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G = D206C4order 160 = 25·5

3rd semidirect product of D20 and C4 acting via C4/C2=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D206C4, C4.9D20, C10.7D8, C20.1D4, C10.7SD16, C4⋊C41D5, C4.1(C4×D5), C52(D4⋊C4), C20.24(C2×C4), C2.2(D4⋊D5), (C2×D20).6C2, (C2×C4).35D10, (C2×C10).30D4, C2.2(Q8⋊D5), (C2×C20).10C22, C2.5(D10⋊C4), C10.14(C22⋊C4), C22.14(C5⋊D4), (C5×C4⋊C4)⋊1C2, (C2×C52C8)⋊1C2, SmallGroup(160,16)

Series: Derived Chief Lower central Upper central

C1C20 — D206C4
C1C5C10C2×C10C2×C20C2×D20 — D206C4
C5C10C20 — D206C4
C1C22C2×C4C4⋊C4

Generators and relations for D206C4
 G = < a,b,c | a20=b2=c4=1, bab=a-1, cac-1=a11, cbc-1=a15b >

20C2
20C2
4C4
10C22
10C22
20C22
20C22
4D5
4D5
2C2×C4
5D4
5D4
10C8
10D4
10C23
2D10
2D10
4D10
4D10
4C20
5C2×C8
5C2×D4
2C52C8
2C2×C20
2C22×D5
2D20
5D4⋊C4

Smallest permutation representation of D206C4
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 23)(2 22)(3 21)(4 40)(5 39)(6 38)(7 37)(8 36)(9 35)(10 34)(11 33)(12 32)(13 31)(14 30)(15 29)(16 28)(17 27)(18 26)(19 25)(20 24)(41 64)(42 63)(43 62)(44 61)(45 80)(46 79)(47 78)(48 77)(49 76)(50 75)(51 74)(52 73)(53 72)(54 71)(55 70)(56 69)(57 68)(58 67)(59 66)(60 65)
(1 66 24 55)(2 77 25 46)(3 68 26 57)(4 79 27 48)(5 70 28 59)(6 61 29 50)(7 72 30 41)(8 63 31 52)(9 74 32 43)(10 65 33 54)(11 76 34 45)(12 67 35 56)(13 78 36 47)(14 69 37 58)(15 80 38 49)(16 71 39 60)(17 62 40 51)(18 73 21 42)(19 64 22 53)(20 75 23 44)

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

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

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

D206C4 is a maximal subgroup of
Dic5.5D8  Dic102D4  C4⋊C4.D10  D5×D4⋊C4  (D4×D5)⋊C4  D4⋊D20  D20.8D4  D203D4  Q8⋊C4⋊D5  Dic10.11D4  Q8⋊Dic5⋊C2  Q8⋊(C4×D5)  Q82D5⋊C4  Q8.D20  D204D4  D20.12D4  Dic58SD16  D10.17SD16  C88D20  C82D20  C4.Q8⋊D5  D4015C4  D20⋊Q8  D20.Q8  D4012C4  D10.13D8  C87D20  C2.D8⋊D5  C83D20  C4021(C2×C4)  D202Q8  D20.2Q8  C4○D209C4  (C2×C10).40D8  C4⋊C4.228D10  C4⋊C436D10  C4○D2010C4  C4⋊C4.236D10  C4×D4⋊D5  C42.48D10  C207D8  D4.1D20  C4×Q8⋊D5  C42.56D10  Q8⋊D20  Q8.1D20  D2016D4  D2017D4  (C2×C10)⋊D8  C4⋊D4⋊D5  D20.36D4  D20.37D4  C52C824D4  C22⋊Q8⋊D5  D20.4Q8  C42.70D10  C42.216D10  D205Q8  D206Q8  C20.D8  C42.82D10  C30.D8  D6015C4  D609C4
D206C4 is a maximal quotient of
(C2×D20)⋊C4  C20.47D8  D204C8  C4.D40  D4014C4  C40.5D4  C10.Q32  D40.6C4  C40.8D4  D40.5C4  C20.31C42  C30.D8  D6015C4  D609C4

34 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D5A5B8A8B8C8D10A···10F20A···20L
order122222444455888810···1020···20
size11112020224422101010102···24···4

34 irreducible representations

dim1111122222222244
type++++++++++++
imageC1C2C2C2C4D4D4D5D8SD16D10C4×D5D20C5⋊D4D4⋊D5Q8⋊D5
kernelD206C4C2×C52C8C5×C4⋊C4C2×D20D20C20C2×C10C4⋊C4C10C10C2×C4C4C4C22C2C2
# reps1111411222244422

Matrix representation of D206C4 in GL4(𝔽41) generated by

35100
54000
00139
00140
,
1100
04000
00402
0001
,
9000
0900
003011
001511
G:=sub<GL(4,GF(41))| [35,5,0,0,1,40,0,0,0,0,1,1,0,0,39,40],[1,0,0,0,1,40,0,0,0,0,40,0,0,0,2,1],[9,0,0,0,0,9,0,0,0,0,30,15,0,0,11,11] >;

D206C4 in GAP, Magma, Sage, TeX

D_{20}\rtimes_6C_4
% in TeX

G:=Group("D20:6C4");
// GroupNames label

G:=SmallGroup(160,16);
// by ID

G=gap.SmallGroup(160,16);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,121,31,579,297,69,4613]);
// Polycyclic

G:=Group<a,b,c|a^20=b^2=c^4=1,b*a*b=a^-1,c*a*c^-1=a^11,c*b*c^-1=a^15*b>;
// generators/relations

Export

Subgroup lattice of D206C4 in TeX

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