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G = D10.D8order 320 = 26·5

2nd non-split extension by D10 of D8 acting via D8/C4=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D82F5, D10.2D8, Dic201C4, Dic5.16SD16, (C5×D8)⋊1C4, C8.1(C2×F5), C40.1(C2×C4), C40⋊C41C2, C51(D82C4), (C4×D5).20D4, C52C8.11D4, C8.F51C2, D83D5.2C2, (C8×D5).9C22, C4.1(C22⋊F5), C20.1(C22⋊C4), C2.6(D20⋊C4), C10.5(D4⋊C4), SmallGroup(320,241)

Series: Derived Chief Lower central Upper central

C1C40 — D10.D8
C1C5C10C20C4×D5C8×D5C40⋊C4 — D10.D8
C5C10C20C40 — D10.D8
C1C2C4C8D8

Generators and relations for D10.D8
 G = < a,b,c,d | a10=b2=1, c8=a5, d2=a-1b, bab=a-1, cac-1=dad-1=a3, cbc-1=a7b, dbd-1=a2b, dcd-1=a4bc7 >

Subgroups: 314 in 58 conjugacy classes, 20 normal (all characteristic)
C1, C2, C2 [×2], C4, C4 [×3], C22 [×2], C5, C8, C8, C2×C4 [×3], D4 [×2], Q8, D5, C10, C10, C16, C4⋊C4, C2×C8, D8, SD16, Q16, C4○D4, Dic5, Dic5, C20, F5, D10, C2×C10, C4.Q8, M5(2), C4○D8, C52C8, C40, Dic10, C4×D5, C2×Dic5, C5⋊D4, C5×D4, C2×F5, D82C4, C5⋊C16, C8×D5, Dic20, D4.D5, C5×D8, C4⋊F5, D42D5, C8.F5, C40⋊C4, D83D5, D10.D8
Quotients: C1, C2 [×3], C4 [×2], C22, C2×C4, D4 [×2], C22⋊C4, D8, SD16, F5, D4⋊C4, C2×F5, D82C4, C22⋊F5, D20⋊C4, D10.D8

Character table of D10.D8

 class 12A2B2C4A4B4C4D4E58A8B8C10A10B10C16A16B16C16D2040A40B
 size 118102104040404410104161620202020888
ρ111111111111111111111111    trivial
ρ211-11111-1111111-1-1-1-1-1-1111    linear of order 2
ρ3111111-11-11111111-1-1-1-1111    linear of order 2
ρ411-1111-1-1-111111-1-11111111    linear of order 2
ρ5111-11-1-i-1i11-1-1111-iii-i111    linear of order 4
ρ611-1-11-1-i1i11-1-11-1-1i-i-ii111    linear of order 4
ρ7111-11-1i-1-i11-1-1111i-i-ii111    linear of order 4
ρ811-1-11-1i1-i11-1-11-1-1-iii-i111    linear of order 4
ρ92202220002-2-2-220000002-2-2    orthogonal lifted from D4
ρ10220-22-20002-22220000002-2-2    orthogonal lifted from D4
ρ112202-2-200020002002-22-2-200    orthogonal lifted from D8
ρ122202-2-20002000200-22-22-200    orthogonal lifted from D8
ρ13220-2-220002000200--2--2-2-2-200    complex lifted from SD16
ρ14220-2-220002000200-2-2--2--2-200    complex lifted from SD16
ρ1544-4040000-1400-1110000-1-1-1    orthogonal lifted from C2×F5
ρ16444040000-1400-1-1-10000-1-1-1    orthogonal lifted from F5
ρ17440040000-1-400-1-550000-111    orthogonal lifted from C22⋊F5
ρ18440040000-1-400-15-50000-111    orthogonal lifted from C22⋊F5
ρ194-4000000040-2-22-2-4000000000    complex lifted from D82C4
ρ204-40000000402-2-2-2-4000000000    complex lifted from D82C4
ρ218800-80000-2000-2000000200    orthogonal lifted from D20⋊C4, Schur index 2
ρ228-80000000-200020000000-1010    symplectic faithful, Schur index 2
ρ238-80000000-20002000000010-10    symplectic faithful, Schur index 2

Smallest permutation representation of D10.D8
On 80 points
Generators in S80
(1 33 28 74 57 9 41 20 66 49)(2 75 42 50 29 10 67 34 58 21)(3 51 68 22 43 11 59 76 30 35)(4 23 60 36 69 12 31 52 44 77)(5 37 32 78 61 13 45 24 70 53)(6 79 46 54 17 14 71 38 62 25)(7 55 72 26 47 15 63 80 18 39)(8 27 64 40 73 16 19 56 48 65)
(1 49)(2 29)(3 35)(4 69)(5 53)(6 17)(7 39)(8 73)(9 57)(10 21)(11 43)(12 77)(13 61)(14 25)(15 47)(16 65)(18 55)(19 48)(20 28)(22 59)(23 36)(24 32)(26 63)(27 40)(30 51)(31 44)(33 66)(37 70)(41 74)(45 78)(50 75)(54 79)(58 67)(62 71)(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)
(2 8 10 16)(3 7)(4 14 12 6)(5 13)(11 15)(17 44 79 52)(18 59 72 43)(19 34 65 58)(20 49 74 33)(21 40 67 64)(22 55 76 39)(23 46 69 54)(24 61 78 45)(25 36 71 60)(26 51 80 35)(27 42 73 50)(28 57 66 41)(29 48 75 56)(30 63 68 47)(31 38 77 62)(32 53 70 37)

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

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

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

Matrix representation of D10.D8 in GL8(𝔽241)

2400000000
0240000000
0024000000
0002400000
0000002401
0000002400
0000102400
0000012400
,
2400000000
0240000000
00100000
42177010000
0000012400
0000102400
0000002400
0000002401
,
1661102030000
16611382030000
10000000
19111640000
0000002400
0000240000
0000000240
0000024000
,
10000000
0240000000
1661102030000
22020922200000
00000010
00001000
00000001
00000100

G:=sub<GL(8,GF(241))| [240,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,240,240,240,240,0,0,0,0,1,0,0,0],[240,0,0,42,0,0,0,0,0,240,0,177,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,240,240,240,240,0,0,0,0,0,0,0,1],[166,166,1,19,0,0,0,0,11,11,0,1,0,0,0,0,0,38,0,11,0,0,0,0,203,203,0,64,0,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,0,240,0,0,0,0,240,0,0,0,0,0,0,0,0,0,240,0],[1,0,166,220,0,0,0,0,0,240,11,209,0,0,0,0,0,0,0,222,0,0,0,0,0,0,203,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0] >;

D10.D8 in GAP, Magma, Sage, TeX

D_{10}.D_8
% in TeX

G:=Group("D10.D8");
// GroupNames label

G:=SmallGroup(320,241);
// by ID

G=gap.SmallGroup(320,241);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,387,675,794,80,1684,851,102,6278,3156]);
// Polycyclic

G:=Group<a,b,c,d|a^10=b^2=1,c^8=a^5,d^2=a^-1*b,b*a*b=a^-1,c*a*c^-1=d*a*d^-1=a^3,c*b*c^-1=a^7*b,d*b*d^-1=a^2*b,d*c*d^-1=a^4*b*c^7>;
// generators/relations

Export

Character table of D10.D8 in TeX

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