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G = C5×C22.11C24order 320 = 26·5

Direct product of C5 and C22.11C24

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

Aliases: C5×C22.11C24, C10.1492+ 1+4, D47(C2×C20), (C4×D4)⋊5C10, (C2×D4)⋊11C20, (D4×C10)⋊35C4, (D4×C20)⋊34C2, C233(C2×C20), C424(C2×C10), (C4×C20)⋊38C22, C2.7(C23×C20), C42⋊C26C10, C24.11(C2×C10), C4.19(C22×C20), C10.80(C23×C4), (C22×C20)⋊5C22, (C22×D4).9C10, (C2×C10).338C24, (C2×C20).709C23, C20.223(C22×C4), C22.2(C22×C20), C2.1(C5×2+ 1+4), (D4×C10).332C22, C23.30(C22×C10), C22.11(C23×C10), (C23×C10).11C22, (C22×C10).254C23, (C2×C4)⋊4(C2×C20), C4⋊C420(C2×C10), (C2×C20)⋊38(C2×C4), (C5×D4)⋊37(C2×C4), (D4×C2×C10).22C2, (C2×C22⋊C4)⋊5C10, (C5×C4⋊C4)⋊77C22, (C22×C4)⋊3(C2×C10), (C10×C22⋊C4)⋊10C2, C22⋊C418(C2×C10), (C22×C10)⋊12(C2×C4), (C2×D4).78(C2×C10), (C5×C42⋊C2)⋊27C2, (C5×C22⋊C4)⋊72C22, (C2×C4).56(C22×C10), (C2×C10).134(C22×C4), SmallGroup(320,1520)

Series: Derived Chief Lower central Upper central

C1C2 — C5×C22.11C24
C1C2C22C2×C10C2×C20C5×C22⋊C4D4×C20 — C5×C22.11C24
C1C2 — C5×C22.11C24
C1C2×C10 — C5×C22.11C24

Generators and relations for C5×C22.11C24
 G = < a,b,c,d,e,f,g | a5=b2=c2=e2=f2=g2=1, d2=c, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, ede=gdg=bd=db, fef=be=eb, bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, cg=gc, df=fd, eg=ge, fg=gf >

Subgroups: 514 in 338 conjugacy classes, 242 normal (14 characteristic)
C1, C2, C2 [×2], C2 [×10], C4 [×4], C4 [×8], C22, C22 [×10], C22 [×18], C5, C2×C4 [×14], C2×C4 [×8], D4 [×16], C23, C23 [×12], C23 [×4], C10, C10 [×2], C10 [×10], C42 [×4], C22⋊C4 [×12], C4⋊C4 [×4], C22×C4, C22×C4 [×8], C2×D4 [×12], C24 [×2], C20 [×4], C20 [×8], C2×C10, C2×C10 [×10], C2×C10 [×18], C2×C22⋊C4 [×4], C42⋊C2 [×2], C4×D4 [×8], C22×D4, C2×C20 [×14], C2×C20 [×8], C5×D4 [×16], C22×C10, C22×C10 [×12], C22×C10 [×4], C22.11C24, C4×C20 [×4], C5×C22⋊C4 [×12], C5×C4⋊C4 [×4], C22×C20, C22×C20 [×8], D4×C10 [×12], C23×C10 [×2], C10×C22⋊C4 [×4], C5×C42⋊C2 [×2], D4×C20 [×8], D4×C2×C10, C5×C22.11C24
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C5, C2×C4 [×28], C23 [×15], C10 [×15], C22×C4 [×14], C24, C20 [×8], C2×C10 [×35], C23×C4, 2+ 1+4 [×2], C2×C20 [×28], C22×C10 [×15], C22.11C24, C22×C20 [×14], C23×C10, C23×C20, C5×2+ 1+4 [×2], C5×C22.11C24

Smallest permutation representation of C5×C22.11C24
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)
(1 35)(2 31)(3 32)(4 33)(5 34)(6 80)(7 76)(8 77)(9 78)(10 79)(11 17)(12 18)(13 19)(14 20)(15 16)(21 27)(22 28)(23 29)(24 30)(25 26)(36 55)(37 51)(38 52)(39 53)(40 54)(41 47)(42 48)(43 49)(44 50)(45 46)(56 75)(57 71)(58 72)(59 73)(60 74)(61 67)(62 68)(63 69)(64 70)(65 66)
(1 21)(2 22)(3 23)(4 24)(5 25)(6 20)(7 16)(8 17)(9 18)(10 19)(11 77)(12 78)(13 79)(14 80)(15 76)(26 34)(27 35)(28 31)(29 32)(30 33)(36 41)(37 42)(38 43)(39 44)(40 45)(46 54)(47 55)(48 51)(49 52)(50 53)(56 61)(57 62)(58 63)(59 64)(60 65)(66 74)(67 75)(68 71)(69 72)(70 73)
(1 61 21 56)(2 62 22 57)(3 63 23 58)(4 64 24 59)(5 65 25 60)(6 54 20 46)(7 55 16 47)(8 51 17 48)(9 52 18 49)(10 53 19 50)(11 42 77 37)(12 43 78 38)(13 44 79 39)(14 45 80 40)(15 41 76 36)(26 74 34 66)(27 75 35 67)(28 71 31 68)(29 72 32 69)(30 73 33 70)
(36 55)(37 51)(38 52)(39 53)(40 54)(41 47)(42 48)(43 49)(44 50)(45 46)(56 75)(57 71)(58 72)(59 73)(60 74)(61 67)(62 68)(63 69)(64 70)(65 66)
(1 36)(2 37)(3 38)(4 39)(5 40)(6 74)(7 75)(8 71)(9 72)(10 73)(11 62)(12 63)(13 64)(14 65)(15 61)(16 67)(17 68)(18 69)(19 70)(20 66)(21 41)(22 42)(23 43)(24 44)(25 45)(26 46)(27 47)(28 48)(29 49)(30 50)(31 51)(32 52)(33 53)(34 54)(35 55)(56 76)(57 77)(58 78)(59 79)(60 80)
(6 80)(7 76)(8 77)(9 78)(10 79)(11 17)(12 18)(13 19)(14 20)(15 16)(56 75)(57 71)(58 72)(59 73)(60 74)(61 67)(62 68)(63 69)(64 70)(65 66)

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

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

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

170 conjugacy classes

class 1 2A2B2C2D···2M4A···4T5A5B5C5D10A···10L10M···10AZ20A···20CB
order12222···24···4555510···1010···1020···20
size11112···22···211111···12···22···2

170 irreducible representations

dim11111111111144
type++++++
imageC1C2C2C2C2C4C5C10C10C10C10C202+ 1+4C5×2+ 1+4
kernelC5×C22.11C24C10×C22⋊C4C5×C42⋊C2D4×C20D4×C2×C10D4×C10C22.11C24C2×C22⋊C4C42⋊C2C4×D4C22×D4C2×D4C10C2
# reps142811641683246428

Matrix representation of C5×C22.11C24 in GL5(𝔽41)

370000
01000
00100
00010
00001
,
10000
040000
004000
000400
000040
,
400000
01000
00100
00010
00001
,
320000
00010
00001
01000
00100
,
400000
01000
004000
000400
00001
,
10000
00100
01000
00001
00010
,
10000
01000
00100
000400
000040

G:=sub<GL(5,GF(41))| [37,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,40],[40,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[32,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,1,0,0,0,0,0,1,0,0],[40,0,0,0,0,0,1,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,1,0],[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] >;

C5×C22.11C24 in GAP, Magma, Sage, TeX

C_5\times C_2^2._{11}C_2^4
% in TeX

G:=Group("C5xC2^2.11C2^4");
// GroupNames label

G:=SmallGroup(320,1520);
// by ID

G=gap.SmallGroup(320,1520);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-5,-2,-2,1120,1149,891,2467]);
// Polycyclic

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

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