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## G = C5×C42⋊C3order 240 = 24·3·5

### Direct product of C5 and C42⋊C3

Aliases: C5×C42⋊C3, C42⋊C15, (C4×C20)⋊C3, (C2×C10).A4, C22.(C5×A4), SmallGroup(240,32)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C42 — C5×C42⋊C3
 Chief series C1 — C22 — C42 — C4×C20 — C5×C42⋊C3
 Lower central C42 — C5×C42⋊C3
 Upper central C1 — C5

Generators and relations for C5×C42⋊C3
G = < a,b,c,d | a5=b4=c4=d3=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=bc-1, dcd-1=b-1c2 >

Smallest permutation representation of C5×C42⋊C3
On 60 points
Generators in S60
(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)
(1 26 21 31)(2 27 22 32)(3 28 23 33)(4 29 24 34)(5 30 25 35)(6 56)(7 57)(8 58)(9 59)(10 60)(11 16)(12 17)(13 18)(14 19)(15 20)(36 46 41 51)(37 47 42 52)(38 48 43 53)(39 49 44 54)(40 50 45 55)
(1 31 21 26)(2 32 22 27)(3 33 23 28)(4 34 24 29)(5 35 25 30)(6 16 56 11)(7 17 57 12)(8 18 58 13)(9 19 59 14)(10 20 60 15)
(1 56 36)(2 57 37)(3 58 38)(4 59 39)(5 60 40)(6 41 21)(7 42 22)(8 43 23)(9 44 24)(10 45 25)(11 46 26)(12 47 27)(13 48 28)(14 49 29)(15 50 30)(16 51 31)(17 52 32)(18 53 33)(19 54 34)(20 55 35)

G:=sub<Sym(60)| (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), (1,26,21,31)(2,27,22,32)(3,28,23,33)(4,29,24,34)(5,30,25,35)(6,56)(7,57)(8,58)(9,59)(10,60)(11,16)(12,17)(13,18)(14,19)(15,20)(36,46,41,51)(37,47,42,52)(38,48,43,53)(39,49,44,54)(40,50,45,55), (1,31,21,26)(2,32,22,27)(3,33,23,28)(4,34,24,29)(5,35,25,30)(6,16,56,11)(7,17,57,12)(8,18,58,13)(9,19,59,14)(10,20,60,15), (1,56,36)(2,57,37)(3,58,38)(4,59,39)(5,60,40)(6,41,21)(7,42,22)(8,43,23)(9,44,24)(10,45,25)(11,46,26)(12,47,27)(13,48,28)(14,49,29)(15,50,30)(16,51,31)(17,52,32)(18,53,33)(19,54,34)(20,55,35)>;

G:=Group( (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), (1,26,21,31)(2,27,22,32)(3,28,23,33)(4,29,24,34)(5,30,25,35)(6,56)(7,57)(8,58)(9,59)(10,60)(11,16)(12,17)(13,18)(14,19)(15,20)(36,46,41,51)(37,47,42,52)(38,48,43,53)(39,49,44,54)(40,50,45,55), (1,31,21,26)(2,32,22,27)(3,33,23,28)(4,34,24,29)(5,35,25,30)(6,16,56,11)(7,17,57,12)(8,18,58,13)(9,19,59,14)(10,20,60,15), (1,56,36)(2,57,37)(3,58,38)(4,59,39)(5,60,40)(6,41,21)(7,42,22)(8,43,23)(9,44,24)(10,45,25)(11,46,26)(12,47,27)(13,48,28)(14,49,29)(15,50,30)(16,51,31)(17,52,32)(18,53,33)(19,54,34)(20,55,35) );

G=PermutationGroup([[(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)], [(1,26,21,31),(2,27,22,32),(3,28,23,33),(4,29,24,34),(5,30,25,35),(6,56),(7,57),(8,58),(9,59),(10,60),(11,16),(12,17),(13,18),(14,19),(15,20),(36,46,41,51),(37,47,42,52),(38,48,43,53),(39,49,44,54),(40,50,45,55)], [(1,31,21,26),(2,32,22,27),(3,33,23,28),(4,34,24,29),(5,35,25,30),(6,16,56,11),(7,17,57,12),(8,18,58,13),(9,19,59,14),(10,20,60,15)], [(1,56,36),(2,57,37),(3,58,38),(4,59,39),(5,60,40),(6,41,21),(7,42,22),(8,43,23),(9,44,24),(10,45,25),(11,46,26),(12,47,27),(13,48,28),(14,49,29),(15,50,30),(16,51,31),(17,52,32),(18,53,33),(19,54,34),(20,55,35)]])

C5×C42⋊C3 is a maximal subgroup of   C42⋊D15  C204D4⋊C3  (C4×C20)⋊C6

40 conjugacy classes

 class 1 2 3A 3B 4A 4B 4C 4D 5A 5B 5C 5D 10A 10B 10C 10D 15A ··· 15H 20A ··· 20P order 1 2 3 3 4 4 4 4 5 5 5 5 10 10 10 10 15 ··· 15 20 ··· 20 size 1 3 16 16 3 3 3 3 1 1 1 1 3 3 3 3 16 ··· 16 3 ··· 3

40 irreducible representations

 dim 1 1 1 1 3 3 3 3 type + + image C1 C3 C5 C15 A4 C42⋊C3 C5×A4 C5×C42⋊C3 kernel C5×C42⋊C3 C4×C20 C42⋊C3 C42 C2×C10 C5 C22 C1 # reps 1 2 4 8 1 4 4 16

Matrix representation of C5×C42⋊C3 in GL3(𝔽61) generated by

 58 0 0 0 58 0 0 0 58
,
 50 0 0 0 50 0 0 0 60
,
 11 0 0 0 1 0 0 0 50
,
 0 1 0 0 0 1 1 0 0
G:=sub<GL(3,GF(61))| [58,0,0,0,58,0,0,0,58],[50,0,0,0,50,0,0,0,60],[11,0,0,0,1,0,0,0,50],[0,0,1,1,0,0,0,1,0] >;

C5×C42⋊C3 in GAP, Magma, Sage, TeX

C_5\times C_4^2\rtimes C_3
% in TeX

G:=Group("C5xC4^2:C3");
// GroupNames label

G:=SmallGroup(240,32);
// by ID

G=gap.SmallGroup(240,32);
# by ID

G:=PCGroup([6,-3,-5,-2,2,-2,2,542,230,3963,69,3604,6485]);
// Polycyclic

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

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