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G = D5×2+ 1+4order 320 = 26·5

Direct product of D5 and 2+ 1+4

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

Aliases: D5×2+ 1+4, D2013C23, C10.15C25, C20.50C24, D10.25C24, Dic1012C23, Dic5.10C24, C4○D410D10, (C2×D4)⋊31D10, (C2×C20)⋊2C23, (C4×D5)⋊3C23, C5⋊D46C23, D46D109C2, Q89(C22×D5), (C5×D4)⋊11C23, D410(C22×D5), (D4×D5)⋊17C22, (C2×C10).6C24, D48D1011C2, (Q8×D5)⋊20C22, (C5×Q8)⋊10C23, C4.47(C23×D5), C2.16(D5×C24), C232(C22×D5), C53(C2×2+ 1+4), C4○D2012C22, (C2×D20)⋊40C22, (D4×C10)⋊25C22, (C22×C10)⋊2C23, (C2×Dic5)⋊6C23, (C22×D5)⋊6C23, D42D515C22, C22.3(C23×D5), Q82D515C22, (C23×D5)⋊19C22, (C5×2+ 1+4)⋊4C2, (C2×D4×D5)⋊29C2, (D5×C4○D4)⋊7C2, (C2×C4×D5)⋊36C22, (C2×C4)⋊2(C22×D5), (C5×C4○D4)⋊10C22, (C2×C5⋊D4)⋊32C22, SmallGroup(320,1622)

Series: Derived Chief Lower central Upper central

C1C10 — D5×2+ 1+4
C1C5C10D10C22×D5C23×D5C2×D4×D5 — D5×2+ 1+4
C5C10 — D5×2+ 1+4
C1C22+ 1+4

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

Subgroups: 3118 in 898 conjugacy classes, 445 normal (8 characteristic)
C1, C2, C2 [×20], C4 [×6], C4 [×6], C22 [×9], C22 [×52], C5, C2×C4 [×9], C2×C4 [×33], D4 [×18], D4 [×54], Q8 [×2], Q8 [×6], C23 [×6], C23 [×39], D5 [×2], D5 [×9], C10, C10 [×9], C22×C4 [×9], C2×D4 [×9], C2×D4 [×81], C2×Q8 [×2], C4○D4 [×6], C4○D4 [×42], C24 [×6], Dic5 [×6], C20 [×6], D10, D10 [×9], D10 [×36], C2×C10 [×9], C2×C10 [×6], C22×D4 [×9], C2×C4○D4 [×6], 2+ 1+4, 2+ 1+4 [×15], Dic10 [×6], C4×D5 [×24], D20 [×18], C2×Dic5 [×9], C5⋊D4 [×36], C2×C20 [×9], C5×D4 [×18], C5×Q8 [×2], C22×D5 [×27], C22×D5 [×12], C22×C10 [×6], C2×2+ 1+4, C2×C4×D5 [×9], C2×D20 [×9], C4○D20 [×18], D4×D5 [×54], D42D5 [×18], Q8×D5 [×2], Q82D5 [×6], C2×C5⋊D4 [×18], D4×C10 [×9], C5×C4○D4 [×6], C23×D5 [×6], C2×D4×D5 [×9], D46D10 [×9], D5×C4○D4 [×6], D48D10 [×6], C5×2+ 1+4, D5×2+ 1+4
Quotients: C1, C2 [×31], C22 [×155], C23 [×155], D5, C24 [×31], D10 [×15], 2+ 1+4 [×2], C25, C22×D5 [×35], C2×2+ 1+4, C23×D5 [×15], D5×C24, 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 14 9 19)(2 15 10 20)(3 11 6 16)(4 12 7 17)(5 13 8 18)(21 31 26 36)(22 32 27 37)(23 33 28 38)(24 34 29 39)(25 35 30 40)
(11 16)(12 17)(13 18)(14 19)(15 20)(31 36)(32 37)(33 38)(34 39)(35 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)

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

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

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

68 conjugacy classes

class 1 2A2B···2J2K2L2M···2U4A···4F4G···4L5A5B10A10B10C···10T20A···20L
order122···2222···24···44···455101010···1020···20
size112···25510···102···210···1022224···44···4

68 irreducible representations

dim11111122248
type+++++++++++
imageC1C2C2C2C2C2D5D10D102+ 1+4D5×2+ 1+4
kernelD5×2+ 1+4C2×D4×D5D46D10D5×C4○D4D48D10C5×2+ 1+42+ 1+4C2×D4C4○D4D5C1
# reps1996612181222

Matrix representation of D5×2+ 1+4 in GL6(𝔽41)

4010000
5350000
001000
000100
000010
000001
,
100000
36400000
001000
000100
000010
000001
,
4000000
0400000
0004000
001000
001112
004004040
,
4000000
0400000
001000
0004000
000010
004004040
,
4000000
0400000
0000400
001112
001000
004040040
,
100000
010000
001000
000100
0000400
004040040

G:=sub<GL(6,GF(41))| [40,5,0,0,0,0,1,35,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,36,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,1,1,40,0,0,40,0,1,0,0,0,0,0,1,40,0,0,0,0,2,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,40,0,0,0,40,0,0,0,0,0,0,1,40,0,0,0,0,0,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,1,1,40,0,0,0,1,0,40,0,0,40,1,0,0,0,0,0,2,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,40,0,0,0,1,0,40,0,0,0,0,40,0,0,0,0,0,0,40] >;

D5×2+ 1+4 in GAP, Magma, Sage, TeX

D_5\times 2_+^{1+4}
% in TeX

G:=Group("D5xES+(2,2)");
// GroupNames label

G:=SmallGroup(320,1622);
// by ID

G=gap.SmallGroup(320,1622);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,297,851,12550]);
// Polycyclic

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

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