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## G = D5×C8○D4order 320 = 26·5

### Direct product of D5 and C8○D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — D5×C8○D4
 Chief series C1 — C5 — C10 — C20 — C4×D5 — C2×C4×D5 — D5×C4○D4 — D5×C8○D4
 Lower central C5 — C10 — D5×C8○D4
 Upper central C1 — C8 — C8○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, C4, C4, C4, C22, C22, C5, C8, C8, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, D5, D5, C10, C10, C2×C8, C2×C8, M4(2), M4(2), C22×C4, C2×D4, C2×Q8, C4○D4, C4○D4, Dic5, Dic5, C20, C20, D10, D10, D10, C2×C10, C22×C8, C2×M4(2), C8○D4, C8○D4, C2×C4○D4, C52C8, C52C8, C40, C40, Dic10, C4×D5, C4×D5, D20, C2×Dic5, C5⋊D4, C2×C20, C5×D4, C5×Q8, C22×D5, C2×C8○D4, C8×D5, C8×D5, C8⋊D5, C2×C52C8, C4.Dic5, C2×C40, C5×M4(2), C2×C4×D5, C4○D20, D4×D5, D42D5, Q8×D5, Q82D5, C5×C4○D4, D5×C2×C8, D20.3C4, D5×M4(2), D20.2C4, D4.Dic5, C5×C8○D4, D5×C4○D4, D5×C8○D4
Quotients: C1, C2, C4, C22, C2×C4, C23, D5, C22×C4, C24, D10, C8○D4, C23×C4, C4×D5, C22×D5, C2×C8○D4, C2×C4×D5, C23×D5, D5×C22×C4, D5×C8○D4

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

80 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 type + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C4 C4 C4 C4 D5 D10 D10 D10 C8○D4 C4×D5 C4×D5 D5×C8○D4 kernel D5×C8○D4 D5×C2×C8 D20.3C4 D5×M4(2) D20.2C4 D4.Dic5 C5×C8○D4 D5×C4○D4 D4×D5 D4⋊2D5 Q8×D5 Q8⋊2D5 C8○D4 C2×C8 M4(2) C4○D4 D5 D4 Q8 C1 # reps 1 3 3 3 3 1 1 1 6 6 2 2 2 6 6 2 8 12 4 8

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

 0 1 0 0 40 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 2 0 0 40 1
,
 1 0 0 0 0 1 0 0 0 0 1 0 0 0 1 40
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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