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## G = C5×C4.10C42order 320 = 26·5

### Direct product of C5 and C4.10C42

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C4 — C5×C4.10C42
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C22×C20 — C10×M4(2) — C5×C4.10C42
 Lower central C1 — C4 — C5×C4.10C42
 Upper central C1 — C20 — C5×C4.10C42

Generators and relations for C5×C4.10C42
G = < a,b,c,d | a5=b4=1, c4=d4=b2, ab=ba, ac=ca, ad=da, dcd-1=bc=cb, bd=db >

Subgroups: 122 in 86 conjugacy classes, 54 normal (12 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C23, C10, C10, C2×C8, M4(2), C22×C4, C20, C20, C2×C10, C2×C10, C2×M4(2), C40, C2×C20, C22×C10, C4.10C42, C2×C40, C5×M4(2), C22×C20, C10×M4(2), C5×C4.10C42
Quotients: C1, C2, C4, C22, C5, C2×C4, D4, Q8, C10, C42, C22⋊C4, C4⋊C4, C20, C2×C10, C2.C42, C2×C20, C5×D4, C5×Q8, C4.10C42, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C5×C2.C42, C5×C4.10C42

Smallest permutation representation of C5×C4.10C42
On 80 points
Generators in S80
(1 57 67 17 27)(2 58 68 18 28)(3 59 69 19 29)(4 60 70 20 30)(5 61 71 21 31)(6 62 72 22 32)(7 63 65 23 25)(8 64 66 24 26)(9 39 42 53 79)(10 40 43 54 80)(11 33 44 55 73)(12 34 45 56 74)(13 35 46 49 75)(14 36 47 50 76)(15 37 48 51 77)(16 38 41 52 78)
(1 3 5 7)(2 4 6 8)(9 15 13 11)(10 16 14 12)(17 19 21 23)(18 20 22 24)(25 27 29 31)(26 28 30 32)(33 39 37 35)(34 40 38 36)(41 47 45 43)(42 48 46 44)(49 55 53 51)(50 56 54 52)(57 59 61 63)(58 60 62 64)(65 67 69 71)(66 68 70 72)(73 79 77 75)(74 80 78 76)
(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 33 7 35 5 37 3 39)(2 36 4 34 6 40 8 38)(9 27 11 25 13 31 15 29)(10 26 16 28 14 30 12 32)(17 73 23 75 21 77 19 79)(18 76 20 74 22 80 24 78)(41 58 47 60 45 62 43 64)(42 57 44 63 46 61 48 59)(49 71 51 69 53 67 55 65)(50 70 56 72 54 66 52 68)

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

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

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

110 conjugacy classes

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E 5A 5B 5C 5D 8A ··· 8L 10A 10B 10C 10D 10E ··· 10P 20A ··· 20H 20I ··· 20T 40A ··· 40AV order 1 2 2 2 2 4 4 4 4 4 5 5 5 5 8 ··· 8 10 10 10 10 10 ··· 10 20 ··· 20 20 ··· 20 40 ··· 40 size 1 1 2 2 2 1 1 2 2 2 1 1 1 1 4 ··· 4 1 1 1 1 2 ··· 2 1 ··· 1 2 ··· 2 4 ··· 4

110 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 4 4 type + + + - image C1 C2 C4 C5 C10 C20 D4 Q8 C5×D4 C5×Q8 C4.10C42 C5×C4.10C42 kernel C5×C4.10C42 C10×M4(2) C2×C40 C4.10C42 C2×M4(2) C2×C8 C2×C20 C22×C10 C2×C4 C23 C5 C1 # reps 1 3 12 4 12 48 3 1 12 4 2 8

Matrix representation of C5×C4.10C42 in GL4(𝔽41) generated by

 18 0 0 0 0 18 0 0 0 0 18 0 0 0 0 18
,
 32 0 0 0 0 32 0 0 0 0 32 0 0 0 0 32
,
 9 8 40 8 32 9 9 0 18 0 32 8 0 23 32 32
,
 0 1 16 37 9 0 36 29 0 0 0 32 0 0 1 0
G:=sub<GL(4,GF(41))| [18,0,0,0,0,18,0,0,0,0,18,0,0,0,0,18],[32,0,0,0,0,32,0,0,0,0,32,0,0,0,0,32],[9,32,18,0,8,9,0,23,40,9,32,32,8,0,8,32],[0,9,0,0,1,0,0,0,16,36,0,1,37,29,32,0] >;

C5×C4.10C42 in GAP, Magma, Sage, TeX

C_5\times C_4._{10}C_4^2
% in TeX

G:=Group("C5xC4.10C4^2");
// GroupNames label

G:=SmallGroup(320,143);
// by ID

G=gap.SmallGroup(320,143);
# by ID

G:=PCGroup([7,-2,-2,-5,-2,-2,-2,-2,280,309,568,248,3511,172,10085,124]);
// Polycyclic

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

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