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## G = C42×F5order 320 = 26·5

### Direct product of C42 and F5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5 — C42×F5
 Chief series C1 — C5 — D5 — D10 — C22×D5 — C22×F5 — C2×C4×F5 — C42×F5
 Lower central C5 — C42×F5
 Upper central C1 — C42

Generators and relations for C42×F5
G = < a,b,c,d | a4=b4=c5=d4=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c3 >

Subgroups: 714 in 258 conjugacy classes, 144 normal (9 characteristic)
C1, C2 [×3], C2 [×4], C4 [×6], C4 [×22], C22, C22 [×6], C5, C2×C4 [×3], C2×C4 [×39], C23, D5 [×4], C10 [×3], C42, C42 [×27], C22×C4 [×7], Dic5 [×6], C20 [×6], F5 [×16], D10 [×6], C2×C10, C2×C42 [×7], C4×D5 [×12], C2×Dic5 [×3], C2×C20 [×3], C2×F5 [×24], C22×D5, C43, C4×Dic5 [×3], C4×C20, C4×F5 [×24], C2×C4×D5 [×3], C22×F5 [×4], D5×C42, C2×C4×F5 [×6], C42×F5
Quotients: C1, C2 [×7], C4 [×28], C22 [×7], C2×C4 [×42], C23, C42 [×28], C22×C4 [×7], F5, C2×C42 [×7], C2×F5 [×3], C43, C4×F5 [×6], C22×F5, C2×C4×F5 [×3], C42×F5

Smallest permutation representation of C42×F5
On 80 points
Generators in S80
(1 46 6 41)(2 47 7 42)(3 48 8 43)(4 49 9 44)(5 50 10 45)(11 56 16 51)(12 57 17 52)(13 58 18 53)(14 59 19 54)(15 60 20 55)(21 66 26 61)(22 67 27 62)(23 68 28 63)(24 69 29 64)(25 70 30 65)(31 76 36 71)(32 77 37 72)(33 78 38 73)(34 79 39 74)(35 80 40 75)
(1 36 11 26)(2 37 12 27)(3 38 13 28)(4 39 14 29)(5 40 15 30)(6 31 16 21)(7 32 17 22)(8 33 18 23)(9 34 19 24)(10 35 20 25)(41 76 51 66)(42 77 52 67)(43 78 53 68)(44 79 54 69)(45 80 55 70)(46 71 56 61)(47 72 57 62)(48 73 58 63)(49 74 59 64)(50 75 60 65)
(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 61 16 76)(2 63 20 79)(3 65 19 77)(4 62 18 80)(5 64 17 78)(6 66 11 71)(7 68 15 74)(8 70 14 72)(9 67 13 75)(10 69 12 73)(21 51 36 46)(22 53 40 49)(23 55 39 47)(24 52 38 50)(25 54 37 48)(26 56 31 41)(27 58 35 44)(28 60 34 42)(29 57 33 45)(30 59 32 43)

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

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

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

80 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A ··· 4L 4M ··· 4BD 5 10A 10B 10C 20A ··· 20L order 1 2 2 2 2 2 2 2 4 ··· 4 4 ··· 4 5 10 10 10 20 ··· 20 size 1 1 1 1 5 5 5 5 1 ··· 1 5 ··· 5 4 4 4 4 4 ··· 4

80 irreducible representations

 dim 1 1 1 1 1 1 4 4 4 type + + + + + image C1 C2 C2 C4 C4 C4 F5 C2×F5 C4×F5 kernel C42×F5 D5×C42 C2×C4×F5 C4×Dic5 C4×C20 C4×F5 C42 C2×C4 C4 # reps 1 1 6 6 2 48 1 3 12

Matrix representation of C42×F5 in GL6(𝔽41)

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

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

C42×F5 in GAP, Magma, Sage, TeX

C_4^2\times F_5
% in TeX

G:=Group("C4^2xF5");
// GroupNames label

G:=SmallGroup(320,1023);
// by ID

G=gap.SmallGroup(320,1023);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,232,100,6278,1595]);
// Polycyclic

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

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