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G = 2- 1+4.D5order 320 = 26·5

The non-split extension by 2- 1+4 of D5 acting faithfully

non-abelian, soluble

Aliases: 2- 1+4.D5, C2.2(C24:D5), 2- 1+4:C5.C2, SmallGroup(320,1581)

Series: Derived Chief Lower central Upper central

C1C22- 1+42- 1+4:C5 — 2- 1+4.D5
C1C22- 1+42- 1+4:C5 — 2- 1+4.D5
2- 1+4:C5 — 2- 1+4.D5
C1C2

Generators and relations for 2- 1+4.D5
 G = < a,b,c,d,e,f | a4=b2=e5=1, c2=d2=f2=a2, bab=a-1, ac=ca, ebe-1=ad=da, eae-1=fdf-1=a-1bd, faf-1=ede-1=bcd, bc=cb, bd=db, fbf-1=acd, dcd-1=a2c, ece-1=a2bc, fcf-1=abc, fef-1=e-1 >

Subgroups: 343 in 48 conjugacy classes, 5 normal (all characteristic)
Quotients: C1, C2, D5, C24:D5, 2- 1+4.D5
10C2
16C5
5C22
5C4
5C4
10C4
10C4
20C4
16C10
5Q8
5C2xC4
5D4
5C2xC4
5Q8
5D4
5C2xC4
10C2xC4
10C2xC4
10Q8
10C8
10C8
10Q8
16Dic5
5C4oD4
5C4oD4
5C2xQ8
5M4(2)
5C2xQ8
5C42
5M4(2)
10Q16
10SD16
10Q16
10C4:C4
10C4:C4
10SD16
5C4wrC2
5C4.10D4
5C8.C22
5C4:Q8
5C8.C22
5C4wrC2
5D4.10D4

Character table of 2- 1+4.D5

 class 12A2B4A4B4C4D4E5A5B8A8B10A10B
 size 11101010202040323240403232
ρ111111111111111    trivial
ρ211111-1-1-111-1-111    linear of order 2
ρ322222000-1-5/2-1+5/200-1-5/2-1+5/2    orthogonal lifted from D5
ρ422222000-1+5/2-1-5/200-1+5/2-1-5/2    orthogonal lifted from D5
ρ54-40002-20-1-10011    symplectic faithful, Schur index 2
ρ64-4000-220-1-10011    symplectic faithful, Schur index 2
ρ75511-311-100-1100    orthogonal lifted from C24:D5
ρ855-31111100-1-100    orthogonal lifted from C24:D5
ρ9551-31-1-1100-1100    orthogonal lifted from C24:D5
ρ105511-3-1-11001-100    orthogonal lifted from C24:D5
ρ1155-311-1-1-1001100    orthogonal lifted from C24:D5
ρ12551-3111-1001-100    orthogonal lifted from C24:D5
ρ138-80000001-5/21+5/200-1+5/2-1-5/2    symplectic faithful, Schur index 2
ρ148-80000001+5/21-5/200-1-5/2-1+5/2    symplectic faithful, Schur index 2

Smallest permutation representation of 2- 1+4.D5
On 64 points
Generators in S64
(1 59 3 45)(2 10 4 20)(5 32 30 7)(6 15 31 26)(8 14 33 24)(9 21 34 11)(12 27 22 16)(13 25 23 19)(17 29 28 18)(35 46 50 55)(36 53 51 38)(37 40 52 63)(39 58 54 49)(41 56 64 47)(42 61 60 43)(44 57 62 48)
(1 42)(2 17)(3 60)(4 28)(5 31)(6 30)(7 26)(8 22)(9 19)(10 18)(11 13)(12 33)(14 27)(15 32)(16 24)(20 29)(21 23)(25 34)(35 44)(36 52)(37 51)(38 63)(39 47)(40 53)(41 49)(43 59)(45 61)(46 48)(50 62)(54 56)(55 57)(58 64)
(1 52 3 37)(2 31 4 6)(5 28 30 17)(7 29 32 18)(8 11 33 21)(9 24 34 14)(10 26 20 15)(12 23 22 13)(16 25 27 19)(35 49 50 58)(36 60 51 42)(38 61 53 43)(39 55 54 46)(40 59 63 45)(41 62 64 44)(47 57 56 48)
(1 47 3 56)(2 22 4 12)(5 21 30 11)(6 13 31 23)(7 9 32 34)(8 28 33 17)(10 16 20 27)(14 18 24 29)(15 25 26 19)(35 53 50 38)(36 46 51 55)(37 57 52 48)(39 60 54 42)(40 62 63 44)(41 45 64 59)(43 49 61 58)
(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 41 42 43 44)(45 46 47 48 49)(50 51 52 53 54)(55 56 57 58 59)(60 61 62 63 64)
(1 4 3 2)(5 54 30 39)(6 53 31 38)(7 52 32 37)(8 51 33 36)(9 50 34 35)(10 46 20 55)(11 45 21 59)(12 49 22 58)(13 48 23 57)(14 47 24 56)(15 41 26 64)(16 40 27 63)(17 44 28 62)(18 43 29 61)(19 42 25 60)

G:=sub<Sym(64)| (1,59,3,45)(2,10,4,20)(5,32,30,7)(6,15,31,26)(8,14,33,24)(9,21,34,11)(12,27,22,16)(13,25,23,19)(17,29,28,18)(35,46,50,55)(36,53,51,38)(37,40,52,63)(39,58,54,49)(41,56,64,47)(42,61,60,43)(44,57,62,48), (1,42)(2,17)(3,60)(4,28)(5,31)(6,30)(7,26)(8,22)(9,19)(10,18)(11,13)(12,33)(14,27)(15,32)(16,24)(20,29)(21,23)(25,34)(35,44)(36,52)(37,51)(38,63)(39,47)(40,53)(41,49)(43,59)(45,61)(46,48)(50,62)(54,56)(55,57)(58,64), (1,52,3,37)(2,31,4,6)(5,28,30,17)(7,29,32,18)(8,11,33,21)(9,24,34,14)(10,26,20,15)(12,23,22,13)(16,25,27,19)(35,49,50,58)(36,60,51,42)(38,61,53,43)(39,55,54,46)(40,59,63,45)(41,62,64,44)(47,57,56,48), (1,47,3,56)(2,22,4,12)(5,21,30,11)(6,13,31,23)(7,9,32,34)(8,28,33,17)(10,16,20,27)(14,18,24,29)(15,25,26,19)(35,53,50,38)(36,46,51,55)(37,57,52,48)(39,60,54,42)(40,62,63,44)(41,45,64,59)(43,49,61,58), (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)(40,41,42,43,44)(45,46,47,48,49)(50,51,52,53,54)(55,56,57,58,59)(60,61,62,63,64), (1,4,3,2)(5,54,30,39)(6,53,31,38)(7,52,32,37)(8,51,33,36)(9,50,34,35)(10,46,20,55)(11,45,21,59)(12,49,22,58)(13,48,23,57)(14,47,24,56)(15,41,26,64)(16,40,27,63)(17,44,28,62)(18,43,29,61)(19,42,25,60)>;

G:=Group( (1,59,3,45)(2,10,4,20)(5,32,30,7)(6,15,31,26)(8,14,33,24)(9,21,34,11)(12,27,22,16)(13,25,23,19)(17,29,28,18)(35,46,50,55)(36,53,51,38)(37,40,52,63)(39,58,54,49)(41,56,64,47)(42,61,60,43)(44,57,62,48), (1,42)(2,17)(3,60)(4,28)(5,31)(6,30)(7,26)(8,22)(9,19)(10,18)(11,13)(12,33)(14,27)(15,32)(16,24)(20,29)(21,23)(25,34)(35,44)(36,52)(37,51)(38,63)(39,47)(40,53)(41,49)(43,59)(45,61)(46,48)(50,62)(54,56)(55,57)(58,64), (1,52,3,37)(2,31,4,6)(5,28,30,17)(7,29,32,18)(8,11,33,21)(9,24,34,14)(10,26,20,15)(12,23,22,13)(16,25,27,19)(35,49,50,58)(36,60,51,42)(38,61,53,43)(39,55,54,46)(40,59,63,45)(41,62,64,44)(47,57,56,48), (1,47,3,56)(2,22,4,12)(5,21,30,11)(6,13,31,23)(7,9,32,34)(8,28,33,17)(10,16,20,27)(14,18,24,29)(15,25,26,19)(35,53,50,38)(36,46,51,55)(37,57,52,48)(39,60,54,42)(40,62,63,44)(41,45,64,59)(43,49,61,58), (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)(40,41,42,43,44)(45,46,47,48,49)(50,51,52,53,54)(55,56,57,58,59)(60,61,62,63,64), (1,4,3,2)(5,54,30,39)(6,53,31,38)(7,52,32,37)(8,51,33,36)(9,50,34,35)(10,46,20,55)(11,45,21,59)(12,49,22,58)(13,48,23,57)(14,47,24,56)(15,41,26,64)(16,40,27,63)(17,44,28,62)(18,43,29,61)(19,42,25,60) );

G=PermutationGroup([[(1,59,3,45),(2,10,4,20),(5,32,30,7),(6,15,31,26),(8,14,33,24),(9,21,34,11),(12,27,22,16),(13,25,23,19),(17,29,28,18),(35,46,50,55),(36,53,51,38),(37,40,52,63),(39,58,54,49),(41,56,64,47),(42,61,60,43),(44,57,62,48)], [(1,42),(2,17),(3,60),(4,28),(5,31),(6,30),(7,26),(8,22),(9,19),(10,18),(11,13),(12,33),(14,27),(15,32),(16,24),(20,29),(21,23),(25,34),(35,44),(36,52),(37,51),(38,63),(39,47),(40,53),(41,49),(43,59),(45,61),(46,48),(50,62),(54,56),(55,57),(58,64)], [(1,52,3,37),(2,31,4,6),(5,28,30,17),(7,29,32,18),(8,11,33,21),(9,24,34,14),(10,26,20,15),(12,23,22,13),(16,25,27,19),(35,49,50,58),(36,60,51,42),(38,61,53,43),(39,55,54,46),(40,59,63,45),(41,62,64,44),(47,57,56,48)], [(1,47,3,56),(2,22,4,12),(5,21,30,11),(6,13,31,23),(7,9,32,34),(8,28,33,17),(10,16,20,27),(14,18,24,29),(15,25,26,19),(35,53,50,38),(36,46,51,55),(37,57,52,48),(39,60,54,42),(40,62,63,44),(41,45,64,59),(43,49,61,58)], [(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),(40,41,42,43,44),(45,46,47,48,49),(50,51,52,53,54),(55,56,57,58,59),(60,61,62,63,64)], [(1,4,3,2),(5,54,30,39),(6,53,31,38),(7,52,32,37),(8,51,33,36),(9,50,34,35),(10,46,20,55),(11,45,21,59),(12,49,22,58),(13,48,23,57),(14,47,24,56),(15,41,26,64),(16,40,27,63),(17,44,28,62),(18,43,29,61),(19,42,25,60)]])

Matrix representation of 2- 1+4.D5 in GL4(F3) generated by

2112
0020
0100
2011
,
1202
0002
1022
0200
,
1010
2012
1020
0110
,
2212
0010
0200
2011
,
1122
1212
1001
1102
,
2202
1120
0122
1011
G:=sub<GL(4,GF(3))| [2,0,0,2,1,0,1,0,1,2,0,1,2,0,0,1],[1,0,1,0,2,0,0,2,0,0,2,0,2,2,2,0],[1,2,1,0,0,0,0,1,1,1,2,1,0,2,0,0],[2,0,0,2,2,0,2,0,1,1,0,1,2,0,0,1],[1,1,1,1,1,2,0,1,2,1,0,0,2,2,1,2],[2,1,0,1,2,1,1,0,0,2,2,1,2,0,2,1] >;

2- 1+4.D5 in GAP, Magma, Sage, TeX

2_-^{1+4}.D_5
% in TeX

G:=Group("ES-(2,2).D5");
// GroupNames label

G:=SmallGroup(320,1581);
// by ID

G=gap.SmallGroup(320,1581);
# by ID

G:=PCGroup([7,-2,-5,-2,2,2,2,-2,1120,113,632,324,5043,850,521,248,3854,2111,718,375,172,2105,3582,1027]);
// Polycyclic

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

Export

Subgroup lattice of 2- 1+4.D5 in TeX
Character table of 2- 1+4.D5 in TeX

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