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## G = C5×C42.3C4order 320 = 26·5

### Direct product of C5 and C42.3C4

direct product, metabelian, nilpotent (class 4), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C4 — C5×C42.3C4
 Chief series C1 — C2 — C22 — C2×C4 — C2×Q8 — Q8×C10 — C5×C4.10D4 — C5×C42.3C4
 Lower central C1 — C2 — C22 — C2×C4 — C5×C42.3C4
 Upper central C1 — C10 — C2×C10 — Q8×C10 — C5×C42.3C4

Generators and relations for C5×C42.3C4
G = < a,b,c,d | a5=b4=c4=1, d4=c2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1c, dcd-1=b2c >

Subgroups: 114 in 60 conjugacy classes, 26 normal (18 characteristic)
C1, C2, C2, C4, C22, C5, C8, C2×C4, C2×C4, C2×C4, Q8, C10, C10, C42, C4⋊C4, M4(2), C2×Q8, C20, C2×C10, C4.10D4, C4⋊Q8, C40, C2×C20, C2×C20, C2×C20, C5×Q8, C42.3C4, C4×C20, C5×C4⋊C4, C5×M4(2), Q8×C10, C5×C4.10D4, C5×C4⋊Q8, C5×C42.3C4
Quotients: C1, C2, C4, C22, C5, C2×C4, D4, C10, C22⋊C4, C20, C2×C10, C23⋊C4, C2×C20, C5×D4, C42.3C4, C5×C22⋊C4, C5×C23⋊C4, C5×C42.3C4

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

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

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

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

65 conjugacy classes

 class 1 2A 2B 4A ··· 4E 4F 5A 5B 5C 5D 8A 8B 8C 8D 10A 10B 10C 10D 10E 10F 10G 10H 20A ··· 20T 20U 20V 20W 20X 40A ··· 40P order 1 2 2 4 ··· 4 4 5 5 5 5 8 8 8 8 10 10 10 10 10 10 10 10 20 ··· 20 20 20 20 20 40 ··· 40 size 1 1 2 4 ··· 4 8 1 1 1 1 8 8 8 8 1 1 1 1 2 2 2 2 4 ··· 4 8 8 8 8 8 ··· 8

65 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 4 4 4 4 type + + + + + - image C1 C2 C2 C4 C4 C5 C10 C10 C20 C20 D4 C5×D4 C23⋊C4 C42.3C4 C5×C23⋊C4 C5×C42.3C4 kernel C5×C42.3C4 C5×C4.10D4 C5×C4⋊Q8 C4×C20 Q8×C10 C42.3C4 C4.10D4 C4⋊Q8 C42 C2×Q8 C2×C20 C2×C4 C10 C5 C2 C1 # reps 1 2 1 2 2 4 8 4 8 8 2 8 1 2 4 8

Matrix representation of C5×C42.3C4 in GL4(𝔽41) generated by

 16 0 0 0 0 16 0 0 0 0 16 0 0 0 0 16
,
 1 0 0 0 0 1 0 0 0 0 0 32 0 0 32 0
,
 0 9 0 0 9 0 0 0 0 0 0 32 0 0 32 0
,
 0 0 1 0 0 0 0 1 0 1 0 0 40 0 0 0
G:=sub<GL(4,GF(41))| [16,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[1,0,0,0,0,1,0,0,0,0,0,32,0,0,32,0],[0,9,0,0,9,0,0,0,0,0,0,32,0,0,32,0],[0,0,0,40,0,0,1,0,1,0,0,0,0,1,0,0] >;

C5×C42.3C4 in GAP, Magma, Sage, TeX

C_5\times C_4^2._3C_4
% in TeX

G:=Group("C5xC4^2.3C4");
// GroupNames label

G:=SmallGroup(320,161);
// by ID

G=gap.SmallGroup(320,161);
# by ID

G:=PCGroup([7,-2,-2,-5,-2,-2,-2,-2,280,309,1128,2803,2530,248,4911,375,172,10085]);
// Polycyclic

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

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