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G = D4.4D20order 320 = 26·5

4th non-split extension by D4 of D20 acting via D20/C20=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D4.4D20, C40.83D4, Q8.4D20, M4(2).34D10, C8○D42D5, (C2×D40)⋊13C2, (C5×D4).21D4, C20.43(C2×D4), C4.20(C2×D20), (C2×C8).81D10, (C5×Q8).21D4, D4⋊D103C2, C4○D4.32D10, C53(D4.4D4), C8.40(C5⋊D4), C40.6C415C2, (C2×C40).67C22, C20.46D414C2, C10.77(C4⋊D4), C2.25(C207D4), (C2×C20).422C23, C22.9(C4○D20), (C2×D20).116C22, C4.Dic5.17C22, (C5×M4(2)).37C22, (C5×C8○D4)⋊2C2, C4.118(C2×C5⋊D4), (C2×C10).7(C4○D4), (C5×C4○D4).37C22, (C2×C4).124(C22×D5), SmallGroup(320,769)

Series: Derived Chief Lower central Upper central

C1C2×C20 — D4.4D20
C1C5C10C20C2×C20C2×D20C2×D40 — D4.4D20
C5C10C2×C20 — D4.4D20
C1C2C2×C4C8○D4

Generators and relations for D4.4D20
 G = < a,b,c,d | a4=b2=d2=1, c20=a2, bab=dad=a-1, ac=ca, bc=cb, dbd=ab, dcd=a2c19 >

Subgroups: 542 in 108 conjugacy classes, 39 normal (31 characteristic)
C1, C2, C2 [×4], C4 [×2], C4, C22, C22 [×5], C5, C8 [×2], C8 [×3], C2×C4, C2×C4, D4, D4 [×5], Q8, C23 [×2], D5 [×2], C10, C10 [×2], C2×C8, C2×C8, M4(2), M4(2) [×3], D8 [×4], SD16 [×2], C2×D4 [×2], C4○D4, C20 [×2], C20, D10 [×4], C2×C10, C2×C10, C4.D4 [×2], C8.C4, C8○D4, C2×D8, C8⋊C22 [×2], C52C8 [×2], C40 [×2], C40, D20 [×4], C2×C20, C2×C20, C5×D4, C5×D4, C5×Q8, C22×D5 [×2], D4.4D4, D40 [×2], C4.Dic5 [×2], D4⋊D5 [×2], Q8⋊D5 [×2], C2×C40, C2×C40, C5×M4(2), C5×M4(2), C2×D20 [×2], C5×C4○D4, C40.6C4, C20.46D4 [×2], C2×D40, D4⋊D10 [×2], C5×C8○D4, D4.4D20
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, D5, C2×D4 [×2], C4○D4, D10 [×3], C4⋊D4, D20 [×2], C5⋊D4 [×2], C22×D5, D4.4D4, C2×D20, C4○D20, C2×C5⋊D4, C207D4, D4.4D20

Smallest permutation representation of D4.4D20
On 80 points
Generators in S80
(1 11 21 31)(2 12 22 32)(3 13 23 33)(4 14 24 34)(5 15 25 35)(6 16 26 36)(7 17 27 37)(8 18 28 38)(9 19 29 39)(10 20 30 40)(41 71 61 51)(42 72 62 52)(43 73 63 53)(44 74 64 54)(45 75 65 55)(46 76 66 56)(47 77 67 57)(48 78 68 58)(49 79 69 59)(50 80 70 60)
(1 66)(2 67)(3 68)(4 69)(5 70)(6 71)(7 72)(8 73)(9 74)(10 75)(11 76)(12 77)(13 78)(14 79)(15 80)(16 41)(17 42)(18 43)(19 44)(20 45)(21 46)(22 47)(23 48)(24 49)(25 50)(26 51)(27 52)(28 53)(29 54)(30 55)(31 56)(32 57)(33 58)(34 59)(35 60)(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 40)(2 39)(3 38)(4 37)(5 36)(6 35)(7 34)(8 33)(9 32)(10 31)(11 30)(12 29)(13 28)(14 27)(15 26)(16 25)(17 24)(18 23)(19 22)(20 21)(41 60)(42 59)(43 58)(44 57)(45 56)(46 55)(47 54)(48 53)(49 52)(50 51)(61 80)(62 79)(63 78)(64 77)(65 76)(66 75)(67 74)(68 73)(69 72)(70 71)

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

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

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

56 conjugacy classes

class 1 2A2B2C2D2E4A4B4C5A5B8A8B8C8D8E8F8G10A10B10C···10H20A20B20C20D20E···20J40A···40H40I···40T
order122222444558888888101010···102020202020···2040···4040···40
size1124404022422224444040224···422224···42···24···4

56 irreducible representations

dim11111122222222222244
type+++++++++++++++++
imageC1C2C2C2C2C2D4D4D4D5C4○D4D10D10D10C5⋊D4D20D20C4○D20D4.4D4D4.4D20
kernelD4.4D20C40.6C4C20.46D4C2×D40D4⋊D10C5×C8○D4C40C5×D4C5×Q8C8○D4C2×C10C2×C8M4(2)C4○D4C8D4Q8C22C5C1
# reps11212121122222844828

Matrix representation of D4.4D20 in GL6(𝔽41)

4000000
0400000
0004000
001000
00114037
00020211
,
4000000
010000
00114037
0000400
0004000
000212040
,
2300000
0250000
00122900
00121200
002929247
0060350
,
010000
100000
00121200
00122900
002929247
001263517

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,1,1,0,0,0,40,0,1,20,0,0,0,0,40,21,0,0,0,0,37,1],[40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,40,21,0,0,40,40,0,20,0,0,37,0,0,40],[23,0,0,0,0,0,0,25,0,0,0,0,0,0,12,12,29,6,0,0,29,12,29,0,0,0,0,0,24,35,0,0,0,0,7,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,12,12,29,12,0,0,12,29,29,6,0,0,0,0,24,35,0,0,0,0,7,17] >;

D4.4D20 in GAP, Magma, Sage, TeX

D_4._4D_{20}
% in TeX

G:=Group("D4.4D20");
// GroupNames label

G:=SmallGroup(320,769);
// by ID

G=gap.SmallGroup(320,769);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,344,254,1123,297,136,1684,102,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^2=d^2=1,c^20=a^2,b*a*b=d*a*d=a^-1,a*c=c*a,b*c=c*b,d*b*d=a*b,d*c*d=a^2*c^19>;
// generators/relations

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