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## G = C5×C4.10D4order 160 = 25·5

### Direct product of C5 and C4.10D4

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C5×C4.10D4
 Chief series C1 — C2 — C4 — C2×C4 — C2×C20 — C5×M4(2) — C5×C4.10D4
 Lower central C1 — C2 — C22 — C5×C4.10D4
 Upper central C1 — C10 — C2×C20 — C5×C4.10D4

Generators and relations for C5×C4.10D4
G = < a,b,c,d | a5=b4=1, c4=b2, d2=cbc-1=b-1, ab=ba, ac=ca, ad=da, bd=db, dcd-1=b-1c3 >

Smallest permutation representation of C5×C4.10D4
On 80 points
Generators in S80
(1 63 71 23 31)(2 64 72 24 32)(3 57 65 17 25)(4 58 66 18 26)(5 59 67 19 27)(6 60 68 20 28)(7 61 69 21 29)(8 62 70 22 30)(9 74 50 34 46)(10 75 51 35 47)(11 76 52 36 48)(12 77 53 37 41)(13 78 54 38 42)(14 79 55 39 43)(15 80 56 40 44)(16 73 49 33 45)
(1 3 5 7)(2 8 6 4)(9 15 13 11)(10 12 14 16)(17 19 21 23)(18 24 22 20)(25 27 29 31)(26 32 30 28)(33 35 37 39)(34 40 38 36)(41 43 45 47)(42 48 46 44)(49 51 53 55)(50 56 54 52)(57 59 61 63)(58 64 62 60)(65 67 69 71)(66 72 70 68)(73 75 77 79)(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 36 7 38 5 40 3 34)(2 37 4 35 6 33 8 39)(9 71 11 69 13 67 15 65)(10 68 16 70 14 72 12 66)(17 74 23 76 21 78 19 80)(18 75 20 73 22 79 24 77)(25 50 31 52 29 54 27 56)(26 51 28 49 30 55 32 53)(41 58 47 60 45 62 43 64)(42 59 44 57 46 63 48 61)

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

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

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

C5×C4.10D4 is a maximal subgroup of   (C2×C4).D20  (C2×Q8).D10  M4(2).21D10  D20.4D4  D20.5D4  D20.6D4  D20.7D4

55 conjugacy classes

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

55 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 4 4 type + + + + - image C1 C2 C2 C4 C5 C10 C10 C20 D4 C5×D4 C4.10D4 C5×C4.10D4 kernel C5×C4.10D4 C5×M4(2) Q8×C10 C2×C20 C4.10D4 M4(2) C2×Q8 C2×C4 C20 C4 C5 C1 # reps 1 2 1 4 4 8 4 16 2 8 1 4

Matrix representation of C5×C4.10D4 in GL4(𝔽41) generated by

 37 0 0 0 0 37 0 0 0 0 37 0 0 0 0 37
,
 9 0 0 0 21 32 0 0 0 0 32 0 0 0 20 9
,
 0 0 1 0 0 0 0 1 9 0 0 0 21 32 0 0
,
 0 0 23 33 0 0 15 18 39 31 0 0 21 2 0 0
G:=sub<GL(4,GF(41))| [37,0,0,0,0,37,0,0,0,0,37,0,0,0,0,37],[9,21,0,0,0,32,0,0,0,0,32,20,0,0,0,9],[0,0,9,21,0,0,0,32,1,0,0,0,0,1,0,0],[0,0,39,21,0,0,31,2,23,15,0,0,33,18,0,0] >;

C5×C4.10D4 in GAP, Magma, Sage, TeX

C_5\times C_4._{10}D_4
% in TeX

G:=Group("C5xC4.10D4");
// GroupNames label

G:=SmallGroup(160,51);
// by ID

G=gap.SmallGroup(160,51);
# by ID

G:=PCGroup([6,-2,-2,-5,-2,-2,-2,240,265,487,2403,1810,88]);
// Polycyclic

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

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