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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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C5×C22.11C24
 Chief series C1 — C2 — C22 — C2×C10 — C2×C20 — C5×C22⋊C4 — D4×C20 — C5×C22.11C24
 Lower central C1 — C2 — C5×C22.11C24
 Upper central C1 — C2×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, C2, C4, C4, C22, C22, C22, C5, C2×C4, C2×C4, D4, C23, C23, C23, C10, C10, C10, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C24, C20, C20, C2×C10, C2×C10, C2×C10, C2×C22⋊C4, C42⋊C2, C4×D4, C22×D4, C2×C20, C2×C20, C5×D4, C22×C10, C22×C10, C22×C10, C22.11C24, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C22×C20, C22×C20, D4×C10, C23×C10, C10×C22⋊C4, C5×C42⋊C2, D4×C20, D4×C2×C10, C5×C22.11C24
Quotients: C1, C2, C4, C22, C5, C2×C4, C23, C10, C22×C4, C24, C20, C2×C10, C23×C4, 2+ 1+4, C2×C20, C22×C10, C22.11C24, C22×C20, C23×C10, C23×C20, C5×2+ 1+4, 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 26)(2 27)(3 28)(4 29)(5 30)(6 16)(7 17)(8 18)(9 19)(10 20)(11 76)(12 77)(13 78)(14 79)(15 80)(21 31)(22 32)(23 33)(24 34)(25 35)(36 46)(37 47)(38 48)(39 49)(40 50)(41 51)(42 52)(43 53)(44 54)(45 55)(56 66)(57 67)(58 68)(59 69)(60 70)(61 71)(62 72)(63 73)(64 74)(65 75)
(1 21)(2 22)(3 23)(4 24)(5 25)(6 76)(7 77)(8 78)(9 79)(10 80)(11 16)(12 17)(13 18)(14 19)(15 20)(26 31)(27 32)(28 33)(29 34)(30 35)(36 41)(37 42)(38 43)(39 44)(40 45)(46 51)(47 52)(48 53)(49 54)(50 55)(56 61)(57 62)(58 63)(59 64)(60 65)(66 71)(67 72)(68 73)(69 74)(70 75)
(1 61 21 56)(2 62 22 57)(3 63 23 58)(4 64 24 59)(5 65 25 60)(6 41 76 36)(7 42 77 37)(8 43 78 38)(9 44 79 39)(10 45 80 40)(11 46 16 51)(12 47 17 52)(13 48 18 53)(14 49 19 54)(15 50 20 55)(26 71 31 66)(27 72 32 67)(28 73 33 68)(29 74 34 69)(30 75 35 70)
(36 46)(37 47)(38 48)(39 49)(40 50)(41 51)(42 52)(43 53)(44 54)(45 55)(56 66)(57 67)(58 68)(59 69)(60 70)(61 71)(62 72)(63 73)(64 74)(65 75)
(1 36)(2 37)(3 38)(4 39)(5 40)(6 61)(7 62)(8 63)(9 64)(10 65)(11 66)(12 67)(13 68)(14 69)(15 70)(16 71)(17 72)(18 73)(19 74)(20 75)(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 16)(7 17)(8 18)(9 19)(10 20)(11 76)(12 77)(13 78)(14 79)(15 80)(56 66)(57 67)(58 68)(59 69)(60 70)(61 71)(62 72)(63 73)(64 74)(65 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), (1,26)(2,27)(3,28)(4,29)(5,30)(6,16)(7,17)(8,18)(9,19)(10,20)(11,76)(12,77)(13,78)(14,79)(15,80)(21,31)(22,32)(23,33)(24,34)(25,35)(36,46)(37,47)(38,48)(39,49)(40,50)(41,51)(42,52)(43,53)(44,54)(45,55)(56,66)(57,67)(58,68)(59,69)(60,70)(61,71)(62,72)(63,73)(64,74)(65,75), (1,21)(2,22)(3,23)(4,24)(5,25)(6,76)(7,77)(8,78)(9,79)(10,80)(11,16)(12,17)(13,18)(14,19)(15,20)(26,31)(27,32)(28,33)(29,34)(30,35)(36,41)(37,42)(38,43)(39,44)(40,45)(46,51)(47,52)(48,53)(49,54)(50,55)(56,61)(57,62)(58,63)(59,64)(60,65)(66,71)(67,72)(68,73)(69,74)(70,75), (1,61,21,56)(2,62,22,57)(3,63,23,58)(4,64,24,59)(5,65,25,60)(6,41,76,36)(7,42,77,37)(8,43,78,38)(9,44,79,39)(10,45,80,40)(11,46,16,51)(12,47,17,52)(13,48,18,53)(14,49,19,54)(15,50,20,55)(26,71,31,66)(27,72,32,67)(28,73,33,68)(29,74,34,69)(30,75,35,70), (36,46)(37,47)(38,48)(39,49)(40,50)(41,51)(42,52)(43,53)(44,54)(45,55)(56,66)(57,67)(58,68)(59,69)(60,70)(61,71)(62,72)(63,73)(64,74)(65,75), (1,36)(2,37)(3,38)(4,39)(5,40)(6,61)(7,62)(8,63)(9,64)(10,65)(11,66)(12,67)(13,68)(14,69)(15,70)(16,71)(17,72)(18,73)(19,74)(20,75)(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), (6,16)(7,17)(8,18)(9,19)(10,20)(11,76)(12,77)(13,78)(14,79)(15,80)(56,66)(57,67)(58,68)(59,69)(60,70)(61,71)(62,72)(63,73)(64,74)(65,75)>;

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

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

170 conjugacy classes

 class 1 2A 2B 2C 2D ··· 2M 4A ··· 4T 5A 5B 5C 5D 10A ··· 10L 10M ··· 10AZ 20A ··· 20CB order 1 2 2 2 2 ··· 2 4 ··· 4 5 5 5 5 10 ··· 10 10 ··· 10 20 ··· 20 size 1 1 1 1 2 ··· 2 2 ··· 2 1 1 1 1 1 ··· 1 2 ··· 2 2 ··· 2

170 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 4 4 type + + + + + + image C1 C2 C2 C2 C2 C4 C5 C10 C10 C10 C10 C20 2+ 1+4 C5×2+ 1+4 kernel C5×C22.11C24 C10×C22⋊C4 C5×C42⋊C2 D4×C20 D4×C2×C10 D4×C10 C22.11C24 C2×C22⋊C4 C42⋊C2 C4×D4 C22×D4 C2×D4 C10 C2 # reps 1 4 2 8 1 16 4 16 8 32 4 64 2 8

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

 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

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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