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G = D20.14D4order 320 = 26·5

14th non-split extension by D20 of D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D20.14D4, C42.66D10, Dic10.14D4, C4.52(D4×D5), C4.4D45D5, (C2×C20).11D4, C20.29(C2×D4), (C2×D4).52D10, C53(D4.8D4), (C2×Q8).41D10, D204C412C2, C10.52C22≀C2, D4.D102C2, C20.10D45C2, (C4×C20).110C22, (C2×C20).380C23, Q8.10D102C2, C4○D20.20C22, (D4×C10).68C22, (Q8×C10).59C22, C2.20(C23⋊D10), C4.Dic5.14C22, (C5×C4.4D4)⋊5C2, (C2×C10).511(C2×D4), (C2×C4).10(C5⋊D4), C22.32(C2×C5⋊D4), (C2×C4).117(C22×D5), SmallGroup(320,689)

Series: Derived Chief Lower central Upper central

C1C2×C20 — D20.14D4
C1C5C10C20C2×C20C4○D20Q8.10D10 — D20.14D4
C5C10C2×C20 — D20.14D4
C1C2C2×C4C4.4D4

Generators and relations for D20.14D4
 G = < a,b,c,d | a20=b2=d2=1, c4=a10, bab=cac-1=a-1, dad=a9, cbc-1=a3b, dbd=a18b, dcd=a10c3 >

Subgroups: 590 in 146 conjugacy classes, 39 normal (19 characteristic)
C1, C2, C2 [×4], C4 [×2], C4 [×5], C22, C22 [×5], C5, C8 [×2], C2×C4, C2×C4 [×2], C2×C4 [×7], D4 [×8], Q8 [×6], C23, D5 [×2], C10, C10 [×2], C42, C22⋊C4 [×2], M4(2) [×2], D8 [×2], SD16 [×2], C2×D4, C2×Q8, C2×Q8 [×2], C4○D4 [×6], Dic5 [×2], C20 [×2], C20 [×3], D10 [×2], C2×C10, C2×C10 [×3], C4.10D4, C4≀C2 [×2], C4.4D4, C8⋊C22 [×2], 2- 1+4, C52C8 [×2], Dic10 [×2], Dic10 [×2], C4×D5 [×6], D20 [×2], D20 [×2], C5⋊D4 [×2], C2×C20, C2×C20 [×2], C2×C20, C5×D4 [×2], C5×Q8 [×2], C22×C10, D4.8D4, C4.Dic5 [×2], D4⋊D5 [×2], D4.D5 [×2], C4×C20, C5×C22⋊C4 [×2], C4○D20 [×2], C4○D20 [×2], Q8×D5 [×2], Q82D5 [×2], D4×C10, Q8×C10, D204C4 [×2], C20.10D4, D4.D10 [×2], C5×C4.4D4, Q8.10D10, D20.14D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, D5, C2×D4 [×3], D10 [×3], C22≀C2, C5⋊D4 [×2], C22×D5, D4.8D4, D4×D5 [×2], C2×C5⋊D4, C23⋊D10, D20.14D4

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

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

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

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

44 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G4H5A5B8A8B10A···10F10G10H10I10J20A···20L20M20N20O20P
order12222244444444558810···101010101020···2020202020
size1128202022444420202240402···288884···48888

44 irreducible representations

dim11111122222222444
type++++++++++++++
imageC1C2C2C2C2C2D4D4D4D5D10D10D10C5⋊D4D4.8D4D4×D5D20.14D4
kernelD20.14D4D204C4C20.10D4D4.D10C5×C4.4D4Q8.10D10Dic10D20C2×C20C4.4D4C42C2×D4C2×Q8C2×C4C5C4C1
# reps12121122222228248

Matrix representation of D20.14D4 in GL6(𝔽41)

010000
40340000
0004000
001000
00202014
003102040
,
30320000
27110000
0000320
002525325
009000
00631816
,
14300000
14270000
00202014
0000400
001000
003313021
,
27110000
27140000
0021214037
0000400
0004000
002881020

G:=sub<GL(6,GF(41))| [0,40,0,0,0,0,1,34,0,0,0,0,0,0,0,1,20,31,0,0,40,0,20,0,0,0,0,0,1,20,0,0,0,0,4,40],[30,27,0,0,0,0,32,11,0,0,0,0,0,0,0,25,9,6,0,0,0,25,0,31,0,0,32,32,0,8,0,0,0,5,0,16],[14,14,0,0,0,0,30,27,0,0,0,0,0,0,20,0,1,33,0,0,20,0,0,13,0,0,1,40,0,0,0,0,4,0,0,21],[27,27,0,0,0,0,11,14,0,0,0,0,0,0,21,0,0,28,0,0,21,0,40,8,0,0,40,40,0,10,0,0,37,0,0,20] >;

D20.14D4 in GAP, Magma, Sage, TeX

D_{20}._{14}D_4
% in TeX

G:=Group("D20.14D4");
// GroupNames label

G:=SmallGroup(320,689);
// by ID

G=gap.SmallGroup(320,689);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,254,219,1123,570,297,136,1684,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^20=b^2=d^2=1,c^4=a^10,b*a*b=c*a*c^-1=a^-1,d*a*d=a^9,c*b*c^-1=a^3*b,d*b*d=a^18*b,d*c*d=a^10*c^3>;
// generators/relations

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