218D10  D2010D4  C4226D10  D2011D4  C10.1452+ 1+4  D10.C24
C2×D4×D5 is a maximal quotient of
C24.27D10  C10.2- 1+4  C4212D10  C42.228D10  D2023D4  D2024D4  Dic1023D4  Dic1024D4  C24.56D10  C243D10  C244D10  C24.33D10  C24.34D10  C20⋊(C4○D4)  C10.682- 1+4  Dic1019D4  Dic1020D4  C10.372+ 1+4  C4⋊C421D10  C10.382+ 1+4  C10.392+ 1+4  D2019D4  C10.402+ 1+4  C10.732- 1+4  D2020D4  C4⋊C426D10  C10.162- 1+4  C10.172- 1+4  D2021D4  D2022D4  Dic1021D4  Dic1022D4  C10.792- 1+4  C10.1202+ 1+4  C10.1212+ 1+4  C10.822- 1+4  C4⋊C428D10  C42.233D10  C4218D10  C42.141D10  D2010D4  Dic1010D4  C4226D10  C42.238D10  D2011D4  Dic1011D4  C42.171D10  C42.240D10  D2012D4  D208Q8  D813D10  D20.29D4  D20.30D4  Q16⋊D10  D815D10  D811D10  D20.47D4  SD16⋊D10  D85D10  D86D10  D40⋊C22  C40.C23  D20.44D4

40 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J2K2L2M2N2O4A4B4C4D5A5B10A···10F10G···10N20A20B20C20D
order122222222222222244445510···1010···1020202020
size11112222555510101010221010222···24···44444

40 irreducible representations

dim1111111222224
type+++++++++++++
imageC1C2C2C2C2C2C2D4D5D10D10D10D4×D5
kernelC2×D4×D5C2×C4×D5C2×D20D4×D5C2×C5⋊D4D4×C10C23×D5D10C2×D4C2×C4D4C23C2
# reps1118212422844

Matrix representation of C2×D4×D5 in GL4(𝔽41) generated by

40000
04000
00400
00040
,
1000
0100
00040
0010
,
1000
0100
0010
00040
,
0100
40600
0010
0001
,
04000
40000
0010
0001
G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,40,0,0,0,0,40],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,40,0],[1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,40],[0,40,0,0,1,6,0,0,0,0,1,0,0,0,0,1],[0,40,0,0,40,0,0,0,0,0,1,0,0,0,0,1] >;

C2×D4×D5 in GAP, Magma, Sage, TeX

C_2\times D_4\times D_5
% in TeX

G:=Group("C2xD4xD5");
// GroupNames label

G:=SmallGroup(160,217);
// by ID

G=gap.SmallGroup(160,217);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,159,4613]);
// Polycyclic

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

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