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G = C6×C11⋊C5order 330 = 2·3·5·11

Direct product of C6 and C11⋊C5

direct product, metacyclic, supersoluble, monomial, Z-group, 5-hyperelementary

Aliases: C6×C11⋊C5, C66⋊C5, C22⋊C15, C334C10, C112C30, SmallGroup(330,4)

Series: Derived Chief Lower central Upper central

C1C11 — C6×C11⋊C5
C1C11C11⋊C5C3×C11⋊C5 — C6×C11⋊C5
C11 — C6×C11⋊C5
C1C6

Generators and relations for C6×C11⋊C5
 G = < a,b,c | a6=b11=c5=1, ab=ba, ac=ca, cbc-1=b3 >

11C5
11C10
11C15
11C30

Smallest permutation representation of C6×C11⋊C5
On 66 points
Generators in S66
(1 45 23 34 12 56)(2 46 24 35 13 57)(3 47 25 36 14 58)(4 48 26 37 15 59)(5 49 27 38 16 60)(6 50 28 39 17 61)(7 51 29 40 18 62)(8 52 30 41 19 63)(9 53 31 42 20 64)(10 54 32 43 21 65)(11 55 33 44 22 66)
(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 61 62 63 64 65 66)
(2 5 6 10 4)(3 9 11 8 7)(13 16 17 21 15)(14 20 22 19 18)(24 27 28 32 26)(25 31 33 30 29)(35 38 39 43 37)(36 42 44 41 40)(46 49 50 54 48)(47 53 55 52 51)(57 60 61 65 59)(58 64 66 63 62)

G:=sub<Sym(66)| (1,45,23,34,12,56)(2,46,24,35,13,57)(3,47,25,36,14,58)(4,48,26,37,15,59)(5,49,27,38,16,60)(6,50,28,39,17,61)(7,51,29,40,18,62)(8,52,30,41,19,63)(9,53,31,42,20,64)(10,54,32,43,21,65)(11,55,33,44,22,66), (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)(56,57,58,59,60,61,62,63,64,65,66), (2,5,6,10,4)(3,9,11,8,7)(13,16,17,21,15)(14,20,22,19,18)(24,27,28,32,26)(25,31,33,30,29)(35,38,39,43,37)(36,42,44,41,40)(46,49,50,54,48)(47,53,55,52,51)(57,60,61,65,59)(58,64,66,63,62)>;

G:=Group( (1,45,23,34,12,56)(2,46,24,35,13,57)(3,47,25,36,14,58)(4,48,26,37,15,59)(5,49,27,38,16,60)(6,50,28,39,17,61)(7,51,29,40,18,62)(8,52,30,41,19,63)(9,53,31,42,20,64)(10,54,32,43,21,65)(11,55,33,44,22,66), (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)(56,57,58,59,60,61,62,63,64,65,66), (2,5,6,10,4)(3,9,11,8,7)(13,16,17,21,15)(14,20,22,19,18)(24,27,28,32,26)(25,31,33,30,29)(35,38,39,43,37)(36,42,44,41,40)(46,49,50,54,48)(47,53,55,52,51)(57,60,61,65,59)(58,64,66,63,62) );

G=PermutationGroup([[(1,45,23,34,12,56),(2,46,24,35,13,57),(3,47,25,36,14,58),(4,48,26,37,15,59),(5,49,27,38,16,60),(6,50,28,39,17,61),(7,51,29,40,18,62),(8,52,30,41,19,63),(9,53,31,42,20,64),(10,54,32,43,21,65),(11,55,33,44,22,66)], [(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),(56,57,58,59,60,61,62,63,64,65,66)], [(2,5,6,10,4),(3,9,11,8,7),(13,16,17,21,15),(14,20,22,19,18),(24,27,28,32,26),(25,31,33,30,29),(35,38,39,43,37),(36,42,44,41,40),(46,49,50,54,48),(47,53,55,52,51),(57,60,61,65,59),(58,64,66,63,62)]])

42 conjugacy classes

class 1  2 3A3B5A5B5C5D6A6B10A10B10C10D11A11B15A···15H22A22B30A···30H33A33B33C33D66A66B66C66D
order123355556610101010111115···15222230···303333333366666666
size11111111111111111111115511···115511···1155555555

42 irreducible representations

dim111111115555
type++
imageC1C2C3C5C6C10C15C30C11⋊C5C2×C11⋊C5C3×C11⋊C5C6×C11⋊C5
kernelC6×C11⋊C5C3×C11⋊C5C2×C11⋊C5C66C11⋊C5C33C22C11C6C3C2C1
# reps112424882244

Matrix representation of C6×C11⋊C5 in GL5(𝔽331)

3000000
0300000
0030000
0003000
0000300
,
00001
1000227
0100330
00101
0001226
,
1031020
106002261
226001040
104002280
2012250

G:=sub<GL(5,GF(331))| [300,0,0,0,0,0,300,0,0,0,0,0,300,0,0,0,0,0,300,0,0,0,0,0,300],[0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,1,227,330,1,226],[103,106,226,104,2,1,0,0,0,0,0,0,0,0,1,2,226,104,228,225,0,1,0,0,0] >;

C6×C11⋊C5 in GAP, Magma, Sage, TeX

C_6\times C_{11}\rtimes C_5
% in TeX

G:=Group("C6xC11:C5");
// GroupNames label

G:=SmallGroup(330,4);
// by ID

G=gap.SmallGroup(330,4);
# by ID

G:=PCGroup([4,-2,-3,-5,-11,331]);
// Polycyclic

G:=Group<a,b,c|a^6=b^11=c^5=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^3>;
// generators/relations

Export

Subgroup lattice of C6×C11⋊C5 in TeX

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