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G = D5×C8○D4order 320 = 26·5

Direct product of D5 and C8○D4

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D5×C8○D4, C20.71C24, C40.48C23, M4(2)⋊27D10, (C2×C8)⋊30D10, (D4×D5).4C4, (Q8×D5).4C4, D4.12(C4×D5), Q8.13(C4×D5), (C2×C40)⋊25C22, C4○D4.42D10, D20.34(C2×C4), D42D5.4C4, (C8×D5)⋊20C22, Q82D5.4C4, C4.70(C23×D5), C8.66(C22×D5), C8⋊D520C22, (D5×M4(2))⋊12C2, D4.Dic514C2, C10.55(C23×C4), C20.73(C22×C4), C52C8.43C23, (C4×D5).96C23, D20.2C414C2, D20.3C416C2, (C2×C20).513C23, Dic10.36(C2×C4), C4○D20.51C22, D10.24(C22×C4), C4.Dic526C22, (C5×M4(2))⋊27C22, Dic5.23(C22×C4), C56(C2×C8○D4), (D5×C2×C8)⋊30C2, C4.38(C2×C4×D5), (C5×C8○D4)⋊8C2, C22.4(C2×C4×D5), C5⋊D4.5(C2×C4), C2.35(D5×C22×C4), (D5×C4○D4).13C2, (C5×D4).30(C2×C4), (C4×D5).54(C2×C4), (C5×Q8).32(C2×C4), (C2×C52C8)⋊34C22, (C2×C4×D5).324C22, (C2×C10).11(C22×C4), (C5×C4○D4).43C22, (C22×D5).84(C2×C4), (C2×C4).606(C22×D5), (C2×Dic5).118(C2×C4), SmallGroup(320,1421)

Series: Derived Chief Lower central Upper central

C1C10 — D5×C8○D4
C1C5C10C20C4×D5C2×C4×D5D5×C4○D4 — D5×C8○D4
C5C10 — D5×C8○D4
C1C8C8○D4

Generators and relations for D5×C8○D4
 G = < a,b,c,d,e | a5=b2=c8=e2=1, d2=c4, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=c4d >

Subgroups: 734 in 266 conjugacy classes, 149 normal (24 characteristic)
C1, C2, C2 [×8], C4, C4 [×3], C4 [×4], C22 [×3], C22 [×10], C5, C8, C8 [×3], C8 [×4], C2×C4 [×3], C2×C4 [×13], D4 [×3], D4 [×9], Q8, Q8 [×3], C23 [×3], D5 [×2], D5 [×3], C10, C10 [×3], C2×C8 [×3], C2×C8 [×13], M4(2) [×3], M4(2) [×9], C22×C4 [×3], C2×D4 [×3], C2×Q8, C4○D4, C4○D4 [×7], Dic5, Dic5 [×3], C20, C20 [×3], D10, D10 [×3], D10 [×6], C2×C10 [×3], C22×C8 [×3], C2×M4(2) [×3], C8○D4, C8○D4 [×7], C2×C4○D4, C52C8, C52C8 [×3], C40, C40 [×3], Dic10 [×3], C4×D5, C4×D5 [×9], D20 [×3], C2×Dic5 [×3], C5⋊D4 [×6], C2×C20 [×3], C5×D4 [×3], C5×Q8, C22×D5 [×3], C2×C8○D4, C8×D5, C8×D5 [×9], C8⋊D5 [×6], C2×C52C8 [×3], C4.Dic5 [×3], C2×C40 [×3], C5×M4(2) [×3], C2×C4×D5 [×3], C4○D20 [×3], D4×D5 [×3], D42D5 [×3], Q8×D5, Q82D5, C5×C4○D4, D5×C2×C8 [×3], D20.3C4 [×3], D5×M4(2) [×3], D20.2C4 [×3], D4.Dic5, C5×C8○D4, D5×C4○D4, D5×C8○D4
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], D5, C22×C4 [×14], C24, D10 [×7], C8○D4 [×2], C23×C4, C4×D5 [×4], C22×D5 [×7], C2×C8○D4, C2×C4×D5 [×6], C23×D5, D5×C22×C4, D5×C8○D4

Smallest permutation representation of D5×C8○D4
On 80 points
Generators in S80
(1 59 46 49 68)(2 60 47 50 69)(3 61 48 51 70)(4 62 41 52 71)(5 63 42 53 72)(6 64 43 54 65)(7 57 44 55 66)(8 58 45 56 67)(9 27 38 17 79)(10 28 39 18 80)(11 29 40 19 73)(12 30 33 20 74)(13 31 34 21 75)(14 32 35 22 76)(15 25 36 23 77)(16 26 37 24 78)
(1 68)(2 69)(3 70)(4 71)(5 72)(6 65)(7 66)(8 67)(9 27)(10 28)(11 29)(12 30)(13 31)(14 32)(15 25)(16 26)(33 74)(34 75)(35 76)(36 77)(37 78)(38 79)(39 80)(40 73)(49 59)(50 60)(51 61)(52 62)(53 63)(54 64)(55 57)(56 58)
(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 32 5 28)(2 25 6 29)(3 26 7 30)(4 27 8 31)(9 67 13 71)(10 68 14 72)(11 69 15 65)(12 70 16 66)(17 45 21 41)(18 46 22 42)(19 47 23 43)(20 48 24 44)(33 61 37 57)(34 62 38 58)(35 63 39 59)(36 64 40 60)(49 76 53 80)(50 77 54 73)(51 78 55 74)(52 79 56 75)
(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)(25 29)(26 30)(27 31)(28 32)(33 37)(34 38)(35 39)(36 40)(73 77)(74 78)(75 79)(76 80)

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

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

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

80 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I4A4B4C4D4E4F4G4H4I4J5A5B8A8B8C8D8E···8J8K8L8M8N8O···8T10A10B10C···10H20A20B20C20D20E···20J40A···40H40I···40T
order122222222244444444445588888···888888···8101010···102020202020···2040···4040···40
size112225510101011222551010102211112···2555510···10224···422224···42···24···4

80 irreducible representations

dim11111111111122222224
type++++++++++++
imageC1C2C2C2C2C2C2C2C4C4C4C4D5D10D10D10C8○D4C4×D5C4×D5D5×C8○D4
kernelD5×C8○D4D5×C2×C8D20.3C4D5×M4(2)D20.2C4D4.Dic5C5×C8○D4D5×C4○D4D4×D5D42D5Q8×D5Q82D5C8○D4C2×C8M4(2)C4○D4D5D4Q8C1
# reps133331116622266281248

Matrix representation of D5×C8○D4 in GL4(𝔽41) generated by

0100
40600
0010
0001
,
0100
1000
0010
0001
,
1000
0100
0030
0003
,
1000
0100
00402
00401
,
1000
0100
0010
00140
G:=sub<GL(4,GF(41))| [0,40,0,0,1,6,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,3,0,0,0,0,3],[1,0,0,0,0,1,0,0,0,0,40,40,0,0,2,1],[1,0,0,0,0,1,0,0,0,0,1,1,0,0,0,40] >;

D5×C8○D4 in GAP, Magma, Sage, TeX

D_5\times C_8\circ D_4
% in TeX

G:=Group("D5xC8oD4");
// GroupNames label

G:=SmallGroup(320,1421);
// by ID

G=gap.SmallGroup(320,1421);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,387,80,102,12550]);
// Polycyclic

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

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