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G = D8.D10order 320 = 26·5

1st non-split extension by D8 of D10 acting via D10/C10=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D8.6D10, C20.21D8, C40.34D4, D4014C22, C40.24C23, Dic2011C22, (C2×D8)⋊7D5, (C10×D8)⋊1C2, C5⋊D165C2, D8.D55C2, C54(C16⋊C22), (C2×C10).42D8, C10.63(C2×D8), (C2×C8).83D10, C8.2(C5⋊D4), C20.4C82C2, D407C22C2, C4.17(D4⋊D5), C52C163C22, C20.160(C2×D4), (C2×C20).180D4, (C5×D8).6C22, C8.30(C22×D5), (C2×C40).31C22, C22.10(D4⋊D5), C4.2(C2×C5⋊D4), C2.18(C2×D4⋊D5), (C2×C4).79(C5⋊D4), SmallGroup(320,774)

Series: Derived Chief Lower central Upper central

C1C40 — D8.D10
C1C5C10C20C40D40D407C2 — D8.D10
C5C10C20C40 — D8.D10
C1C2C2×C4C2×C8C2×D8

Generators and relations for D8.D10
 G = < a,b,c,d | a8=b2=1, c10=d2=a4, bab=dad-1=a-1, ac=ca, cbc-1=a4b, dbd-1=ab, dcd-1=c9 >

Subgroups: 398 in 90 conjugacy classes, 35 normal (25 characteristic)
C1, C2, C2 [×4], C4 [×2], C4, C22, C22 [×5], C5, C8 [×2], C2×C4, C2×C4, D4 [×5], Q8, C23, D5, C10, C10 [×3], C16 [×2], C2×C8, D8 [×2], D8 [×2], SD16, Q16, C2×D4, C4○D4, Dic5, C20 [×2], D10, C2×C10, C2×C10 [×4], M5(2), D16 [×2], SD32 [×2], C2×D8, C4○D8, C40 [×2], Dic10, C4×D5, D20, C5⋊D4, C2×C20, C5×D4 [×3], C22×C10, C16⋊C22, C52C16 [×2], C40⋊C2, D40, Dic20, C2×C40, C5×D8 [×2], C5×D8, C4○D20, D4×C10, C20.4C8, C5⋊D16 [×2], D8.D5 [×2], D407C2, C10×D8, D8.D10
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, D5, D8 [×2], C2×D4, D10 [×3], C2×D8, C5⋊D4 [×2], C22×D5, C16⋊C22, D4⋊D5 [×2], C2×C5⋊D4, C2×D4⋊D5, D8.D10

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

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

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

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

44 conjugacy classes

class 1 2A2B2C2D2E4A4B4C5A5B8A8B8C10A···10F10G···10N16A16B16C16D20A20B20C20D40A···40H
order1222224445588810···1010···10161616162020202040···40
size11288402240222242···28···82020202044444···4

44 irreducible representations

dim1111112222222224444
type++++++++++++++++
imageC1C2C2C2C2C2D4D4D5D8D8D10D10C5⋊D4C5⋊D4C16⋊C22D4⋊D5D4⋊D5D8.D10
kernelD8.D10C20.4C8C5⋊D16D8.D5D407C2C10×D8C40C2×C20C2×D8C20C2×C10C2×C8D8C8C2×C4C5C4C22C1
# reps1122111122224442228

Matrix representation of D8.D10 in GL6(𝔽241)

100000
010000
0021921900
0011000
0000022
0000230219
,
24000000
02400000
0002200
0011000
00002400
000011
,
3600000
921540000
001200
0024024000
000012
0000240240
,
187460000
120540000
000012
0000240240
001200
0024024000

G:=sub<GL(6,GF(241))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,219,11,0,0,0,0,219,0,0,0,0,0,0,0,0,230,0,0,0,0,22,219],[240,0,0,0,0,0,0,240,0,0,0,0,0,0,0,11,0,0,0,0,22,0,0,0,0,0,0,0,240,1,0,0,0,0,0,1],[36,92,0,0,0,0,0,154,0,0,0,0,0,0,1,240,0,0,0,0,2,240,0,0,0,0,0,0,1,240,0,0,0,0,2,240],[187,120,0,0,0,0,46,54,0,0,0,0,0,0,0,0,1,240,0,0,0,0,2,240,0,0,1,240,0,0,0,0,2,240,0,0] >;

D8.D10 in GAP, Magma, Sage, TeX

D_8.D_{10}
% in TeX

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

G:=SmallGroup(320,774);
// by ID

G=gap.SmallGroup(320,774);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,254,387,675,185,192,1684,438,102,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^8=b^2=1,c^10=d^2=a^4,b*a*b=d*a*d^-1=a^-1,a*c=c*a,c*b*c^-1=a^4*b,d*b*d^-1=a*b,d*c*d^-1=c^9>;
// generators/relations

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