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G = Dic20⋊C4order 320 = 26·5

3rd semidirect product of Dic20 and C4 acting faithfully

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Q163F5, Dic203C4, C8.8(C2×F5), C40.6(C2×C4), C5⋊(Q16⋊C4), (C5×Q16)⋊2C4, C5⋊Q164C4, (C2×F5).7D4, C2.24(D4×F5), (Q8×F5).2C2, Q8.4(C2×F5), 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), (C4×D5).32C23, (C8×D5).15C22, Dic5.6(C4○D4), D5.3(C8.C22), (C5×Q8).4(C2×C4), C52C8.12(C2×C4), SmallGroup(320,1077)

Series: Derived Chief Lower central Upper central

C1C20 — Dic20⋊C4
C1C5C10D10C4×D5C4×F5Q8×F5 — Dic20⋊C4
C5C10C20 — Dic20⋊C4
C1C2C4Q16

Generators and relations for Dic20⋊C4
 G = < a,b,c | a40=c4=1, b2=a20, bab-1=a-1, cac-1=a13, cbc-1=a20b >

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, C52C8, 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

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
ρ1922-2-2-200-2i2-2i2i2i000000200002-20000    complex lifted from C4○D4
ρ2022-2-2-2002i22i-2i-2i000000200002-20000    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
ρ288-80000000000000000-200002000-1010    symplectic faithful, Schur index 2
ρ298-80000000000000000-20000200010-10    symplectic faithful, Schur index 2

Smallest permutation representation of Dic20⋊C4
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 50 21 70)(2 49 22 69)(3 48 23 68)(4 47 24 67)(5 46 25 66)(6 45 26 65)(7 44 27 64)(8 43 28 63)(9 42 29 62)(10 41 30 61)(11 80 31 60)(12 79 32 59)(13 78 33 58)(14 77 34 57)(15 76 35 56)(16 75 36 55)(17 74 37 54)(18 73 38 53)(19 72 39 52)(20 71 40 51)
(1 75 21 55)(2 72 30 68)(3 69 39 41)(4 66 8 54)(5 63 17 67)(6 60 26 80)(7 57 35 53)(9 51 13 79)(10 48 22 52)(11 45 31 65)(12 42 40 78)(14 76 18 64)(15 73 27 77)(16 70 36 50)(19 61 23 49)(20 58 32 62)(24 46 28 74)(25 43 37 47)(29 71 33 59)(34 56 38 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), (1,50,21,70)(2,49,22,69)(3,48,23,68)(4,47,24,67)(5,46,25,66)(6,45,26,65)(7,44,27,64)(8,43,28,63)(9,42,29,62)(10,41,30,61)(11,80,31,60)(12,79,32,59)(13,78,33,58)(14,77,34,57)(15,76,35,56)(16,75,36,55)(17,74,37,54)(18,73,38,53)(19,72,39,52)(20,71,40,51), (1,75,21,55)(2,72,30,68)(3,69,39,41)(4,66,8,54)(5,63,17,67)(6,60,26,80)(7,57,35,53)(9,51,13,79)(10,48,22,52)(11,45,31,65)(12,42,40,78)(14,76,18,64)(15,73,27,77)(16,70,36,50)(19,61,23,49)(20,58,32,62)(24,46,28,74)(25,43,37,47)(29,71,33,59)(34,56,38,44)>;

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

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

Matrix representation of Dic20⋊C4 in 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] >;

Dic20⋊C4 in GAP, Magma, Sage, TeX

{\rm 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

Export

Character table of Dic20⋊C4 in TeX

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