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G = C5×C23⋊C8order 320 = 26·5

Direct product of C5 and C23⋊C8

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C5×C23⋊C8
 Chief series C1 — C2 — C22 — C2×C4 — C22×C4 — C22×C20 — C5×C22⋊C8 — C5×C23⋊C8
 Lower central C1 — C2 — C22 — C5×C23⋊C8
 Upper central C1 — C2×C10 — C22×C20 — C5×C23⋊C8

Generators and relations for C5×C23⋊C8
G = < a,b,c,d,e | a5=b2=c2=d2=e8=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, ebe-1=bcd, ece-1=cd=dc, de=ed >

Subgroups: 218 in 98 conjugacy classes, 38 normal (26 characteristic)
C1, C2 [×3], C2 [×4], C4 [×3], C22 [×3], C22 [×10], C5, C8 [×2], C2×C4 [×2], C2×C4 [×3], C23, C23 [×2], C23 [×4], C10 [×3], C10 [×4], C22⋊C4 [×2], C2×C8 [×2], C22×C4 [×2], C24, C20 [×3], C2×C10 [×3], C2×C10 [×10], C22⋊C8 [×2], C2×C22⋊C4, C40 [×2], C2×C20 [×2], C2×C20 [×3], C22×C10, C22×C10 [×2], C22×C10 [×4], C23⋊C8, C5×C22⋊C4 [×2], C2×C40 [×2], C22×C20 [×2], C23×C10, C5×C22⋊C8 [×2], C10×C22⋊C4, C5×C23⋊C8
Quotients: C1, C2 [×3], C4 [×2], C22, C5, C8 [×2], C2×C4, D4 [×2], C10 [×3], C22⋊C4, C2×C8, M4(2), C20 [×2], C2×C10, C22⋊C8, C23⋊C4, C4.D4, C40 [×2], C2×C20, C5×D4 [×2], C23⋊C8, C5×C22⋊C4, C2×C40, C5×M4(2), C5×C22⋊C8, C5×C23⋊C4, C5×C4.D4, C5×C23⋊C8

Smallest permutation representation of C5×C23⋊C8
On 80 points
Generators in S80
(1 54 11 23 63)(2 55 12 24 64)(3 56 13 17 57)(4 49 14 18 58)(5 50 15 19 59)(6 51 16 20 60)(7 52 9 21 61)(8 53 10 22 62)(25 47 73 33 65)(26 48 74 34 66)(27 41 75 35 67)(28 42 76 36 68)(29 43 77 37 69)(30 44 78 38 70)(31 45 79 39 71)(32 46 80 40 72)
(2 28)(3 25)(4 8)(6 32)(7 29)(9 77)(10 14)(12 76)(13 73)(16 80)(17 33)(18 22)(20 40)(21 37)(24 36)(26 30)(34 38)(42 55)(43 52)(44 48)(46 51)(47 56)(49 53)(57 65)(58 62)(60 72)(61 69)(64 68)(66 70)(74 78)
(1 5)(2 28)(3 7)(4 30)(6 32)(8 26)(9 13)(10 74)(11 15)(12 76)(14 78)(16 80)(17 21)(18 38)(19 23)(20 40)(22 34)(24 36)(25 29)(27 31)(33 37)(35 39)(41 45)(42 55)(43 47)(44 49)(46 51)(48 53)(50 54)(52 56)(57 61)(58 70)(59 63)(60 72)(62 66)(64 68)(65 69)(67 71)(73 77)(75 79)
(1 31)(2 32)(3 25)(4 26)(5 27)(6 28)(7 29)(8 30)(9 77)(10 78)(11 79)(12 80)(13 73)(14 74)(15 75)(16 76)(17 33)(18 34)(19 35)(20 36)(21 37)(22 38)(23 39)(24 40)(41 50)(42 51)(43 52)(44 53)(45 54)(46 55)(47 56)(48 49)(57 65)(58 66)(59 67)(60 68)(61 69)(62 70)(63 71)(64 72)
(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)

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

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

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

110 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E 4F 5A 5B 5C 5D 8A ··· 8H 10A ··· 10L 10M ··· 10T 10U ··· 10AB 20A ··· 20P 20Q ··· 20X 40A ··· 40AF order 1 2 2 2 2 2 2 2 4 4 4 4 4 4 5 5 5 5 8 ··· 8 10 ··· 10 10 ··· 10 10 ··· 10 20 ··· 20 20 ··· 20 40 ··· 40 size 1 1 1 1 2 2 4 4 2 2 2 2 4 4 1 1 1 1 4 ··· 4 1 ··· 1 2 ··· 2 4 ··· 4 2 ··· 2 4 ··· 4 4 ··· 4

110 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 4 4 4 4 type + + + + + + image C1 C2 C2 C4 C4 C5 C8 C10 C10 C20 C20 C40 D4 M4(2) C5×D4 C5×M4(2) C23⋊C4 C4.D4 C5×C23⋊C4 C5×C4.D4 kernel C5×C23⋊C8 C5×C22⋊C8 C10×C22⋊C4 C22×C20 C23×C10 C23⋊C8 C22×C10 C22⋊C8 C2×C22⋊C4 C22×C4 C24 C23 C2×C20 C2×C10 C2×C4 C22 C10 C10 C2 C2 # reps 1 2 1 2 2 4 8 8 4 8 8 32 2 2 8 8 1 1 4 4

Matrix representation of C5×C23⋊C8 in GL6(𝔽41)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 10 0 0 0 0 0 0 10 0 0 0 0 0 0 10 0 0 0 0 0 0 10
,
 1 0 0 0 0 0 24 40 0 0 0 0 0 0 1 0 0 0 0 0 0 40 0 0 0 0 0 0 1 0 0 0 0 0 0 40
,
 40 0 0 0 0 0 0 40 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 40 0 0 0 0 0 0 40
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40
,
 24 39 0 0 0 0 17 17 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 1 0 0 0

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,10,0,0,0,0,0,0,10,0,0,0,0,0,0,10,0,0,0,0,0,0,10],[1,24,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[24,17,0,0,0,0,39,17,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0] >;

C5×C23⋊C8 in GAP, Magma, Sage, TeX

C_5\times C_2^3\rtimes C_8
% in TeX

G:=Group("C5xC2^3:C8");
// GroupNames label

G:=SmallGroup(320,128);
// by ID

G=gap.SmallGroup(320,128);
# by ID

G:=PCGroup([7,-2,-2,-5,-2,-2,-2,-2,280,309,2803,2111,172]);
// Polycyclic

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

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