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G = Q16⋊F5order 320 = 26·5

4th semidirect product of Q16 and F5 acting via F5/D5=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D403C4, Q164F5, C5⋊C8.3D4, Q8⋊D54C4, C8.9(C2×F5), C40.7(C2×C4), C8⋊F53C2, (C5×Q16)⋊3C4, Q8.F54C2, C2.26(D4×F5), Q8.6(C2×F5), C53(C8.26D4), Q82F54C2, D20.6(C2×C4), C10.25(C4×D4), C40.C43C2, D5⋊C8.6C22, (C4×F5).6C22, C4.12(C22×F5), D10.6(C4○D4), Q8.D10.3C2, C4.F5.6C22, C20.12(C22×C4), Dic5.77(C2×D4), (C8×D5).16C22, (C4×D5).34C23, Q82D5.7C22, (C5×Q8).6(C2×C4), C52C8.14(C2×C4), SmallGroup(320,1079)

Series: Derived Chief Lower central Upper central

C1C20 — Q16⋊F5
C1C5C10Dic5C4×D5D5⋊C8Q8.F5 — Q16⋊F5
C5C10C20 — Q16⋊F5
C1C2C4Q16

Generators and relations for Q16⋊F5
 G = < a,b,c,d | a8=c5=d4=1, b2=a4, bab-1=a-1, ac=ca, dad-1=a5, bc=cb, dbd-1=a6b, dcd-1=c3 >

Subgroups: 418 in 104 conjugacy classes, 40 normal (22 characteristic)
C1, C2, C2 [×3], C4, C4 [×4], C22 [×3], C5, C8, C8 [×5], C2×C4 [×4], D4 [×4], Q8 [×2], D5 [×3], C10, C42, C2×C8 [×4], M4(2) [×4], D8, SD16 [×2], Q16, C4○D4 [×2], Dic5, C20, C20 [×2], F5, D10, D10 [×2], C8⋊C4, C4≀C2 [×2], C8.C4, C8○D4 [×2], C4○D8, C52C8, C40, C5⋊C8 [×2], C5⋊C8 [×2], C4×D5, C4×D5 [×2], D20 [×2], D20 [×2], C5×Q8 [×2], C2×F5, C8.26D4, C8×D5, D40, Q8⋊D5 [×2], C5×Q16, D5⋊C8, D5⋊C8 [×2], C4.F5 [×2], C4.F5 [×2], C4×F5, Q82D5 [×2], C8⋊F5, C40.C4, Q82F5 [×2], Q8.D10, Q8.F5 [×2], Q16⋊F5
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], C23, C22×C4, C2×D4, C4○D4, F5, C4×D4, C2×F5 [×3], C8.26D4, C22×F5, D4×F5, Q16⋊F5

Character table of Q16⋊F5

 class 12A2B2C2D4A4B4C4D4E4F4G58A8B8C8D8E8F8G8H8I8J1020A20B20C40A40B
 size 111020202445520204410101010202020202048161688
ρ111111111111111111111111111111    trivial
ρ21111-11-1111111-1-1-1-1-1-111-1-111-11-1-1    linear of order 2
ρ31111111111-1-111-1-1-1-11-1-1-1-1111111    linear of order 2
ρ41111-11-1111-1-11-11111-1-1-11111-11-1-1    linear of order 2
ρ5111-1-11-1-111111111111-1-1-1-111-1-111    linear of order 2
ρ6111-1111-111111-1-1-1-1-1-1-1-111111-1-1-1    linear of order 2
ρ7111-1-11-1-111-1-111-1-1-1-11111111-1-111    linear of order 2
ρ8111-1111-111-1-11-11111-111-1-1111-1-1-1    linear of order 2
ρ911-1-1-1111-1-1-ii11-ii-ii-1i-ii-i111111    linear of order 4
ρ1011-1-111-11-1-1-ii1-1i-ii-i1i-i-ii11-11-1-1    linear of order 4
ρ1111-11-111-1-1-1-ii1-1i-ii-i1-iii-i111-1-1-1    linear of order 4
ρ1211-1111-1-1-1-1-ii11-ii-ii-1-ii-ii11-1-111    linear of order 4
ρ1311-1-111-11-1-1i-i1-1-ii-ii1-iii-i11-11-1-1    linear of order 4
ρ1411-1-1-1111-1-1i-i11i-ii-i-1-ii-ii111111    linear of order 4
ρ1511-1111-1-1-1-1i-i11i-ii-i-1i-ii-i11-1-111    linear of order 4
ρ1611-11-111-1-1-1i-i1-1-ii-ii1i-i-ii111-1-1-1    linear of order 4
ρ1722-200-20022002022-2-2000002-20000    orthogonal lifted from D4
ρ1822-200-200220020-2-222000002-20000    orthogonal lifted from D4
ρ1922200-200-2-20020-2i2i2i-2i000002-20000    complex lifted from C4○D4
ρ2022200-200-2-200202i-2i-2i2i000002-20000    complex lifted from C4○D4
ρ214400044-40000-1-4000000000-1-1-1111    orthogonal lifted from C2×F5
ρ22440004-4-40000-14000000000-1-111-1-1    orthogonal lifted from C2×F5
ρ23440004440000-14000000000-1-1-1-1-1-1    orthogonal lifted from F5
ρ24440004-440000-1-4000000000-1-11-111    orthogonal lifted from C2×F5
ρ254-40000004i-4i0040000000000-400000    complex lifted from C8.26D4
ρ264-4000000-4i4i0040000000000-400000    complex lifted from C8.26D4
ρ2788000-8000000-20000000000-220000    orthogonal lifted from D4×F5
ρ288-80000000000-200000000002000-1010    orthogonal faithful
ρ298-80000000000-20000000000200010-10    orthogonal faithful

Smallest permutation representation of Q16⋊F5
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 60 5 64)(2 59 6 63)(3 58 7 62)(4 57 8 61)(9 35 13 39)(10 34 14 38)(11 33 15 37)(12 40 16 36)(17 45 21 41)(18 44 22 48)(19 43 23 47)(20 42 24 46)(25 75 29 79)(26 74 30 78)(27 73 31 77)(28 80 32 76)(49 71 53 67)(50 70 54 66)(51 69 55 65)(52 68 56 72)
(1 75 41 12 66)(2 76 42 13 67)(3 77 43 14 68)(4 78 44 15 69)(5 79 45 16 70)(6 80 46 9 71)(7 73 47 10 72)(8 74 48 11 65)(17 40 50 60 29)(18 33 51 61 30)(19 34 52 62 31)(20 35 53 63 32)(21 36 54 64 25)(22 37 55 57 26)(23 38 56 58 27)(24 39 49 59 28)
(2 6)(4 8)(9 76 46 67)(10 73 47 72)(11 78 48 69)(12 75 41 66)(13 80 42 71)(14 77 43 68)(15 74 44 65)(16 79 45 70)(17 56 36 31)(18 53 37 28)(19 50 38 25)(20 55 39 30)(21 52 40 27)(22 49 33 32)(23 54 34 29)(24 51 35 26)(57 59 61 63)(58 64 62 60)

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

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

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

Matrix representation of Q16⋊F5 in GL8(𝔽41)

400000000
040000000
004000000
000400000
000000033
0000020310
000004210
00004000
,
10000000
01000000
00100000
00010000
000001360
0000400037
000000032
000000320
,
000400000
100400000
010400000
001400000
00001000
00000100
00000010
00000001
,
00100000
10000000
00010000
01000000
00001000
000009330
000000320
000000040

G:=sub<GL(8,GF(41))| [40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,20,4,0,0,0,0,0,0,31,21,0,0,0,0,0,33,0,0,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,0,36,0,0,32,0,0,0,0,0,37,32,0],[0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,40,40,40,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,33,32,0,0,0,0,0,0,0,0,40] >;

Q16⋊F5 in GAP, Magma, Sage, TeX

Q_{16}\rtimes F_5
% in TeX

G:=Group("Q16:F5");
// GroupNames label

G:=SmallGroup(320,1079);
// by ID

G=gap.SmallGroup(320,1079);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,232,758,219,184,136,851,438,102,6278,1595]);
// Polycyclic

G:=Group<a,b,c,d|a^8=c^5=d^4=1,b^2=a^4,b*a*b^-1=a^-1,a*c=c*a,d*a*d^-1=a^5,b*c=c*b,d*b*d^-1=a^6*b,d*c*d^-1=c^3>;
// generators/relations

Export

Character table of Q16⋊F5 in TeX

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