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G = D4.F5order 160 = 25·5

The non-split extension by D4 of F5 acting through Inn(D4)

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D4.F5, Dic10.C4, Dic5.12C23, (C5×D4).C4, C5⋊D4.C4, D5⋊C82C2, C51(C8○D4), C4.F53C2, C4.5(C2×F5), C20.5(C2×C4), C5⋊C8.1C22, D10.1(C2×C4), C22.F52C2, C2.8(C22×F5), C22.1(C2×F5), D42D5.3C2, C10.7(C22×C4), Dic5.1(C2×C4), (C4×D5).11C22, (C2×Dic5).25C22, (C2×C5⋊C8)⋊4C2, (C2×C10).(C2×C4), SmallGroup(160,206)

Series: Derived Chief Lower central Upper central

C1C10 — D4.F5
C1C5C10Dic5C5⋊C8C2×C5⋊C8 — D4.F5
C5C10 — D4.F5
C1C2D4

Generators and relations for D4.F5
 G = < a,b,c,d | a4=b2=c5=1, d4=a2, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c3 >

Subgroups: 164 in 62 conjugacy classes, 34 normal (18 characteristic)
C1, C2, C2 [×3], C4, C4 [×3], C22 [×2], C22, C5, C8 [×4], C2×C4 [×3], D4, D4 [×2], Q8, D5, C10, C10 [×2], C2×C8 [×3], M4(2) [×3], C4○D4, Dic5, Dic5 [×2], C20, D10, C2×C10 [×2], C8○D4, C5⋊C8 [×2], C5⋊C8 [×2], Dic10, C4×D5, C2×Dic5 [×2], C5⋊D4 [×2], C5×D4, D5⋊C8, C4.F5, C2×C5⋊C8 [×2], C22.F5 [×2], D42D5, D4.F5
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], C23, C22×C4, F5, C8○D4, C2×F5 [×3], C22×F5, D4.F5

Character table of D4.F5

 class 12A2B2C2D4A4B4C4D4E58A8B8C8D8E8F8G8H8I8J10A10B10C20
 size 1122102551010455551010101010104888
ρ11111111111111111111111111    trivial
ρ211-11-1-1111-11-1-1-1-11-1-11111-11-1    linear of order 2
ρ311-1-11111-1-1111111-1-1-1-111-1-11    linear of order 2
ρ4111-1-1-111-111-1-1-1-1111-1-1111-1-1    linear of order 2
ρ5111-1-1-111-1111111-1-1-111-111-1-1    linear of order 2
ρ611-1-11111-1-11-1-1-1-1-11111-11-1-11    linear of order 2
ρ711-11-1-1111-111111-111-1-1-11-11-1    linear of order 2
ρ811111111111-1-1-1-1-1-1-1-1-1-11111    linear of order 2
ρ9111-11-1-1-11-11-i-iii-ii-i-iii11-1-1    linear of order 4
ρ101111-11-1-1-1-11-i-iiii-ii-ii-i1111    linear of order 4
ρ11111-11-1-1-11-11ii-i-ii-iii-i-i11-1-1    linear of order 4
ρ121111-11-1-1-1-11ii-i-i-ii-ii-ii1111    linear of order 4
ρ1311-111-1-1-1-111ii-i-iii-i-ii-i1-11-1    linear of order 4
ρ1411-1-1-11-1-1111ii-i-i-i-ii-iii1-1-11    linear of order 4
ρ1511-111-1-1-1-111-i-iii-i-iii-ii1-11-1    linear of order 4
ρ1611-1-1-11-1-1111-i-iiiii-ii-i-i1-1-11    linear of order 4
ρ172-200002i-2i0028588783000000-2000    complex lifted from C8○D4
ρ182-200002i-2i0028858387000000-2000    complex lifted from C8○D4
ρ192-20000-2i2i0028387885000000-2000    complex lifted from C8○D4
ρ202-20000-2i2i0028783858000000-2000    complex lifted from C8○D4
ρ21444-40-40000-10000000000-1-111    orthogonal lifted from C2×F5
ρ2244-440-40000-10000000000-11-11    orthogonal lifted from C2×F5
ρ234444040000-10000000000-1-1-1-1    orthogonal lifted from F5
ρ2444-4-4040000-10000000000-111-1    orthogonal lifted from C2×F5
ρ258-800000000-200000000002000    symplectic faithful, Schur index 2

Smallest permutation representation of D4.F5
On 80 points
Generators in S80
(1 7 5 3)(2 8 6 4)(9 55 13 51)(10 56 14 52)(11 49 15 53)(12 50 16 54)(17 79 21 75)(18 80 22 76)(19 73 23 77)(20 74 24 78)(25 43 29 47)(26 44 30 48)(27 45 31 41)(28 46 32 42)(33 63 37 59)(34 64 38 60)(35 57 39 61)(36 58 40 62)(65 67 69 71)(66 68 70 72)
(1 71)(2 72)(3 65)(4 66)(5 67)(6 68)(7 69)(8 70)(9 57)(10 58)(11 59)(12 60)(13 61)(14 62)(15 63)(16 64)(17 27)(18 28)(19 29)(20 30)(21 31)(22 32)(23 25)(24 26)(33 53)(34 54)(35 55)(36 56)(37 49)(38 50)(39 51)(40 52)(41 79)(42 80)(43 73)(44 74)(45 75)(46 76)(47 77)(48 78)
(1 33 75 19 57)(2 20 34 58 76)(3 59 21 77 35)(4 78 60 36 22)(5 37 79 23 61)(6 24 38 62 80)(7 63 17 73 39)(8 74 64 40 18)(9 71 53 45 29)(10 46 72 30 54)(11 31 47 55 65)(12 56 32 66 48)(13 67 49 41 25)(14 42 68 26 50)(15 27 43 51 69)(16 52 28 70 44)
(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)

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

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

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

D4.F5 is a maximal subgroup of
D85F5  D8⋊F5  SD163F5  SD162F5  Dic5.C24  Dic5.21C24  Dic5.22C24  D12.2F5  D12.F5  C5⋊C8.D6  D15⋊C8⋊C2  Dic10.Dic3
D4.F5 is a maximal quotient of
Dic5.C42  C5⋊C88D4  C5⋊C8⋊D4  D10⋊M4(2)  Dic5⋊M4(2)  C20⋊C8⋊C2  C23.(C2×F5)  D10.C42  Dic10⋊C8  C4⋊C4.7F5  Dic5.M4(2)  C4⋊C4.9F5  C20.M4(2)  D4×C5⋊C8  C5⋊C87D4  C202M4(2)  (C2×D4).7F5  (C2×D4).8F5  D12.2F5  D12.F5  C5⋊C8.D6  D15⋊C8⋊C2  Dic10.Dic3

Matrix representation of D4.F5 in GL6(𝔽41)

900000
28320000
001000
000100
000010
000001
,
29370000
5120000
0040000
0004000
0000400
0000040
,
100000
010000
0040404040
001000
000100
000010
,
1400000
0140000
003201818
001818032
002314230
00927279

G:=sub<GL(6,GF(41))| [9,28,0,0,0,0,0,32,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[29,5,0,0,0,0,37,12,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,1,0,0,0,0,40,0,1,0,0,0,40,0,0,1,0,0,40,0,0,0],[14,0,0,0,0,0,0,14,0,0,0,0,0,0,32,18,23,9,0,0,0,18,14,27,0,0,18,0,23,27,0,0,18,32,0,9] >;

D4.F5 in GAP, Magma, Sage, TeX

D_4.F_5
% in TeX

G:=Group("D4.F5");
// GroupNames label

G:=SmallGroup(160,206);
// by ID

G=gap.SmallGroup(160,206);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,48,188,69,2309,599]);
// Polycyclic

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

Export

Character table of D4.F5 in TeX

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