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G = D208C4order 160 = 25·5

5th semidirect product of D20 and C4 acting via C4/C2=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D208C4, Dic55D4, C54(C4×D4), C4⋊C48D5, C41(C4×D5), C205(C2×C4), C2.4(D4×D5), D104(C2×C4), (C4×Dic5)⋊3C2, (C2×D20).8C2, (C2×C4).31D10, C10.24(C2×D4), D10⋊C412C2, C10.33(C4○D4), (C2×C10).34C23, (C2×C20).24C22, C10.24(C22×C4), C2.2(Q82D5), C22.18(C22×D5), (C2×Dic5).63C22, (C22×D5).23C22, (C5×C4⋊C4)⋊4C2, (C2×C4×D5)⋊12C2, C2.13(C2×C4×D5), SmallGroup(160,114)

Series: Derived Chief Lower central Upper central

C1C10 — D208C4
C1C5C10C2×C10C22×D5C2×D20 — D208C4
C5C10 — D208C4
C1C22C4⋊C4

Generators and relations for D208C4
 G = < a,b,c | a20=b2=c4=1, bab=a-1, cac-1=a11, cbc-1=a10b >

Subgroups: 312 in 94 conjugacy classes, 43 normal (19 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×5], C22, C22 [×8], C5, C2×C4, C2×C4 [×2], C2×C4 [×6], D4 [×4], C23 [×2], D5 [×4], C10 [×3], C42, C22⋊C4 [×2], C4⋊C4, C22×C4 [×2], C2×D4, Dic5 [×2], Dic5, C20 [×2], C20 [×2], D10 [×4], D10 [×4], C2×C10, C4×D4, C4×D5 [×4], D20 [×4], C2×Dic5 [×2], C2×C20, C2×C20 [×2], C22×D5 [×2], C4×Dic5, D10⋊C4 [×2], C5×C4⋊C4, C2×C4×D5 [×2], C2×D20, D208C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], C23, D5, C22×C4, C2×D4, C4○D4, D10 [×3], C4×D4, C4×D5 [×2], C22×D5, C2×C4×D5, D4×D5, Q82D5, D208C4

Smallest permutation representation of D208C4
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 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 20)(17 19)(21 25)(22 24)(26 40)(27 39)(28 38)(29 37)(30 36)(31 35)(32 34)(41 55)(42 54)(43 53)(44 52)(45 51)(46 50)(47 49)(56 60)(57 59)(61 71)(62 70)(63 69)(64 68)(65 67)(72 80)(73 79)(74 78)(75 77)
(1 64 41 21)(2 75 42 32)(3 66 43 23)(4 77 44 34)(5 68 45 25)(6 79 46 36)(7 70 47 27)(8 61 48 38)(9 72 49 29)(10 63 50 40)(11 74 51 31)(12 65 52 22)(13 76 53 33)(14 67 54 24)(15 78 55 35)(16 69 56 26)(17 80 57 37)(18 71 58 28)(19 62 59 39)(20 73 60 30)

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

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

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

D208C4 is a maximal subgroup of
D20⋊C8  Dic54D8  D4⋊D56C4  D203D4  D20.D4  Dic57SD16  Q8⋊D56C4  Dic5⋊SD16  D20.12D4  Dic58SD16  D4015C4  D20⋊Q8  D20.Q8  D4012C4  C4021(C2×C4)  D202Q8  D20.2Q8  D202C8  D102M4(2)  C20⋊M4(2)  C10.82+ 1+4  C10.2- 1+4  C10.112+ 1+4  C427D10  C42.188D10  C42.95D10  C42.97D10  C4×D4×D5  C4211D10  Dic1024D4  C42.114D10  C42.122D10  C4×Q82D5  C42.126D10  C42.136D10  C20⋊(C4○D4)  D2019D4  D2020D4  C10.472+ 1+4  C22⋊Q825D5  C4⋊C426D10  D2021D4  D2022D4  Dic1022D4  C10.532+ 1+4  C10.202- 1+4  C10.242- 1+4  C10.1212+ 1+4  C4⋊C428D10  C10.612+ 1+4  C10.642+ 1+4  D207Q8  C42.237D10  C42.150D10  C42.151D10  C42.153D10  C42.156D10  C4223D10  C4224D10  C42.189D10  C42.163D10  C42.240D10  D208Q8  D209Q8  C42.178D10  Dic1513D4  D6017C4  D208Dic3  C1520(C4×D4)  D6011C4
D208C4 is a maximal quotient of
C4⋊Dic515C4  (C2×C4)⋊9D20  D102C42  C10.54(C4×D4)  D205C8  D105M4(2)  C206M4(2)  Dic58SD16  Dic2015C4  D4015C4  D4012C4  Dic55Q16  C4021(C2×C4)  D4016C4  D4013C4  C204(C4⋊C4)  C4⋊C4×Dic5  (C2×D20)⋊22C4  D105(C4⋊C4)  C10.90(C4×D4)  Dic1513D4  D6017C4  D208Dic3  C1520(C4×D4)  D6011C4

40 conjugacy classes

class 1 2A2B2C2D2E2F2G4A···4F4G4H4I4J4K4L5A5B10A···10F20A···20L
order122222224···44444445510···1020···20
size1111101010102···255551010222···24···4

40 irreducible representations

dim11111112222244
type+++++++++++
imageC1C2C2C2C2C2C4D4D5C4○D4D10C4×D5D4×D5Q82D5
kernelD208C4C4×Dic5D10⋊C4C5×C4⋊C4C2×C4×D5C2×D20D20Dic5C4⋊C4C10C2×C4C4C2C2
# reps11212182226822

Matrix representation of D208C4 in GL5(𝔽41)

400000
064000
036100
0004021
000371
,
400000
0404000
00100
000400
000371
,
320000
01000
00100
000916
000032

G:=sub<GL(5,GF(41))| [40,0,0,0,0,0,6,36,0,0,0,40,1,0,0,0,0,0,40,37,0,0,0,21,1],[40,0,0,0,0,0,40,0,0,0,0,40,1,0,0,0,0,0,40,37,0,0,0,0,1],[32,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,9,0,0,0,0,16,32] >;

D208C4 in GAP, Magma, Sage, TeX

D_{20}\rtimes_8C_4
% in TeX

G:=Group("D20:8C4");
// GroupNames label

G:=SmallGroup(160,114);
// by ID

G=gap.SmallGroup(160,114);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,217,103,188,50,4613]);
// 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^10*b>;
// generators/relations

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