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## G = C4×C3⋊F5order 240 = 24·3·5

### Direct product of C4 and C3⋊F5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C15 — C4×C3⋊F5
 Chief series C1 — C5 — C15 — C3×D5 — C6×D5 — C2×C3⋊F5 — C4×C3⋊F5
 Lower central C15 — C4×C3⋊F5
 Upper central C1 — C4

Generators and relations for C4×C3⋊F5
G = < a,b,c,d | a4=b3=c5=d4=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c3 >

Subgroups: 240 in 60 conjugacy classes, 29 normal (19 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, C6, C6, C2×C4, D5, C10, Dic3, C12, C12, C2×C6, C15, C42, Dic5, C20, F5, D10, C2×Dic3, C2×C12, C3×D5, C30, C4×D5, C2×F5, C4×Dic3, C3×Dic5, C60, C3⋊F5, C6×D5, C4×F5, D5×C12, C2×C3⋊F5, C4×C3⋊F5
Quotients: C1, C2, C4, C22, S3, C2×C4, Dic3, D6, C42, F5, C4×S3, C2×Dic3, C2×F5, C4×Dic3, C3⋊F5, C4×F5, C2×C3⋊F5, C4×C3⋊F5

Smallest permutation representation of C4×C3⋊F5
On 60 points
Generators in S60
(1 49 19 34)(2 50 20 35)(3 46 16 31)(4 47 17 32)(5 48 18 33)(6 51 21 36)(7 52 22 37)(8 53 23 38)(9 54 24 39)(10 55 25 40)(11 56 26 41)(12 57 27 42)(13 58 28 43)(14 59 29 44)(15 60 30 45)
(1 9 14)(2 10 15)(3 6 11)(4 7 12)(5 8 13)(16 21 26)(17 22 27)(18 23 28)(19 24 29)(20 25 30)(31 36 41)(32 37 42)(33 38 43)(34 39 44)(35 40 45)(46 51 56)(47 52 57)(48 53 58)(49 54 59)(50 55 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)
(2 3 5 4)(6 13 7 15)(8 12 10 11)(9 14)(16 18 17 20)(21 28 22 30)(23 27 25 26)(24 29)(31 33 32 35)(36 43 37 45)(38 42 40 41)(39 44)(46 48 47 50)(51 58 52 60)(53 57 55 56)(54 59)

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

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

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

C4×C3⋊F5 is a maximal subgroup of
C30.C42  C30.4C42  D124F5  D602C4  C24⋊F5  Dic10⋊Dic3  D202Dic3  Dic65F5  (C4×S3)⋊F5  C4×S3×F5  D603C4  (C2×C12)⋊6F5
C4×C3⋊F5 is a maximal quotient of
C24⋊F5  C30.11C42  D10.10D12

36 conjugacy classes

 class 1 2A 2B 2C 3 4A 4B 4C 4D 4E ··· 4L 5 6A 6B 6C 10 12A 12B 12C 12D 15A 15B 20A 20B 30A 30B 60A 60B 60C 60D order 1 2 2 2 3 4 4 4 4 4 ··· 4 5 6 6 6 10 12 12 12 12 15 15 20 20 30 30 60 60 60 60 size 1 1 5 5 2 1 1 5 5 15 ··· 15 4 2 10 10 4 2 2 10 10 4 4 4 4 4 4 4 4 4 4

36 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 4 4 4 4 4 4 type + + + + - - + + + image C1 C2 C2 C4 C4 C4 S3 Dic3 Dic3 D6 C4×S3 F5 C2×F5 C3⋊F5 C4×F5 C2×C3⋊F5 C4×C3⋊F5 kernel C4×C3⋊F5 D5×C12 C2×C3⋊F5 C3×Dic5 C60 C3⋊F5 C4×D5 Dic5 C20 D10 D5 C12 C6 C4 C3 C2 C1 # reps 1 1 2 2 2 8 1 1 1 1 4 1 1 2 2 2 4

Matrix representation of C4×C3⋊F5 in GL4(𝔽61) generated by

 50 0 0 0 0 50 0 0 0 0 50 0 0 0 0 50
,
 27 0 55 55 6 33 6 0 0 6 33 6 55 55 0 27
,
 0 1 0 0 0 0 1 0 0 0 0 1 60 60 60 60
,
 1 0 0 0 0 0 0 1 0 1 0 0 60 60 60 60
G:=sub<GL(4,GF(61))| [50,0,0,0,0,50,0,0,0,0,50,0,0,0,0,50],[27,6,0,55,0,33,6,55,55,6,33,0,55,0,6,27],[0,0,0,60,1,0,0,60,0,1,0,60,0,0,1,60],[1,0,0,60,0,0,1,60,0,0,0,60,0,1,0,60] >;

C4×C3⋊F5 in GAP, Magma, Sage, TeX

C_4\times C_3\rtimes F_5
% in TeX

G:=Group("C4xC3:F5");
// GroupNames label

G:=SmallGroup(240,120);
// by ID

G=gap.SmallGroup(240,120);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-5,24,55,964,5189,1745]);
// Polycyclic

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

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