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G = C62.67C23  order 288 = 25·32

62nd non-split extension by C62 of C23 acting via C23/C2=C22

metabelian, supersoluble, monomial

Aliases: C62.67C23, C6.51(S3×D4), Dic3⋊C4⋊7S3, (C2×C12).25D6, (C6×C12).7C22, C6.13(C4○D12), C3⋊1(D6.D4), (C2×Dic3).28D6, C6.D12⋊17C2, C2.10(Dic3⋊D6), C6.14(Q8⋊3S3), C2.16(D6.6D6), (C6×Dic3).82C22, C32⋊8(C22.D4), (C2×C4).30S32, (C2×C3⋊S3).20D4, (C3×C6).98(C2×D4), C22.111(C2×S32), (C2×C12⋊S3).3C2, (C3×Dic3⋊C4)⋊20C2, (C3×C6).40(C4○D4), (C2×C6.D6)⋊10C2, (C2×C6).86(C22×S3), (C22×C3⋊S3).19C22, SmallGroup(288,545)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C62 — C62.67C23
Chief series
C1 — C3 — C32 — C3×C6 — C62 — C6×Dic3 — C2×C6.D6 — C62.67C23
Lower central
C32 — C62 — C62.67C23
Upper central
C1 — C22 — C2×C4

Generators and relations for C62.67C23
 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=a3c, ede-1=a3b3d >

Subgroups: 898 in 183 conjugacy classes, 46 normal (14 characteristic)
C1, C2, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, C23, C32, Dic3, C12, D6, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C3⋊S3, C3×C6, C3×C6, C4×S3, D12, C2×Dic3, C2×C12, C2×C12, C22×S3, C22.D4, C3×Dic3, C3×C12, C2×C3⋊S3, C2×C3⋊S3, C62, Dic3⋊C4, D6⋊C4, C3×C4⋊C4, S3×C2×C4, C2×D12, C6.D6, C6×Dic3, C12⋊S3, C6×C12, C22×C3⋊S3, D6.D4, C6.D12, C6.D12, C3×Dic3⋊C4, C2×C6.D6, C2×C12⋊S3, C62.67C23
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, C22×S3, C22.D4, S32, C4○D12, S3×D4, Q8⋊3S3, C2×S32, D6.D4, D6.6D6, Dic3⋊D6, C62.67C23

Smallest permutation representation of C62.67C23
►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 43 9 45 11 47)(8 44 10 46 12 48)(19 30 23 28 21 26)(20 25 24 29 22 27)(31 40 33 42 35 38)(32 41 34 37 36 39)

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

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

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

42 conjugacy classes

class 1 2A2B2C2D2E2F3A3B3C4A4B4C4D4E4F4G6A···6F6G6H6I12A···12H12I···12P
order122222233344444446···666612···1212···12
size11111818362244666612122···24444···412···12

42 irreducible representations

dim11111222222444444
type+++++++++++++++
imageC1C2C2C2C2S3D4D6D6C4○D4C4○D12S32S3×D4Q8⋊3S3C2×S32D6.6D6Dic3⋊D6
kernelC62.67C23C6.D12C3×Dic3⋊C4C2×C6.D6C2×C12⋊S3Dic3⋊C4C2×C3⋊S3C2×Dic3C2×C12C3×C6C6C2×C4C6C6C22C2C2
# reps13211224248122142

Matrix representation of C62.67C23 ►in GL8(𝔽13)

120000000
012000000
00100000
00010000
000012100
000012000
000000120
000000012
,
120000000
012000000
001210000
001200000
00001000
00000100
00000010
00000001
,
80000000
05000000
000120000
001200000
000012000
000001200
000000120
00000001
,
80000000
08000000
001200000
000120000
000001200
000012000
00000010
000000012
,
01000000
120000000
001200000
000120000
000012000
000001200
00000001
00000010

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],[12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[8,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1],[8,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12],[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] >;
 

C62.67C23 in GAP, Magma, Sage, TeX 

C_6^2._{67}C_2^3
 
% in TeX
 
G:=Group("C6^2.67C2^3");
 
// GroupNames label
 
G:=SmallGroup(288,545);
 
// by ID
 
G=gap.SmallGroup(288,545);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,120,254,303,100,1356,9414]);
 
// 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=a^3*c,e*d*e^-1=a^3*b^3*d>;
 
// generators/relations
 

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