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## G = C3×C22⋊F5order 240 = 24·3·5

### Direct product of C3 and C22⋊F5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C3×C22⋊F5
 Chief series C1 — C5 — C10 — D10 — C6×D5 — C6×F5 — C3×C22⋊F5
 Lower central C5 — C10 — C3×C22⋊F5
 Upper central C1 — C6 — C2×C6

Generators and relations for C3×C22⋊F5
G = < a,b,c,d,e | a3=b2=c2=d5=e4=1, ab=ba, ac=ca, ad=da, ae=ea, ebe-1=bc=cb, bd=db, cd=dc, ce=ec, ede-1=d3 >

Subgroups: 236 in 68 conjugacy classes, 28 normal (20 characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, C6, C6, C2×C4, C23, D5, D5, C10, C10, C12, C2×C6, C2×C6, C15, C22⋊C4, F5, D10, D10, C2×C10, C2×C12, C22×C6, C3×D5, C3×D5, C30, C30, C2×F5, C22×D5, C3×C22⋊C4, C3×F5, C6×D5, C6×D5, C2×C30, C22⋊F5, C6×F5, D5×C2×C6, C3×C22⋊F5
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, C12, C2×C6, C22⋊C4, F5, C2×C12, C3×D4, C2×F5, C3×C22⋊C4, C3×F5, C22⋊F5, C6×F5, C3×C22⋊F5

Smallest permutation representation of C3×C22⋊F5
On 60 points
Generators in S60
(1 24 14)(2 25 15)(3 21 11)(4 22 12)(5 23 13)(6 26 16)(7 27 17)(8 28 18)(9 29 19)(10 30 20)(31 51 41)(32 52 42)(33 53 43)(34 54 44)(35 55 45)(36 56 46)(37 57 47)(38 58 48)(39 59 49)(40 60 50)
(1 34)(2 35)(3 31)(4 32)(5 33)(6 36)(7 37)(8 38)(9 39)(10 40)(11 41)(12 42)(13 43)(14 44)(15 45)(16 46)(17 47)(18 48)(19 49)(20 50)(21 51)(22 52)(23 53)(24 54)(25 55)(26 56)(27 57)(28 58)(29 59)(30 60)
(1 9)(2 10)(3 6)(4 7)(5 8)(11 16)(12 17)(13 18)(14 19)(15 20)(21 26)(22 27)(23 28)(24 29)(25 30)(31 36)(32 37)(33 38)(34 39)(35 40)(41 46)(42 47)(43 48)(44 49)(45 50)(51 56)(52 57)(53 58)(54 59)(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)
(1 34 9 39)(2 31 8 37)(3 33 7 40)(4 35 6 38)(5 32 10 36)(11 43 17 50)(12 45 16 48)(13 42 20 46)(14 44 19 49)(15 41 18 47)(21 53 27 60)(22 55 26 58)(23 52 30 56)(24 54 29 59)(25 51 28 57)

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

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

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

C3×C22⋊F5 is a maximal subgroup of   D10.D12  D10.4D12  C22⋊F5.S3  C3⋊D4⋊F5  C3×D4×F5

42 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 4A 4B 4C 4D 5 6A 6B 6C 6D 6E 6F 6G 6H 6I 6J 10A 10B 10C 12A ··· 12H 15A 15B 30A ··· 30F order 1 2 2 2 2 2 3 3 4 4 4 4 5 6 6 6 6 6 6 6 6 6 6 10 10 10 12 ··· 12 15 15 30 ··· 30 size 1 1 2 5 5 10 1 1 10 10 10 10 4 1 1 2 2 5 5 5 5 10 10 4 4 4 10 ··· 10 4 4 4 ··· 4

42 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 4 4 4 4 4 4 type + + + + + + + image C1 C2 C2 C3 C4 C4 C6 C6 C12 C12 D4 C3×D4 F5 C2×F5 C3×F5 C22⋊F5 C6×F5 C3×C22⋊F5 kernel C3×C22⋊F5 C6×F5 D5×C2×C6 C22⋊F5 C6×D5 C2×C30 C2×F5 C22×D5 D10 C2×C10 C3×D5 D5 C2×C6 C6 C22 C3 C2 C1 # reps 1 2 1 2 2 2 4 2 4 4 2 4 1 1 2 2 2 4

Matrix representation of C3×C22⋊F5 in GL6(𝔽61)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 13 0 0 0 0 0 0 13 0 0 0 0 0 0 13 0 0 0 0 0 0 13
,
 1 59 0 0 0 0 0 60 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 60 0 0 0 0 0 0 60 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 60 60 60 60
,
 60 2 0 0 0 0 60 1 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0

G:=sub<GL(6,GF(61))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,13],[1,0,0,0,0,0,59,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,60,0,0,1,0,0,60,0,0,0,1,0,60,0,0,0,0,1,60],[60,60,0,0,0,0,2,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,1,0] >;

C3×C22⋊F5 in GAP, Magma, Sage, TeX

C_3\times C_2^2\rtimes F_5
% in TeX

G:=Group("C3xC2^2:F5");
// GroupNames label

G:=SmallGroup(240,117);
// by ID

G=gap.SmallGroup(240,117);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-2,-5,72,313,3461,599]);
// Polycyclic

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

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