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G = 2+ 1+4.D5order 320 = 26·5

1st non-split extension by 2+ 1+4 of D5 acting via D5/C5=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: 2+ 1+4.1D5, (C5×D4).33D4, (C5×Q8).33D4, C4○D4.10D10, (C2×D4).83D10, C20.218(C2×D4), C55(D4.9D4), C10.81C22≀C2, D4.9D105C2, C20.17D48C2, C20.D412C2, D4.15(C5⋊D4), (C2×C20).22C23, Q8.15(C5⋊D4), (C4×Dic5)⋊9C22, (C22×C10).25D4, D42Dic512C2, C23.13(C5⋊D4), C4.Dic511C22, (C2×Dic10)⋊16C22, (D4×C10).108C22, C2.15(C242D5), (C5×2+ 1+4).1C2, C4.65(C2×C5⋊D4), (C2×C10).43(C2×D4), (C2×C4).22(C22×D5), C22.15(C2×C5⋊D4), (C5×C4○D4).20C22, SmallGroup(320,869)

Series: Derived Chief Lower central Upper central

C1C2×C20 — 2+ 1+4.D5
C1C5C10C2×C10C2×C20C2×Dic10D4.9D10 — 2+ 1+4.D5
C5C10C2×C20 — 2+ 1+4.D5
C1C2C2×C42+ 1+4

Generators and relations for 2+ 1+4.D5
 G = < a,b,c,d,e,f | a4=b2=d2=e5=1, c2=f2=a2, bab=faf-1=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, fbf-1=ab, dcd=fcf-1=a2c, ce=ec, de=ed, fdf-1=cd, fef-1=e-1 >

Subgroups: 478 in 152 conjugacy classes, 43 normal (17 characteristic)
C1, C2, C2 [×5], C4 [×2], C4 [×4], C22, C22 [×8], C5, C8 [×2], C2×C4, C2×C4 [×6], D4 [×2], D4 [×8], Q8 [×2], Q8 [×2], C23 [×2], C23 [×2], C10, C10 [×5], C42, C22⋊C4 [×2], M4(2) [×2], SD16 [×2], Q16 [×2], C2×D4, C2×D4 [×4], C2×Q8, C4○D4 [×2], C4○D4 [×2], Dic5 [×2], C20 [×2], C20 [×2], C2×C10, C2×C10 [×8], C4.D4, C4≀C2 [×2], C4.4D4, C8.C22 [×2], 2+ 1+4, C52C8 [×2], Dic10 [×2], C2×Dic5 [×2], C2×C20, C2×C20 [×4], C5×D4 [×2], C5×D4 [×8], C5×Q8 [×2], C22×C10 [×2], C22×C10 [×2], D4.9D4, C4.Dic5 [×2], C4×Dic5, D4.D5 [×2], C5⋊Q16 [×2], C23.D5 [×2], C2×Dic10, D4×C10, D4×C10 [×4], C5×C4○D4 [×2], C5×C4○D4 [×2], C20.D4, D42Dic5 [×2], C20.17D4, D4.9D10 [×2], C5×2+ 1+4, 2+ 1+4.D5
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, D5, C2×D4 [×3], D10 [×3], C22≀C2, C5⋊D4 [×6], C22×D5, D4.9D4, C2×C5⋊D4 [×3], C242D5, 2+ 1+4.D5

Smallest permutation representation of 2+ 1+4.D5
On 80 points
Generators in S80
(1 19 9 14)(2 20 10 15)(3 16 6 11)(4 17 7 12)(5 18 8 13)(21 31 26 36)(22 32 27 37)(23 33 28 38)(24 34 29 39)(25 35 30 40)(41 56 46 51)(42 57 47 52)(43 58 48 53)(44 59 49 54)(45 60 50 55)(61 71 66 76)(62 72 67 77)(63 73 68 78)(64 74 69 79)(65 75 70 80)
(1 79)(2 80)(3 76)(4 77)(5 78)(6 71)(7 72)(8 73)(9 74)(10 75)(11 61)(12 62)(13 63)(14 64)(15 65)(16 66)(17 67)(18 68)(19 69)(20 70)(21 51)(22 52)(23 53)(24 54)(25 55)(26 56)(27 57)(28 58)(29 59)(30 60)(31 46)(32 47)(33 48)(34 49)(35 50)(36 41)(37 42)(38 43)(39 44)(40 45)
(1 19 9 14)(2 20 10 15)(3 16 6 11)(4 17 7 12)(5 18 8 13)(21 31 26 36)(22 32 27 37)(23 33 28 38)(24 34 29 39)(25 35 30 40)(41 51 46 56)(42 52 47 57)(43 53 48 58)(44 54 49 59)(45 55 50 60)(61 76 66 71)(62 77 67 72)(63 78 68 73)(64 79 69 74)(65 80 70 75)
(1 59)(2 60)(3 56)(4 57)(5 58)(6 51)(7 52)(8 53)(9 54)(10 55)(11 41)(12 42)(13 43)(14 44)(15 45)(16 46)(17 47)(18 48)(19 49)(20 50)(21 71)(22 72)(23 73)(24 74)(25 75)(26 76)(27 77)(28 78)(29 79)(30 80)(31 66)(32 67)(33 68)(34 69)(35 70)(36 61)(37 62)(38 63)(39 64)(40 65)
(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)(41 42 43 44 45)(46 47 48 49 50)(51 52 53 54 55)(56 57 58 59 60)(61 62 63 64 65)(66 67 68 69 70)(71 72 73 74 75)(76 77 78 79 80)
(1 23 9 28)(2 22 10 27)(3 21 6 26)(4 25 7 30)(5 24 8 29)(11 31 16 36)(12 35 17 40)(13 34 18 39)(14 33 19 38)(15 32 20 37)(41 76 46 71)(42 80 47 75)(43 79 48 74)(44 78 49 73)(45 77 50 72)(51 61 56 66)(52 65 57 70)(53 64 58 69)(54 63 59 68)(55 62 60 67)

G:=sub<Sym(80)| (1,19,9,14)(2,20,10,15)(3,16,6,11)(4,17,7,12)(5,18,8,13)(21,31,26,36)(22,32,27,37)(23,33,28,38)(24,34,29,39)(25,35,30,40)(41,56,46,51)(42,57,47,52)(43,58,48,53)(44,59,49,54)(45,60,50,55)(61,71,66,76)(62,72,67,77)(63,73,68,78)(64,74,69,79)(65,75,70,80), (1,79)(2,80)(3,76)(4,77)(5,78)(6,71)(7,72)(8,73)(9,74)(10,75)(11,61)(12,62)(13,63)(14,64)(15,65)(16,66)(17,67)(18,68)(19,69)(20,70)(21,51)(22,52)(23,53)(24,54)(25,55)(26,56)(27,57)(28,58)(29,59)(30,60)(31,46)(32,47)(33,48)(34,49)(35,50)(36,41)(37,42)(38,43)(39,44)(40,45), (1,19,9,14)(2,20,10,15)(3,16,6,11)(4,17,7,12)(5,18,8,13)(21,31,26,36)(22,32,27,37)(23,33,28,38)(24,34,29,39)(25,35,30,40)(41,51,46,56)(42,52,47,57)(43,53,48,58)(44,54,49,59)(45,55,50,60)(61,76,66,71)(62,77,67,72)(63,78,68,73)(64,79,69,74)(65,80,70,75), (1,59)(2,60)(3,56)(4,57)(5,58)(6,51)(7,52)(8,53)(9,54)(10,55)(11,41)(12,42)(13,43)(14,44)(15,45)(16,46)(17,47)(18,48)(19,49)(20,50)(21,71)(22,72)(23,73)(24,74)(25,75)(26,76)(27,77)(28,78)(29,79)(30,80)(31,66)(32,67)(33,68)(34,69)(35,70)(36,61)(37,62)(38,63)(39,64)(40,65), (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)(41,42,43,44,45)(46,47,48,49,50)(51,52,53,54,55)(56,57,58,59,60)(61,62,63,64,65)(66,67,68,69,70)(71,72,73,74,75)(76,77,78,79,80), (1,23,9,28)(2,22,10,27)(3,21,6,26)(4,25,7,30)(5,24,8,29)(11,31,16,36)(12,35,17,40)(13,34,18,39)(14,33,19,38)(15,32,20,37)(41,76,46,71)(42,80,47,75)(43,79,48,74)(44,78,49,73)(45,77,50,72)(51,61,56,66)(52,65,57,70)(53,64,58,69)(54,63,59,68)(55,62,60,67)>;

G:=Group( (1,19,9,14)(2,20,10,15)(3,16,6,11)(4,17,7,12)(5,18,8,13)(21,31,26,36)(22,32,27,37)(23,33,28,38)(24,34,29,39)(25,35,30,40)(41,56,46,51)(42,57,47,52)(43,58,48,53)(44,59,49,54)(45,60,50,55)(61,71,66,76)(62,72,67,77)(63,73,68,78)(64,74,69,79)(65,75,70,80), (1,79)(2,80)(3,76)(4,77)(5,78)(6,71)(7,72)(8,73)(9,74)(10,75)(11,61)(12,62)(13,63)(14,64)(15,65)(16,66)(17,67)(18,68)(19,69)(20,70)(21,51)(22,52)(23,53)(24,54)(25,55)(26,56)(27,57)(28,58)(29,59)(30,60)(31,46)(32,47)(33,48)(34,49)(35,50)(36,41)(37,42)(38,43)(39,44)(40,45), (1,19,9,14)(2,20,10,15)(3,16,6,11)(4,17,7,12)(5,18,8,13)(21,31,26,36)(22,32,27,37)(23,33,28,38)(24,34,29,39)(25,35,30,40)(41,51,46,56)(42,52,47,57)(43,53,48,58)(44,54,49,59)(45,55,50,60)(61,76,66,71)(62,77,67,72)(63,78,68,73)(64,79,69,74)(65,80,70,75), (1,59)(2,60)(3,56)(4,57)(5,58)(6,51)(7,52)(8,53)(9,54)(10,55)(11,41)(12,42)(13,43)(14,44)(15,45)(16,46)(17,47)(18,48)(19,49)(20,50)(21,71)(22,72)(23,73)(24,74)(25,75)(26,76)(27,77)(28,78)(29,79)(30,80)(31,66)(32,67)(33,68)(34,69)(35,70)(36,61)(37,62)(38,63)(39,64)(40,65), (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)(41,42,43,44,45)(46,47,48,49,50)(51,52,53,54,55)(56,57,58,59,60)(61,62,63,64,65)(66,67,68,69,70)(71,72,73,74,75)(76,77,78,79,80), (1,23,9,28)(2,22,10,27)(3,21,6,26)(4,25,7,30)(5,24,8,29)(11,31,16,36)(12,35,17,40)(13,34,18,39)(14,33,19,38)(15,32,20,37)(41,76,46,71)(42,80,47,75)(43,79,48,74)(44,78,49,73)(45,77,50,72)(51,61,56,66)(52,65,57,70)(53,64,58,69)(54,63,59,68)(55,62,60,67) );

G=PermutationGroup([(1,19,9,14),(2,20,10,15),(3,16,6,11),(4,17,7,12),(5,18,8,13),(21,31,26,36),(22,32,27,37),(23,33,28,38),(24,34,29,39),(25,35,30,40),(41,56,46,51),(42,57,47,52),(43,58,48,53),(44,59,49,54),(45,60,50,55),(61,71,66,76),(62,72,67,77),(63,73,68,78),(64,74,69,79),(65,75,70,80)], [(1,79),(2,80),(3,76),(4,77),(5,78),(6,71),(7,72),(8,73),(9,74),(10,75),(11,61),(12,62),(13,63),(14,64),(15,65),(16,66),(17,67),(18,68),(19,69),(20,70),(21,51),(22,52),(23,53),(24,54),(25,55),(26,56),(27,57),(28,58),(29,59),(30,60),(31,46),(32,47),(33,48),(34,49),(35,50),(36,41),(37,42),(38,43),(39,44),(40,45)], [(1,19,9,14),(2,20,10,15),(3,16,6,11),(4,17,7,12),(5,18,8,13),(21,31,26,36),(22,32,27,37),(23,33,28,38),(24,34,29,39),(25,35,30,40),(41,51,46,56),(42,52,47,57),(43,53,48,58),(44,54,49,59),(45,55,50,60),(61,76,66,71),(62,77,67,72),(63,78,68,73),(64,79,69,74),(65,80,70,75)], [(1,59),(2,60),(3,56),(4,57),(5,58),(6,51),(7,52),(8,53),(9,54),(10,55),(11,41),(12,42),(13,43),(14,44),(15,45),(16,46),(17,47),(18,48),(19,49),(20,50),(21,71),(22,72),(23,73),(24,74),(25,75),(26,76),(27,77),(28,78),(29,79),(30,80),(31,66),(32,67),(33,68),(34,69),(35,70),(36,61),(37,62),(38,63),(39,64),(40,65)], [(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),(41,42,43,44,45),(46,47,48,49,50),(51,52,53,54,55),(56,57,58,59,60),(61,62,63,64,65),(66,67,68,69,70),(71,72,73,74,75),(76,77,78,79,80)], [(1,23,9,28),(2,22,10,27),(3,21,6,26),(4,25,7,30),(5,24,8,29),(11,31,16,36),(12,35,17,40),(13,34,18,39),(14,33,19,38),(15,32,20,37),(41,76,46,71),(42,80,47,75),(43,79,48,74),(44,78,49,73),(45,77,50,72),(51,61,56,66),(52,65,57,70),(53,64,58,69),(54,63,59,68),(55,62,60,67)])

50 conjugacy classes

class 1 2A2B2C2D2E2F4A4B4C4D4E4F4G5A5B8A8B10A10B10C···10T20A···20L
order122222244444445588101010···1020···20
size11244442244202040224040224···44···4

50 irreducible representations

dim11111122222222248
type++++++++++++-
imageC1C2C2C2C2C2D4D4D4D5D10D10C5⋊D4C5⋊D4C5⋊D4D4.9D42+ 1+4.D5
kernel2+ 1+4.D5C20.D4D42Dic5C20.17D4D4.9D10C5×2+ 1+4C5×D4C5×Q8C22×C102+ 1+4C2×D4C4○D4D4Q8C23C5C1
# reps11212122222488822

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

4000000
0400000
00323200
000900
00003232
000009
,
0400000
4000000
00003232
0000189
00323200
0018900
,
100000
010000
00323200
000900
000099
0000032
,
4000000
0400000
000099
0000032
00323200
000900
,
17400000
40170000
001000
000100
000010
000001
,
4000000
010000
001100
00394000
0000320
0000189

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,32,0,0,0,0,0,32,9,0,0,0,0,0,0,32,0,0,0,0,0,32,9],[0,40,0,0,0,0,40,0,0,0,0,0,0,0,0,0,32,18,0,0,0,0,32,9,0,0,32,18,0,0,0,0,32,9,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,32,0,0,0,0,0,32,9,0,0,0,0,0,0,9,0,0,0,0,0,9,32],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,32,0,0,0,0,0,32,9,0,0,9,0,0,0,0,0,9,32,0,0],[17,40,0,0,0,0,40,17,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,39,0,0,0,0,1,40,0,0,0,0,0,0,32,18,0,0,0,0,0,9] >;

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(320,869);
// by ID

G=gap.SmallGroup(320,869);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,253,254,570,1684,851,438,102,12550]);
// Polycyclic

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

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