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## G = C22×C4×F5order 320 = 26·5

### Direct product of C22×C4 and F5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5 — C22×C4×F5
 Chief series C1 — C5 — D5 — D10 — C2×F5 — C22×F5 — C23×F5 — C22×C4×F5
 Lower central C5 — C22×C4×F5
 Upper central C1 — C22×C4

Generators and relations for C22×C4×F5
G = < a,b,c,d,e | a2=b2=c4=d5=e4=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=d3 >

Subgroups: 1386 in 498 conjugacy classes, 276 normal (13 characteristic)
C1, C2, C2 [×6], C2 [×8], C4 [×4], C4 [×20], C22 [×7], C22 [×28], C5, C2×C4 [×6], C2×C4 [×78], C23, C23 [×14], D5 [×8], C10, C10 [×6], C42 [×16], C22×C4, C22×C4 [×41], C24, Dic5 [×4], C20 [×4], F5 [×16], D10, D10 [×27], C2×C10 [×7], C2×C42 [×12], C23×C4 [×3], C4×D5 [×16], C2×Dic5 [×6], C2×C20 [×6], C2×F5 [×56], C22×D5 [×14], C22×C10, C22×C42, C4×F5 [×16], C2×C4×D5 [×12], C22×Dic5, C22×C20, C22×F5 [×28], C23×D5, C2×C4×F5 [×12], D5×C22×C4, C23×F5 [×2], C22×C4×F5
Quotients: C1, C2 [×15], C4 [×24], C22 [×35], C2×C4 [×84], C23 [×15], C42 [×16], C22×C4 [×42], C24, F5, C2×C42 [×12], C23×C4 [×3], C2×F5 [×7], C22×C42, C4×F5 [×4], C22×F5 [×7], C2×C4×F5 [×6], C23×F5, C22×C4×F5

Smallest permutation representation of C22×C4×F5
On 80 points
Generators in S80
(1 46)(2 47)(3 48)(4 49)(5 50)(6 41)(7 42)(8 43)(9 44)(10 45)(11 56)(12 57)(13 58)(14 59)(15 60)(16 51)(17 52)(18 53)(19 54)(20 55)(21 66)(22 67)(23 68)(24 69)(25 70)(26 61)(27 62)(28 63)(29 64)(30 65)(31 76)(32 77)(33 78)(34 79)(35 80)(36 71)(37 72)(38 73)(39 74)(40 75)
(1 26)(2 27)(3 28)(4 29)(5 30)(6 21)(7 22)(8 23)(9 24)(10 25)(11 36)(12 37)(13 38)(14 39)(15 40)(16 31)(17 32)(18 33)(19 34)(20 35)(41 66)(42 67)(43 68)(44 69)(45 70)(46 61)(47 62)(48 63)(49 64)(50 65)(51 76)(52 77)(53 78)(54 79)(55 80)(56 71)(57 72)(58 73)(59 74)(60 75)
(1 11 6 16)(2 12 7 17)(3 13 8 18)(4 14 9 19)(5 15 10 20)(21 31 26 36)(22 32 27 37)(23 33 28 38)(24 34 29 39)(25 35 30 40)(41 51 46 56)(42 52 47 57)(43 53 48 58)(44 54 49 59)(45 55 50 60)(61 71 66 76)(62 72 67 77)(63 73 68 78)(64 74 69 79)(65 75 70 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 71 6 76)(2 73 10 79)(3 75 9 77)(4 72 8 80)(5 74 7 78)(11 66 16 61)(12 68 20 64)(13 70 19 62)(14 67 18 65)(15 69 17 63)(21 51 26 56)(22 53 30 59)(23 55 29 57)(24 52 28 60)(25 54 27 58)(31 46 36 41)(32 48 40 44)(33 50 39 42)(34 47 38 45)(35 49 37 43)

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

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

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

80 conjugacy classes

 class 1 2A ··· 2G 2H ··· 2O 4A ··· 4H 4I ··· 4AV 5 10A ··· 10G 20A ··· 20H order 1 2 ··· 2 2 ··· 2 4 ··· 4 4 ··· 4 5 10 ··· 10 20 ··· 20 size 1 1 ··· 1 5 ··· 5 1 ··· 1 5 ··· 5 4 4 ··· 4 4 ··· 4

80 irreducible representations

 dim 1 1 1 1 1 1 1 1 4 4 4 4 type + + + + + + + image C1 C2 C2 C2 C4 C4 C4 C4 F5 C2×F5 C2×F5 C4×F5 kernel C22×C4×F5 C2×C4×F5 D5×C22×C4 C23×F5 C2×C4×D5 C22×Dic5 C22×C20 C22×F5 C22×C4 C2×C4 C23 C22 # reps 1 12 1 2 12 2 2 32 1 6 1 8

Matrix representation of C22×C4×F5 in GL6(𝔽41)

 40 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 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 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 40 0 0 0 0 0 0 32 0 0 0 0 0 0 32 0 0 0 0 0 0 32 0 0 0 0 0 0 32
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 40 0 0 1 0 0 40 0 0 0 1 0 40 0 0 0 0 1 40
,
 9 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 32 0 0 0 32 0 0 0 0 0 0 0 0 32 0 0 0 32 0 0

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,40,0,0,0,0,0,0,32,0,0,0,0,0,0,32,0,0,0,0,0,0,32,0,0,0,0,0,0,32],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,40,40,40,40],[9,0,0,0,0,0,0,9,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,32,0,0,32,0,0,0,0,0,0,0,32,0] >;

C22×C4×F5 in GAP, Magma, Sage, TeX

C_2^2\times C_4\times F_5
% in TeX

G:=Group("C2^2xC4xF5");
// GroupNames label

G:=SmallGroup(320,1590);
// by ID

G=gap.SmallGroup(320,1590);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,136,6278,818]);
// Polycyclic

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

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