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## G = C22×C3⋊F5order 240 = 24·3·5

### Direct product of C22 and C3⋊F5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C15 — C22×C3⋊F5
 Chief series C1 — C5 — C15 — C3×D5 — C3⋊F5 — C2×C3⋊F5 — C22×C3⋊F5
 Lower central C15 — C22×C3⋊F5
 Upper central C1 — C22

Generators and relations for C22×C3⋊F5
G = < a,b,c,d,e | a2=b2=c3=d5=e4=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece-1=c-1, ede-1=d3 >

Subgroups: 416 in 108 conjugacy classes, 53 normal (13 characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, C6, C6, C2×C4, C23, D5, D5, C10, Dic3, C2×C6, C2×C6, C15, C22×C4, F5, D10, C2×C10, C2×Dic3, C22×C6, C3×D5, C3×D5, C30, C2×F5, C22×D5, C22×Dic3, C3⋊F5, C6×D5, C2×C30, C22×F5, C2×C3⋊F5, D5×C2×C6, C22×C3⋊F5
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, Dic3, D6, C22×C4, F5, C2×Dic3, C22×S3, C2×F5, C22×Dic3, C3⋊F5, C22×F5, C2×C3⋊F5, C22×C3⋊F5

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

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

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

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

C22×C3⋊F5 is a maximal subgroup of   D10.20D12  D10.10D12  C2×Dic3×F5  C3⋊D4⋊F5  C22×S3×F5
C22×C3⋊F5 is a maximal quotient of   C60.59(C2×C4)  (C2×C12)⋊6F5  Dic10.Dic3  D20.Dic3

36 conjugacy classes

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

36 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 4 4 4 4 type + + + + - + - + + image C1 C2 C2 C4 C4 S3 Dic3 D6 Dic3 F5 C2×F5 C3⋊F5 C2×C3⋊F5 kernel C22×C3⋊F5 C2×C3⋊F5 D5×C2×C6 C6×D5 C2×C30 C22×D5 D10 D10 C2×C10 C2×C6 C6 C22 C2 # reps 1 6 1 6 2 1 3 3 1 1 3 2 6

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

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 60 0 0 0 0 0 0 60 0 0 0 0 0 0 60 0 0 0 0 0 0 60
,
 60 0 0 0 0 0 0 60 0 0 0 0 0 0 60 0 0 0 0 0 0 60 0 0 0 0 0 0 60 0 0 0 0 0 0 60
,
 1 56 0 0 0 0 25 59 0 0 0 0 0 0 27 6 0 55 0 0 0 33 6 55 0 0 55 6 33 0 0 0 55 0 6 27
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 60 1 0 0 0 0 60 0 1 0 0 0 60 0 0 1 0 0 60 0 0 0
,
 11 6 0 0 0 0 0 50 0 0 0 0 0 0 0 0 60 0 0 0 60 0 0 0 0 0 0 0 0 60 0 0 0 60 0 0

G:=sub<GL(6,GF(61))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[1,25,0,0,0,0,56,59,0,0,0,0,0,0,27,0,55,55,0,0,6,33,6,0,0,0,0,6,33,6,0,0,55,55,0,27],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,60,60,60,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[11,0,0,0,0,0,6,50,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,60,0,0,60,0,0,0,0,0,0,0,60,0] >;

C22×C3⋊F5 in GAP, Magma, Sage, TeX

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

G:=Group("C2^2xC3:F5");
// GroupNames label

G:=SmallGroup(240,201);
// by ID

G=gap.SmallGroup(240,201);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-5,48,964,5189,887]);
// Polycyclic

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

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