Copied to
clipboard

G = D5.Q32order 320 = 26·5

The non-split extension by D5 of Q32 acting via Q32/Q16=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Q161F5, D5.2Q32, Dic204C4, D10.20D8, D5.3SD32, Dic5.5SD16, C5⋊(C2.Q32), (C5×Q16)⋊4C4, C8.12(C2×F5), C40.10(C2×C4), D5⋊C16.1C2, (C4×D5).25D4, C52C8.16D4, D5.D8.1C2, (D5×Q16).4C2, C4.6(C22⋊F5), C20.6(C22⋊C4), (C8×D5).19C22, C2.11(D20⋊C4), C10.10(D4⋊C4), SmallGroup(320,246)

Series: Derived Chief Lower central Upper central

C1C40 — D5.Q32
C1C5C10C20C4×D5C8×D5D5.D8 — D5.Q32
C5C10C20C40 — D5.Q32
C1C2C4C8Q16

Generators and relations for D5.Q32
 G = < a,b,c,d | a5=b2=c16=1, d2=c8, bab=a-1, cac-1=a3, ad=da, cbc-1=a2b, bd=db, dcd-1=a-1bc-1 >

Subgroups: 298 in 58 conjugacy classes, 22 normal (all characteristic)
C1, C2, C2 [×2], C4, C4 [×4], C22, C5, C8, C8, C2×C4 [×3], Q8 [×3], D5 [×2], C10, C16, C4⋊C4, C2×C8, Q16, Q16 [×2], C2×Q8, Dic5, Dic5, C20, C20, F5, D10, C2.D8, C2×C16, C2×Q16, C52C8, C40, Dic10 [×2], C4×D5, C4×D5, C5×Q8, C2×F5, C2.Q32, C5⋊C16, C8×D5, Dic20, C5⋊Q16, C5×Q16, C4⋊F5, Q8×D5, D5⋊C16, D5.D8, D5×Q16, D5.Q32
Quotients: C1, C2 [×3], C4 [×2], C22, C2×C4, D4 [×2], C22⋊C4, D8, SD16, F5, D4⋊C4, SD32, Q32, C2×F5, C2.Q32, C22⋊F5, D20⋊C4, D5.Q32

Character table of D5.Q32

 class 12A2B2C4A4B4C4D4E4F58A8B8C8D1016A16B16C16D16E16F16G16H20A20B20C40A40B
 size 115528104040404221010410101010101010108161688
ρ111111111111111111111111111111    trivial
ρ21111111-11-1111111-1-1-1-1-1-1-1-111111    linear of order 2
ρ311111-111-11111111-1-1-1-1-1-1-1-11-1-111    linear of order 2
ρ411111-11-1-1-1111111111111111-1-111    linear of order 2
ρ511-1-11-1-1-i1i111-1-11i-i-i-i-iiii1-1-111    linear of order 4
ρ611-1-111-1-i-1i111-1-11-iiiii-i-i-i11111    linear of order 4
ρ711-1-11-1-1i1-i111-1-11-iiiii-i-i-i1-1-111    linear of order 4
ρ811-1-111-1i-1-i111-1-11i-i-i-i-iiii11111    linear of order 4
ρ922222020002-2-2-2-2200000000200-2-2    orthogonal lifted from D4
ρ1022-2-220-20002-2-222200000000200-2-2    orthogonal lifted from D4
ρ112222-20-2000200002-222-2-222-2-20000    orthogonal lifted from D8
ρ122222-20-20002000022-2-222-2-22-20000    orthogonal lifted from D8
ρ132-2-2200000022-2-22-2ζ16716165163ζ165163ζ1671616716ζ165163165163167160002-2    symplectic lifted from Q32, Schur index 2
ρ142-2-2200000022-2-22-216716ζ16516316516316716ζ16716165163ζ165163ζ167160002-2    symplectic lifted from Q32, Schur index 2
ρ152-2-220000002-222-2-2ζ165163ζ1671616716ζ16516316516316716ζ16716165163000-22    symplectic lifted from Q32, Schur index 2
ρ162-2-220000002-222-2-216516316716ζ16716165163ζ165163ζ1671616716ζ165163000-22    symplectic lifted from Q32, Schur index 2
ρ1722-2-2-202000200002-2-2-2--2--2--2--2-2-20000    complex lifted from SD16
ρ182-22-20000002-22-22-2ζ16716ζ16131611ζ165163ζ1615169ζ16716ζ16131611ζ165163ζ1615169000-22    complex lifted from SD32
ρ1922-2-2-202000200002--2--2--2-2-2-2-2--2-20000    complex lifted from SD16
ρ202-22-200000022-22-2-2ζ165163ζ16716ζ1615169ζ16131611ζ165163ζ16716ζ1615169ζ161316110002-2    complex lifted from SD32
ρ212-22-20000002-22-22-2ζ1615169ζ165163ζ16131611ζ16716ζ1615169ζ165163ζ16131611ζ16716000-22    complex lifted from SD32
ρ222-22-200000022-22-2-2ζ16131611ζ1615169ζ16716ζ165163ζ16131611ζ1615169ζ16716ζ1651630002-2    complex lifted from SD32
ρ2344004-40000-14400-100000000-111-1-1    orthogonal lifted from C2×F5
ρ244400440000-14400-100000000-1-1-1-1-1    orthogonal lifted from F5
ρ254400400000-1-4-400-100000000-15-511    orthogonal lifted from C22⋊F5
ρ264400400000-1-4-400-100000000-1-5511    orthogonal lifted from C22⋊F5
ρ278800-800000-20000-20000000020000    orthogonal lifted from D20⋊C4, Schur index 2
ρ288-800000000-2-4242002000000000002-2    symplectic faithful, Schur index 2
ρ298-800000000-242-4200200000000000-22    symplectic faithful, Schur index 2

Smallest permutation representation of D5.Q32
On 80 points
Generators in S80
(1 22 59 78 34)(2 79 23 35 60)(3 36 80 61 24)(4 62 37 25 65)(5 26 63 66 38)(6 67 27 39 64)(7 40 68 49 28)(8 50 41 29 69)(9 30 51 70 42)(10 71 31 43 52)(11 44 72 53 32)(12 54 45 17 73)(13 18 55 74 46)(14 75 19 47 56)(15 48 76 57 20)(16 58 33 21 77)
(1 42)(2 52)(3 32)(4 73)(5 46)(6 56)(7 20)(8 77)(9 34)(10 60)(11 24)(12 65)(13 38)(14 64)(15 28)(16 69)(17 62)(18 66)(19 27)(21 50)(22 70)(23 31)(25 54)(26 74)(29 58)(30 78)(33 41)(35 71)(36 53)(37 45)(39 75)(40 57)(43 79)(44 61)(47 67)(48 49)(51 59)(55 63)(68 76)(72 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 2 9 10)(3 16 11 8)(4 7 12 15)(5 14 13 6)(17 57 25 49)(18 67 26 75)(19 55 27 63)(20 65 28 73)(21 53 29 61)(22 79 30 71)(23 51 31 59)(24 77 32 69)(33 72 41 80)(34 60 42 52)(35 70 43 78)(36 58 44 50)(37 68 45 76)(38 56 46 64)(39 66 47 74)(40 54 48 62)

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

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

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

Matrix representation of D5.Q32 in GL6(𝔽241)

100000
010000
00240100
00240010
00240001
00240000
,
100000
010000
00240000
00240001
00240010
00240100
,
852140000
27850000
000010
001000
000001
000100
,
138410000
411030000
001000
000100
000010
000001

G:=sub<GL(6,GF(241))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,240,240,240,240,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,240,240,240,240,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1,0,0],[85,27,0,0,0,0,214,85,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,1,0],[138,41,0,0,0,0,41,103,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

D5.Q32 in GAP, Magma, Sage, TeX

D_5.Q_{32}
% in TeX

G:=Group("D5.Q32");
// GroupNames label

G:=SmallGroup(320,246);
// by ID

G=gap.SmallGroup(320,246);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,232,675,346,192,1684,851,102,6278,3156]);
// Polycyclic

G:=Group<a,b,c,d|a^5=b^2=c^16=1,d^2=c^8,b*a*b=a^-1,c*a*c^-1=a^3,a*d=d*a,c*b*c^-1=a^2*b,b*d=d*b,d*c*d^-1=a^-1*b*c^-1>;
// generators/relations

Export

Character table of D5.Q32 in TeX

׿
×
𝔽