Copied to
clipboard

?

G = Dic20⋊C4  order 320 = 26·5

3rd semidirect product of Dic20 and C4 acting faithfully

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Q16⋊3F5, Dic20⋊3C4, C8.8(C2×F5), C40.6(C2×C4), C5⋊(Q16⋊C4), (C5×Q16)⋊2C4, C5⋊Q16⋊4C4, (C2×F5).7D4, C2.24(D4×F5), Q8.4(C2×F5), (Q8×F5).2C2, C10.23(C4×D4), C40⋊C4.1C2, C8⋊F5.1C2, (D5×Q16).3C2, C4⋊F5.6C22, D10.68(C2×D4), D5⋊C8.5C22, Q8⋊F5.2C2, (C4×F5).5C22, C4.10(C22×F5), (Q8×D5).7C22, C20.10(C22×C4), Dic10.6(C2×C4), (C8×D5).15C22, (C4×D5).32C23, Dic5.6(C4○D4), D5.3(C8.C22), (C5×Q8).4(C2×C4), C5⋊2C8.12(C2×C4), SmallGroup(320,1077)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C20 — Dic20⋊C4
Chief series
C1 — C5 — C10 — D10 — C4×D5 — C4×F5 — Q8×F5 — Dic20⋊C4
Lower central
C5 — C10 — C20 — Dic20⋊C4
Upper central
C1 — C2 — C4 — Q16

Subgroups: 418 in 108 conjugacy classes, 42 normal (22 characteristic)
C1, C2, C2 [×2], C4, C4 [×9], C22, C5, C8, C8 [×2], C2×C4 [×7], Q8 [×2], Q8 [×4], D5 [×2], C10, C42 [×3], C4⋊C4 [×4], C2×C8 [×2], Q16, Q16 [×3], C2×Q8 [×2], Dic5, Dic5 [×2], C20, C20 [×2], F5 [×4], D10, C8⋊C4, Q8⋊C4 [×2], C4.Q8, C4×Q8 [×2], C2×Q16, C5⋊2C8, C40, C5⋊C8, Dic10 [×2], Dic10 [×2], C4×D5, C4×D5 [×2], C5×Q8 [×2], C2×F5 [×2], C2×F5 [×2], Q16⋊C4, C8×D5, Dic20, C5⋊Q16 [×2], C5×Q16, D5⋊C8, C4×F5, C4×F5 [×2], C4⋊F5 [×2], C4⋊F5 [×2], Q8×D5 [×2], C8⋊F5, C40⋊C4, Q8⋊F5 [×2], D5×Q16, Q8×F5 [×2], Dic20⋊C4

Quotients:
C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], C23, C22×C4, C2×D4, C4○D4, F5, C4×D4, C8.C22 [×2], C2×F5 [×3], Q16⋊C4, C22×F5, D4×F5, Dic20⋊C4

Generators and relations
 G = < a,b,c | a40=c4=1, b2=a20, bab-1=a-1, cac-1=a13, cbc-1=a20b >

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

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

Matrix representation ►G ⊆ GL8(𝔽41)

401000000
400100000
400010000
400000000
0000239270
000029292713
000021604
000037391210
,
00010000
00100000
01000000
10000000
000000400
00001112
00001000
00004040040
,
00100000
10000000
00010000
01000000
000025393737
00003925037
0000392140
00001614218

G:=sub<GL(8,GF(41))| [40,40,40,40,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,2,29,2,37,0,0,0,0,39,29,16,39,0,0,0,0,27,27,0,12,0,0,0,0,0,13,4,10],[0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,1,40,0,0,0,0,0,1,0,40,0,0,0,0,40,1,0,0,0,0,0,0,0,2,0,40],[0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,25,39,39,16,0,0,0,0,39,25,2,14,0,0,0,0,37,0,14,2,0,0,0,0,37,37,0,18] >;
 

Character table of Dic20⋊C4

 class 12A2B2C4A4B4C4D4E4F4G4H4I4J4K4L4M4N58A8B8C8D1020A20B20C40A40B
 size 115524410101010102020202020204420202048161688
ρ111111111111111111111111111111    trivial
ρ211111-11-11-1-1-11-111-1-11-1-111111-1-1-1    linear of order 2
ρ3111111-1-11-1-1-1-11-1-1111-1-11111-11-1-1    linear of order 2
ρ411111-1-111111-1-1-1-1-1-11111111-1-111    linear of order 2
ρ511111-11111111-1-1-1111-1-1-1-1111-1-1-1    linear of order 2
ρ61111111-11-1-1-111-1-1-1-1111-1-1111111    linear of order 2
ρ711111-1-1-11-1-1-1-1-11111111-1-111-1-111    linear of order 2
ρ8111111-111111-1111-1-11-1-1-1-111-11-1-1    linear of order 2
ρ911-1-11-11-i-1ii-i-11-iii-i1-11-ii111-1-1-1    linear of order 4
ρ1011-1-11-1-1i-1-i-ii11i-ii-i11-1-ii11-1-111    linear of order 4
ρ1111-1-111-1-i-1ii-i1-1i-i-ii1-11-ii11-11-1-1    linear of order 4
ρ1211-1-1111i-1-i-ii-1-1-ii-ii11-1-ii111111    linear of order 4
ρ1311-1-111-1i-1-i-ii1-1-iii-i1-11i-i11-11-1-1    linear of order 4
ρ1411-1-1111-i-1ii-i-1-1i-ii-i11-1i-i111111    linear of order 4
ρ1511-1-11-11i-1-i-ii-11i-i-ii1-11i-i111-1-1-1    linear of order 4
ρ1611-1-11-1-1-i-1ii-i11-ii-ii11-1i-i11-1-111    linear of order 4
ρ172222-200-2-22-22000000200002-20000    orthogonal lifted from D4
ρ182222-2002-2-22-2000000200002-20000    orthogonal lifted from D4
ρ1922-2-2-2002i22i2i2i000000200002-20000    complex lifted from C4○D4
ρ2022-2-2-2002i22i2i2i000000200002-20000    complex lifted from C4○D4
ρ21440044-400000000000-1-4000-1-11-111    orthogonal lifted from C2×F5
ρ2244004-4-400000000000-14000-1-111-1-1    orthogonal lifted from C2×F5
ρ2344004-4400000000000-1-4000-1-1-1111    orthogonal lifted from C2×F5
ρ24440044400000000000-14000-1-1-1-1-1-1    orthogonal lifted from F5
ρ254-44-40000000000000040000-400000    symplectic lifted from C8.C22, Schur index 2
ρ264-4-440000000000000040000-400000    symplectic lifted from C8.C22, Schur index 2
ρ278800-80000000000000-20000-220000    orthogonal lifted from D4×F5
ρ288-80000000000000000-200002000√10√10    symplectic faithful, Schur index 2
ρ298-80000000000000000-200002000√10√10    symplectic faithful, Schur index 2

In GAP, Magma, Sage, TeX 

Dic_{20}\rtimes C_4
 
% in TeX
 
G:=Group("Dic20:C4");
 
// GroupNames label
 
G:=SmallGroup(320,1077);
 
// by ID
 
G=gap.SmallGroup(320,1077);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,232,758,219,184,851,438,102,6278,1595]);
 
// Polycyclic
 
G:=Group<a,b,c|a^40=c^4=1,b^2=a^20,b*a*b^-1=a^-1,c*a*c^-1=a^13,c*b*c^-1=a^20*b>;
 
// generators/relations
 

׿
×
⋊
ℤ
𝔽
○
≀