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## G = D5×C8.C22order 320 = 26·5

### Direct product of D5 and C8.C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — D5×C8.C22
 Chief series C1 — C5 — C10 — C20 — C4×D5 — C2×C4×D5 — C2×Q8×D5 — D5×C8.C22
 Lower central C5 — C10 — C20 — D5×C8.C22
 Upper central C1 — C2 — C2×C4 — C8.C22

Generators and relations for D5×C8.C22
G = < a,b,c,d,e | a5=b2=c8=d2=e2=1, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd=c3, ece=c5, ede=c4d >

Subgroups: 942 in 258 conjugacy classes, 101 normal (51 characteristic)
C1, C2, C2 [×6], C4 [×2], C4 [×8], C22, C22 [×8], C5, C8 [×2], C8 [×2], C2×C4, C2×C4 [×16], D4, D4 [×6], Q8, Q8 [×2], Q8 [×10], C23 [×2], D5 [×2], D5 [×2], C10, C10 [×2], C2×C8 [×2], M4(2), M4(2) [×3], SD16 [×2], SD16 [×6], Q16 [×2], Q16 [×6], C22×C4 [×3], C2×D4 [×2], C2×Q8, C2×Q8 [×9], C4○D4, C4○D4 [×5], Dic5 [×2], Dic5 [×3], C20 [×2], C20 [×3], D10 [×2], D10 [×5], C2×C10, C2×C10, C2×M4(2), C2×SD16 [×2], C2×Q16 [×2], C8.C22, C8.C22 [×7], C22×Q8, C2×C4○D4, C52C8 [×2], C40 [×2], Dic10, Dic10 [×2], Dic10 [×6], C4×D5 [×4], C4×D5 [×7], D20, D20, C2×Dic5, C2×Dic5 [×2], C5⋊D4 [×3], C2×C20, C2×C20 [×2], C5×D4, C5×D4, C5×Q8, C5×Q8 [×2], C5×Q8, C22×D5, C22×D5, C2×C8.C22, C8×D5 [×2], C8⋊D5 [×2], C40⋊C2 [×2], Dic20 [×2], C4.Dic5, D4.D5 [×2], Q8⋊D5 [×2], C5⋊Q16 [×4], C5×M4(2), C5×SD16 [×2], C5×Q16 [×2], C2×Dic10, C2×Dic10, C2×C4×D5, C2×C4×D5 [×2], C4○D20, C4○D20, D4×D5, D4×D5, D42D5, D42D5, Q8×D5, Q8×D5 [×4], Q8×D5 [×2], Q82D5, Q8×C10, C5×C4○D4, D5×M4(2), C8.D10, D5×SD16 [×2], SD16⋊D5 [×2], D5×Q16 [×2], Q16⋊D5 [×2], C20.C23, D4.9D10, C5×C8.C22, C2×Q8×D5, D5×C4○D4, D5×C8.C22
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C24, D10 [×7], C8.C22 [×2], C22×D4, C22×D5 [×7], C2×C8.C22, D4×D5 [×2], C23×D5, C2×D4×D5, D5×C8.C22

Smallest permutation representation of D5×C8.C22
On 80 points
Generators in S80
(1 75 62 35 11)(2 76 63 36 12)(3 77 64 37 13)(4 78 57 38 14)(5 79 58 39 15)(6 80 59 40 16)(7 73 60 33 9)(8 74 61 34 10)(17 26 69 52 44)(18 27 70 53 45)(19 28 71 54 46)(20 29 72 55 47)(21 30 65 56 48)(22 31 66 49 41)(23 32 67 50 42)(24 25 68 51 43)
(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 9)(8 10)(17 52)(18 53)(19 54)(20 55)(21 56)(22 49)(23 50)(24 51)(25 68)(26 69)(27 70)(28 71)(29 72)(30 65)(31 66)(32 67)(33 73)(34 74)(35 75)(36 76)(37 77)(38 78)(39 79)(40 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 4)(3 7)(6 8)(9 13)(10 16)(12 14)(17 23)(19 21)(20 24)(25 29)(26 32)(28 30)(33 37)(34 40)(36 38)(42 44)(43 47)(46 48)(50 52)(51 55)(54 56)(57 63)(59 61)(60 64)(65 71)(67 69)(68 72)(73 77)(74 80)(76 78)
(1 68)(2 65)(3 70)(4 67)(5 72)(6 69)(7 66)(8 71)(9 31)(10 28)(11 25)(12 30)(13 27)(14 32)(15 29)(16 26)(17 40)(18 37)(19 34)(20 39)(21 36)(22 33)(23 38)(24 35)(41 60)(42 57)(43 62)(44 59)(45 64)(46 61)(47 58)(48 63)(49 73)(50 78)(51 75)(52 80)(53 77)(54 74)(55 79)(56 76)

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

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

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

44 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 5A 5B 8A 8B 8C 8D 10A 10B 10C 10D 10E 10F 20A 20B 20C 20D 20E ··· 20J 40A 40B 40C 40D order 1 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 4 4 5 5 8 8 8 8 10 10 10 10 10 10 20 20 20 20 20 ··· 20 40 40 40 40 size 1 1 2 4 5 5 10 20 2 2 4 4 4 10 10 20 20 20 2 2 4 4 20 20 2 2 4 4 8 8 4 4 4 4 8 ··· 8 8 8 8 8

44 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 8 type + + + + + + + + + + + + + + + + + + + + + - + + - image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 D4 D4 D4 D5 D10 D10 D10 D10 D10 C8.C22 D4×D5 D4×D5 D5×C8.C22 kernel D5×C8.C22 D5×M4(2) C8.D10 D5×SD16 SD16⋊D5 D5×Q16 Q16⋊D5 C20.C23 D4.9D10 C5×C8.C22 C2×Q8×D5 D5×C4○D4 C4×D5 C2×Dic5 C22×D5 C8.C22 M4(2) SD16 Q16 C2×Q8 C4○D4 D5 C4 C22 C1 # reps 1 1 1 2 2 2 2 1 1 1 1 1 2 1 1 2 2 4 4 2 2 2 2 2 2

Matrix representation of D5×C8.C22 in GL8(𝔽41)

 0 1 0 0 0 0 0 0 40 34 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 40 34 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 20 31 39 4 0 0 0 0 9 26 0 10 0 0 0 0 2 30 0 21 0 0 0 0 36 3 10 36
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 5 40 0 0 0 0 0 0 0 38 40 2 0 0 0 0 28 38 0 1
,
 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 39 20 21 20 0 0 0 0 31 16 32 18 0 0 0 0 20 31 39 4 0 0 0 0 36 17 5 29

G:=sub<GL(8,GF(41))| [0,40,0,0,0,0,0,0,1,34,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,1,34,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,20,9,2,36,0,0,0,0,31,26,30,3,0,0,0,0,39,0,0,10,0,0,0,0,4,10,21,36],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,5,0,28,0,0,0,0,0,40,38,38,0,0,0,0,0,0,40,0,0,0,0,0,0,0,2,1],[40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,39,31,20,36,0,0,0,0,20,16,31,17,0,0,0,0,21,32,39,5,0,0,0,0,20,18,4,29] >;

D5×C8.C22 in GAP, Magma, Sage, TeX

D_5\times C_8.C_2^2
% in TeX

G:=Group("D5xC8.C2^2");
// GroupNames label

G:=SmallGroup(320,1448);
// by ID

G=gap.SmallGroup(320,1448);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,184,570,185,438,235,102,12550]);
// Polycyclic

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

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