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G = 2- (1+4).D5  order 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

Derived series
C1 — C2 — 2- (1+4) — 2- (1+4)⋊C5 — 2- (1+4).D5
Chief series
C1 — C2 — 2- (1+4) — 2- (1+4)⋊C5 — 2- (1+4).D5

Subgroups: 343 in 48 conjugacy classes, 5 normal (all characteristic)
C1, C2, C2, C4 [×5], C22, C5, C8 [×2], C2×C4 [×5], D4 [×2], Q8 [×4], C10, C42, C4⋊C4 [×2], M4(2) [×2], SD16 [×2], Q16 [×2], C2×Q8 [×2], C4○D4 [×2], Dic5, C4.10D4, C4≀C2 [×2], C4⋊Q8, C8.C22 [×2], 2- (1+4), D4.10D4, 2- (1+4)⋊C5, 2- (1+4).D5

Quotients:
C1, C2, D5, C24⋊D5, 2- (1+4).D5

Generators and relations
 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 >

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

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

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

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

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

Matrix representation ►G ⊆ 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] >;
 

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

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
 

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