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G = C62.35C23  order 288 = 25·32

30th non-split extension by C62 of C23 acting via C23/C2=C22

metabelian, supersoluble, monomial

Aliases: C62.35C23, C6.5(S3×Q8), C6.47(S3×D4), Dic3⋊C4⋊11S3, C3⋊1(D6⋊Q8), (C2×C12).191D6, C3⋊Dic3.55D4, C6.25(C4○D12), C32⋊6(C22⋊Q8), C2.8(Dic3⋊D6), (C2×Dic3).13D6, (C6×C12).218C22, C62.C22⋊22C2, C6.D12.3C2, C2.7(Dic3.D6), C2.12(D6.D6), (C6×Dic3).77C22, (C2×C4).91S32, (C2×C3⋊S3)⋊5Q8, C22.92(C2×S32), (C3×C6).86(C2×D4), (C3×C6).19(C2×Q8), (C3×Dic3⋊C4)⋊8C2, (C2×C32⋊2Q8)⋊3C2, (C3×C6).22(C4○D4), (C2×C6).54(C22×S3), (C22×C3⋊S3).65C22, (C2×C3⋊Dic3).120C22, (C2×C4×C3⋊S3).17C2, SmallGroup(288,513)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C62 — C62.35C23
Chief series
C1 — C3 — C32 — C3×C6 — C62 — C6×Dic3 — C62.C22 — C62.35C23
Lower central
C32 — C62 — C62.35C23
Upper central
C1 — C22 — C2×C4

Generators and relations for C62.35C23
 G = < a,b,c,d,e | a6=b6=1, c2=d2=b3, e2=a3, ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc-1=b-1, bd=db, be=eb, dcd-1=a3c, ece-1=b3c, ede-1=b3d >

Subgroups: 722 in 175 conjugacy classes, 48 normal (18 characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, Q8, C23, C32, 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, C22×S3, C22⋊Q8, C3×Dic3, C3⋊Dic3, C3×C12, C2×C3⋊S3, C2×C3⋊S3, C62, Dic3⋊C4, Dic3⋊C4, D6⋊C4, C3×C4⋊C4, C2×Dic6, S3×C2×C4, C32⋊2Q8, C6×Dic3, C4×C3⋊S3, C2×C3⋊Dic3, C6×C12, C22×C3⋊S3, D6⋊Q8, C6.D12, C62.C22, C3×Dic3⋊C4, C2×C32⋊2Q8, C2×C4×C3⋊S3, C62.35C23
Quotients: C1, C2, C22, S3, D4, Q8, C23, D6, C2×D4, C2×Q8, C4○D4, C22×S3, C22⋊Q8, S32, C4○D12, S3×D4, S3×Q8, C2×S32, D6⋊Q8, Dic3.D6, D6.D6, Dic3⋊D6, C62.35C23

Smallest permutation representation of C62.35C23
►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 46 16 10)(2 47 17 11)(3 48 18 12)(4 43 13 7)(5 44 14 8)(6 45 15 9)(19 36 26 37)(20 31 27 38)(21 32 28 39)(22 33 29 40)(23 34 30 41)(24 35 25 42)

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

42 conjugacy classes

class 1 2A2B2C2D2E3A3B3C4A4B4C4D4E4F4G4H6A···6F6G6H6I12A···12H12I···12P
order122222333444444446···666612···1212···12
size11111818224221212121218182···24444···412···12

42 irreducible representations

dim11111122222224444444
type++++++++-++++-++
imageC1C2C2C2C2C2S3D4Q8D6D6C4○D4C4○D12S32S3×D4S3×Q8C2×S32Dic3.D6D6.D6Dic3⋊D6
kernelC62.35C23C6.D12C62.C22C3×Dic3⋊C4C2×C32⋊2Q8C2×C4×C3⋊S3Dic3⋊C4C3⋊Dic3C2×C3⋊S3C2×Dic3C2×C12C3×C6C6C2×C4C6C6C22C2C2C2
# reps12121122242281221222

Matrix representation of C62.35C23 ►in GL8(𝔽13)

120000000
012000000
00100000
00010000
000012100
000012000
000000120
000000012
,
120000000
012000000
00010000
0012120000
00001000
00000100
00000010
00000001
,
05000000
50000000
00100000
0012120000
00001000
00000100
00000008
00000050
,
01000000
120000000
001200000
000120000
000001200
000012000
00000001
00000010
,
80000000
05000000
001200000
000120000
000012000
000001200
00000080
00000008

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

C62.35C23 in GAP, Magma, Sage, TeX 

C_6^2._{35}C_2^3
 
% in TeX
 
G:=Group("C6^2.35C2^3");
 
// GroupNames label
 
G:=SmallGroup(288,513);
 
// by ID
 
G=gap.SmallGroup(288,513);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,141,120,422,219,58,1356,9414]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^6=b^6=1,c^2=d^2=b^3,e^2=a^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,d*c*d^-1=a^3*c,e*c*e^-1=b^3*c,e*d*e^-1=b^3*d>;
 
// generators/relations
 

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