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G = C5×C24⋊3C4order 320 = 26·5

Direct product of C5 and C24⋊3C4

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C5×C24⋊3C4
 Chief series C1 — C2 — C22 — C23 — C22×C10 — C22×C20 — C10×C22⋊C4 — C5×C24⋊3C4
 Lower central C1 — C22 — C5×C24⋊3C4
 Upper central C1 — C22×C10 — C5×C24⋊3C4

Generators and relations for C5×C243C4
G = < a,b,c,d,e,f | a5=b2=c2=d2=e2=f4=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, fbf-1=bd=db, be=eb, cd=dc, fcf-1=ce=ec, de=ed, df=fd, ef=fe >

Subgroups: 898 in 506 conjugacy classes, 130 normal (10 characteristic)
C1, C2, C2, C4, C22, C22, C22, C5, C2×C4, C23, C23, C23, C10, C10, C22⋊C4, C22×C4, C24, C24, C20, C2×C10, C2×C10, C2×C10, C2×C22⋊C4, C25, C2×C20, C22×C10, C22×C10, C22×C10, C243C4, C5×C22⋊C4, C22×C20, C23×C10, C23×C10, C10×C22⋊C4, C24×C10, C5×C243C4
Quotients: C1, C2, C4, C22, C5, C2×C4, D4, C23, C10, C22⋊C4, C22×C4, C2×D4, C20, C2×C10, C2×C22⋊C4, C22≀C2, C2×C20, C5×D4, C22×C10, C243C4, C5×C22⋊C4, C22×C20, D4×C10, C10×C22⋊C4, C5×C22≀C2, C5×C243C4

Smallest permutation representation of C5×C243C4
On 80 points
Generators in S80
(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)
(6 60)(7 56)(8 57)(9 58)(10 59)(16 61)(17 62)(18 63)(19 64)(20 65)(21 75)(22 71)(23 72)(24 73)(25 74)(31 76)(32 77)(33 78)(34 79)(35 80)
(6 23)(7 24)(8 25)(9 21)(10 22)(16 33)(17 34)(18 35)(19 31)(20 32)(56 73)(57 74)(58 75)(59 71)(60 72)(61 78)(62 79)(63 80)(64 76)(65 77)
(1 42)(2 43)(3 44)(4 45)(5 41)(6 60)(7 56)(8 57)(9 58)(10 59)(11 53)(12 54)(13 55)(14 51)(15 52)(16 61)(17 62)(18 63)(19 64)(20 65)(21 75)(22 71)(23 72)(24 73)(25 74)(26 69)(27 70)(28 66)(29 67)(30 68)(31 76)(32 77)(33 78)(34 79)(35 80)(36 46)(37 47)(38 48)(39 49)(40 50)
(1 66)(2 67)(3 68)(4 69)(5 70)(6 23)(7 24)(8 25)(9 21)(10 22)(11 36)(12 37)(13 38)(14 39)(15 40)(16 33)(17 34)(18 35)(19 31)(20 32)(26 45)(27 41)(28 42)(29 43)(30 44)(46 53)(47 54)(48 55)(49 51)(50 52)(56 73)(57 74)(58 75)(59 71)(60 72)(61 78)(62 79)(63 80)(64 76)(65 77)
(1 65 12 7)(2 61 13 8)(3 62 14 9)(4 63 15 10)(5 64 11 6)(16 55 57 43)(17 51 58 44)(18 52 59 45)(19 53 60 41)(20 54 56 42)(21 68 79 39)(22 69 80 40)(23 70 76 36)(24 66 77 37)(25 67 78 38)(26 35 50 71)(27 31 46 72)(28 32 47 73)(29 33 48 74)(30 34 49 75)

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

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

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

140 conjugacy classes

 class 1 2A ··· 2G 2H ··· 2S 4A ··· 4H 5A 5B 5C 5D 10A ··· 10AB 10AC ··· 10BX 20A ··· 20AF order 1 2 ··· 2 2 ··· 2 4 ··· 4 5 5 5 5 10 ··· 10 10 ··· 10 20 ··· 20 size 1 1 ··· 1 2 ··· 2 4 ··· 4 1 1 1 1 1 ··· 1 2 ··· 2 4 ··· 4

140 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 type + + + + image C1 C2 C2 C4 C5 C10 C10 C20 D4 C5×D4 kernel C5×C24⋊3C4 C10×C22⋊C4 C24×C10 C23×C10 C24⋊3C4 C2×C22⋊C4 C25 C24 C22×C10 C23 # reps 1 6 1 8 4 24 4 32 12 48

Matrix representation of C5×C243C4 in GL5(𝔽41)

 1 0 0 0 0 0 16 0 0 0 0 0 16 0 0 0 0 0 1 0 0 0 0 0 1
,
 40 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 39 40
,
 40 0 0 0 0 0 1 0 0 0 0 0 40 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 40 0 0 0 0 0 40
,
 1 0 0 0 0 0 40 0 0 0 0 0 40 0 0 0 0 0 1 0 0 0 0 0 1
,
 9 0 0 0 0 0 0 1 0 0 0 40 0 0 0 0 0 0 40 40 0 0 0 0 1

G:=sub<GL(5,GF(41))| [1,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,1,0,0,0,0,0,1],[40,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,39,0,0,0,0,40],[40,0,0,0,0,0,1,0,0,0,0,0,40,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,40,0,0,0,0,0,40],[1,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,1,0,0,0,0,0,1],[9,0,0,0,0,0,0,40,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,40,1] >;

C5×C243C4 in GAP, Magma, Sage, TeX

C_5\times C_2^4\rtimes_3C_4
% in TeX

G:=Group("C5xC2^4:3C4");
// GroupNames label

G:=SmallGroup(320,880);
// by ID

G=gap.SmallGroup(320,880);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,1120,589,1766]);
// Polycyclic

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

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