Copied to
clipboard

G = C12xF5order 240 = 24·3·5

Direct product of C12 and F5

direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary

Aliases: C12xF5, C60:4C4, C20:2C12, C15:3C42, Dic5:2C12, C5:(C4xC12), D5.(C2xC12), C2.2(C6xF5), (C4xD5).6C6, (C2xF5).2C6, (C6xF5).4C2, C6.17(C2xF5), C10.3(C2xC12), C30.17(C2xC4), (C3xDic5):6C4, D10.4(C2xC6), (D5xC12).13C2, (C6xD5).23C22, (C3xD5).3(C2xC4), SmallGroup(240,113)

Series: Derived Chief Lower central Upper central

C1C5 — C12xF5
C1C5C10D10C6xD5C6xF5 — C12xF5
C5 — C12xF5
C1C12

Generators and relations for C12xF5
 G = < a,b,c | a12=b5=c4=1, ab=ba, ac=ca, cbc-1=b3 >

Subgroups: 156 in 60 conjugacy classes, 36 normal (20 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, C6, C6, C2xC4, D5, C10, C12, C12, C2xC6, C15, C42, Dic5, C20, F5, D10, C2xC12, C3xD5, C30, C4xD5, C2xF5, C4xC12, C3xDic5, C60, C3xF5, C6xD5, C4xF5, D5xC12, C6xF5, C12xF5
Quotients: C1, C2, C3, C4, C22, C6, C2xC4, C12, C2xC6, C42, F5, C2xC12, C2xF5, C4xC12, C3xF5, C4xF5, C6xF5, C12xF5

Smallest permutation representation of C12xF5
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 17 54 33 43)(2 18 55 34 44)(3 19 56 35 45)(4 20 57 36 46)(5 21 58 25 47)(6 22 59 26 48)(7 23 60 27 37)(8 24 49 28 38)(9 13 50 29 39)(10 14 51 30 40)(11 15 52 31 41)(12 16 53 32 42)
(1 7)(2 8)(3 9)(4 10)(5 11)(6 12)(13 56 39 35)(14 57 40 36)(15 58 41 25)(16 59 42 26)(17 60 43 27)(18 49 44 28)(19 50 45 29)(20 51 46 30)(21 52 47 31)(22 53 48 32)(23 54 37 33)(24 55 38 34)

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

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

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

C12xF5 is a maximal subgroup of   C30.3C42  D12:2F5  D60:5C4  (C4xS3):F5

60 conjugacy classes

class 1 2A2B2C3A3B4A4B4C···4L 5 6A6B6C6D6E6F 10 12A12B12C12D12E···12X15A15B20A20B30A30B60A60B60C60D
order122233444···45666666101212121212···1215152020303060606060
size115511115···54115555411115···54444444444

60 irreducible representations

dim111111111111444444
type+++++
imageC1C2C2C3C4C4C4C6C6C12C12C12F5C2xF5C3xF5C4xF5C6xF5C12xF5
kernelC12xF5D5xC12C6xF5C4xF5C3xDic5C60C3xF5C4xD5C2xF5Dic5C20F5C12C6C4C3C2C1
# reps1122228244416112224

Matrix representation of C12xF5 in GL5(F61)

500000
013000
001300
000130
000013
,
10000
000060
010060
001060
000160
,
110000
00010
01000
00001
00100

G:=sub<GL(5,GF(61))| [50,0,0,0,0,0,13,0,0,0,0,0,13,0,0,0,0,0,13,0,0,0,0,0,13],[1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,60,60,60,60],[11,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,1,0,0,0,0,0,0,1,0] >;

C12xF5 in GAP, Magma, Sage, TeX

C_{12}\times F_5
% in TeX

G:=Group("C12xF5");
// GroupNames label

G:=SmallGroup(240,113);
// by ID

G=gap.SmallGroup(240,113);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-2,-5,72,151,3461,599]);
// Polycyclic

G:=Group<a,b,c|a^12=b^5=c^4=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^3>;
// generators/relations

׿
x
:
Z
F
o
wr
Q
<