Copied to
clipboard

G = C62.58C23  order 288 = 25·32

53rd non-split extension by C62 of C23 acting via C23/C2=C22

metabelian, supersoluble, monomial

Aliases: C62.58C23, C6.10(S3×Q8), (C2×Dic6)⋊5S3, C6.141(S3×D4), Dic3⋊C4⋊18S3, C3⋊1(D6⋊3Q8), C3⋊6(D6⋊Q8), (C6×Dic6)⋊10C2, (C2×C12).227D6, (C3×Dic3).7D4, C6.11(C4○D12), C32⋊9(C22⋊Q8), (C2×Dic3).23D6, Dic3⋊Dic3⋊25C2, (C6×C12).183C22, C6.12(Q8⋊3S3), C62.C22⋊19C2, C6.D12.8C2, C6.11D12.5C2, Dic3.9(C3⋊D4), C2.15(D6.6D6), (C6×Dic3).37C22, C2.11(Dic3.D6), (C2×C4).24S32, (C2×C3⋊S3)⋊1Q8, (C3×C6).96(C2×D4), C2.15(S3×C3⋊D4), C6.35(C2×C3⋊D4), (C3×C6).28(C2×Q8), C22.105(C2×S32), (C3×Dic3⋊C4)⋊16C2, (C3×C6).35(C4○D4), (C2×C6).77(C22×S3), (C2×C6.D6).3C2, (C22×C3⋊S3).15C22, (C2×C3⋊Dic3).42C22, SmallGroup(288,536)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C62 — C62.58C23
Chief series
C1 — C3 — C32 — C3×C6 — C62 — C6×Dic3 — C2×C6.D6 — C62.58C23
Lower central
C32 — C62 — C62.58C23
Upper central
C1 — C22 — C2×C4

Generators and relations for C62.58C23
 G = < a,b,c,d,e | a6=b6=1, c2=d2=e2=b3, ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc-1=b-1, bd=db, be=eb, cd=dc, ece-1=b3c, ede-1=a3d >

Subgroups: 666 in 165 conjugacy classes, 50 normal (44 characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, Q8, C23, C32, Dic3, Dic3, C12, D6, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C22×C4, C2×Q8, C3⋊S3, C3×C6, Dic6, C4×S3, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C3×Q8, C22×S3, C22⋊Q8, C3×Dic3, C3×Dic3, C3⋊Dic3, C3×C12, C2×C3⋊S3, C2×C3⋊S3, C62, Dic3⋊C4, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C3×C4⋊C4, C2×Dic6, S3×C2×C4, C6×Q8, C6.D6, C3×Dic6, C6×Dic3, C2×C3⋊Dic3, C6×C12, C22×C3⋊S3, D6⋊Q8, D6⋊3Q8, C6.D12, Dic3⋊Dic3, C62.C22, C3×Dic3⋊C4, C6.11D12, C2×C6.D6, C6×Dic6, C62.58C23
Quotients: C1, C2, C22, S3, D4, Q8, C23, D6, C2×D4, C2×Q8, C4○D4, C3⋊D4, C22×S3, C22⋊Q8, S32, C4○D12, S3×D4, S3×Q8, Q8⋊3S3, C2×C3⋊D4, C2×S32, D6⋊Q8, D6⋊3Q8, Dic3.D6, D6.6D6, S3×C3⋊D4, C62.58C23

Smallest permutation representation of C62.58C23
►On 48 points
Generators in S48
(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)

(1 18 5 16 3 14)(2 13 6 17 4 15)(7 47 9 43 11 45)(8 48 10 44 12 46)(19 28 23 26 21 30)(20 29 24 27 22 25)(31 42 33 38 35 40)(32 37 34 39 36 41)

(1 41 16 34)(2 42 17 35)(3 37 18 36)(4 38 13 31)(5 39 14 32)(6 40 15 33)(7 20 43 27)(8 21 44 28)(9 22 45 29)(10 23 46 30)(11 24 47 25)(12 19 48 26)

(1 41 16 34)(2 40 17 33)(3 39 18 32)(4 38 13 31)(5 37 14 36)(6 42 15 35)(7 30 43 23)(8 29 44 22)(9 28 45 21)(10 27 46 20)(11 26 47 19)(12 25 48 24)

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

42 conjugacy classes

class 1 2A2B2C2D2E3A3B3C4A4B4C4D4E4F4G4H6A···6F6G6H6I12A···12H12I···12P
order122222333444444446···666612···1212···12
size11111818224466661212362···24444···412···12

42 irreducible representations

dim1111111122222222244444444
type+++++++++++-++++-+++
imageC1C2C2C2C2C2C2C2S3S3D4Q8D6D6C4○D4C3⋊D4C4○D12S32S3×D4S3×Q8Q8⋊3S3C2×S32Dic3.D6D6.6D6S3×C3⋊D4
kernelC62.58C23C6.D12Dic3⋊Dic3C62.C22C3×Dic3⋊C4C6.11D12C2×C6.D6C6×Dic6Dic3⋊C4C2×Dic6C3×Dic3C2×C3⋊S3C2×Dic3C2×C12C3×C6Dic3C6C2×C4C6C6C6C22C2C2C2
# reps1111111111224224411211222

Matrix representation of C62.58C23 ►in GL8(𝔽13)

10000000
01000000
00100000
00010000
000012100
000012000
000000120
000000012
,
120000000
012000000
001210000
001200000
00001000
00000100
00000010
00000001
,
50000000
08000000
00010000
00100000
00001000
00000100
000000120
000000012
,
50000000
05000000
00100000
00010000
00000100
00001000
000000120
00000001
,
01000000
120000000
001200000
000120000
000012000
000001200
00000001
00000010

G:=sub<GL(8,GF(13))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12],[12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[5,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12],[5,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1],[0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0] >;
 

C62.58C23 in GAP, Magma, Sage, TeX 

C_6^2._{58}C_2^3
 
% in TeX
 
G:=Group("C6^2.58C2^3");
 
// GroupNames label
 
G:=SmallGroup(288,536);
 
// by ID
 
G=gap.SmallGroup(288,536);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,120,422,135,100,1356,9414]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^6=b^6=1,c^2=d^2=e^2=b^3,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,a*e=e*a,c*b*c^-1=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e^-1=b^3*c,e*d*e^-1=a^3*d>;
 
// generators/relations
 

׿
×
⋊
ℤ
𝔽
○
≀
ℚ
◁