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G = D5.2+ 1+4  order 320 = 26·5

The non-split extension by D5 of 2+ 1+4 acting via 2+ 1+4/C4○D4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D10.19C24, D5.22- 1+4, D5.22+ 1+4, C4○D4⋊5F5, D4⋊9(C2×F5), (D4×F5)⋊4C2, Q8⋊8(C2×F5), (Q8×F5)⋊4C2, C4○D20⋊6C4, D20⋊10(C2×C4), D4⋊2D5⋊11C4, Q8⋊2D5⋊11C4, Dic10⋊9(C2×C4), C4⋊F5.14C22, (C4×F5).8C22, (C2×F5).8C23, C2.19(C23×F5), C4.34(C22×F5), C10.18(C23×C4), C20.35(C22×C4), (D4×D5).19C22, D10.9(C22×C4), (C4×D5).58C23, (Q8×D5).17C22, C22⋊F5.5C22, C22.6(C22×F5), C5⋊(C23.33C23), Dic5.8(C22×C4), (C22×F5).11C22, (C22×D5).154C23, D10.C23⋊10C2, (C2×C4⋊F5)⋊5C2, (C2×C4)⋊5(C2×F5), (C2×C20)⋊7(C2×C4), (C5×C4○D4)⋊6C4, (C4×D5)⋊8(C2×C4), C5⋊D4⋊4(C2×C4), (C5×Q8)⋊9(C2×C4), (C5×D4)⋊10(C2×C4), (D5×C4○D4).12C2, (C2×Dic5)⋊20(C2×C4), (C2×C10).7(C22×C4), (C2×C4×D5).225C22, SmallGroup(320,1604)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C10 — D5.2+ 1+4
Chief series
C1 — C5 — D5 — D10 — C2×F5 — C22×F5 — D4×F5 — D5.2+ 1+4
Lower central
C5 — C10 — D5.2+ 1+4
Upper central
C1 — C2 — C4○D4

Generators and relations for D5.2+ 1+4
 G = < a,b,c,d,e,f | a5=b2=c4=d2=1, e2=c2, f2=a-1b, bab=a-1, ac=ca, ad=da, ae=ea, faf-1=a3, bc=cb, bd=db, be=eb, fbf-1=a2b, dcd=c-1, ce=ec, cf=fc, de=ed, df=fd, fef-1=c2e >

Subgroups: 1018 in 294 conjugacy classes, 138 normal (20 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C5, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, D5, D5, C10, C10, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, C4○D4, C4○D4, Dic5, Dic5, C20, C20, F5, D10, D10, D10, C2×C10, C2×C4⋊C4, C42⋊C2, C4×D4, C4×Q8, C2×C4○D4, Dic10, C4×D5, C4×D5, D20, C2×Dic5, C5⋊D4, C2×C20, C5×D4, C5×Q8, C2×F5, C2×F5, C22×D5, C23.33C23, C4×F5, C4⋊F5, C4⋊F5, C22⋊F5, C2×C4×D5, C4○D20, D4×D5, D4⋊2D5, Q8×D5, Q8⋊2D5, C5×C4○D4, C22×F5, C2×C4⋊F5, D10.C23, D4×F5, Q8×F5, D5×C4○D4, D5.2+ 1+4
Quotients: C1, C2, C4, C22, C2×C4, C23, C22×C4, C24, F5, C23×C4, 2+ 1+4, 2- 1+4, C2×F5, C23.33C23, C22×F5, C23×F5, D5.2+ 1+4

Smallest permutation representation of D5.2+ 1+4
►On 40 points
Generators in S40
(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 5)(2 4)(7 10)(8 9)(12 15)(13 14)(17 20)(18 19)(22 25)(23 24)(27 30)(28 29)(32 35)(33 34)(37 40)(38 39)

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

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

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

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

44 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I4A4B4C4D4E···4X 5 10A10B10C10D20A20B20C20D20E
order122222222244444···45101010102020202020
size1122255101010222210···104488844888

44 irreducible representations

dim11111111114444448
type++++++++-+++
imageC1C2C2C2C2C2C4C4C4C4F52+ 1+42- 1+4C2×F5C2×F5C2×F5D5.2+ 1+4
kernelD5.2+ 1+4C2×C4⋊F5D10.C23D4×F5Q8×F5D5×C4○D4C4○D20D4⋊2D5Q8⋊2D5C5×C4○D4C4○D4D5D5C2×C4D4Q8C1
# reps13362166221113312

Matrix representation of D5.2+ 1+4 ►in GL8(ℤ)

01000000
00100000
00010000
-1-1-1-10000
00001000
00000100
00000010
00000001
,
01000000
10000000
-1-1-1-10000
00010000
00001000
00000100
00000010
00000001
,
10000000
01000000
00100000
00010000
0000000-1
00000010
00000-100
00001000
,
10000000
01000000
00100000
00010000
00000001
000000-10
00000-100
00001000
,
-10000000
0-1000000
00-100000
000-10000
00000100
0000-1000
00000001
000000-10
,
10000000
00010000
01000000
-1-1-1-10000
00001000
00000-100
000000-10
00000001

G:=sub<GL(8,Integers())| [0,0,0,-1,0,0,0,0,1,0,0,-1,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,1,-1,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0],[-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,-1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0],[1,0,0,-1,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,1,0,-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,-1,0,0,0,0,0,0,0,0,1] >;
 

D5.2+ 1+4 in GAP, Magma, Sage, TeX 

D_5.2_+^{1+4}
 
% in TeX
 
G:=Group("D5.ES+(2,2)");
 
// GroupNames label
 
G:=SmallGroup(320,1604);
 
// by ID
 
G=gap.SmallGroup(320,1604);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,387,1123,136,6278,818]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^5=b^2=c^4=d^2=1,e^2=c^2,f^2=a^-1*b,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,f*a*f^-1=a^3,b*c=c*b,b*d=d*b,b*e=e*b,f*b*f^-1=a^2*b,d*c*d=c^-1,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=c^2*e>;
 
// generators/relations
 

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