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G = C62.51C23  order 288 = 25·32

46th non-split extension by C62 of C23 acting via C23/C2=C22

metabelian, supersoluble, monomial

Aliases: C62.51C23, D6⋊C4⋊9S3, D6⋊2(C4×S3), Dic32⋊1C2, C32⋊4(C4×D4), C6.48(S3×D4), C3⋊D12⋊2C4, C3⋊Dic3⋊11D4, Dic3⋊1(C4×S3), Dic3⋊C4⋊13S3, (C2×C12).198D6, C3⋊2(Dic3⋊5D4), C6.7(D4⋊2S3), C2.1(Dic3⋊D6), (C2×Dic3).64D6, (C22×S3).33D6, C3⋊2(Dic3⋊4D4), C2.4(D12⋊S3), C6.D12⋊12C2, (C6×C12).229C22, C6.30(Q8⋊3S3), (C6×Dic3).59C22, C2.17(C4×S32), (C2×C4).94S32, C6.16(S3×C2×C4), (S3×C6)⋊3(C2×C4), (C2×S3×Dic3)⋊9C2, (C3×D6⋊C4)⋊22C2, C22.31(C2×S32), (C3×C6).92(C2×D4), (C3×Dic3)⋊2(C2×C4), (S3×C2×C6).14C22, (C3×Dic3⋊C4)⋊12C2, (C2×C3⋊D12).7C2, (C3×C6).30(C4○D4), (C3×C6).15(C22×C4), (C2×C6).70(C22×S3), (C22×C3⋊S3).68C22, (C2×C3⋊Dic3).128C22, (C2×C4×C3⋊S3)⋊12C2, (C2×C3⋊S3)⋊7(C2×C4), SmallGroup(288,529)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3×C6 — C62.51C23
Chief series
C1 — C3 — C32 — C3×C6 — C62 — S3×C2×C6 — C2×S3×Dic3 — C62.51C23
Lower central
C32 — C3×C6 — C62.51C23
Upper central
C1 — C22 — C2×C4

Generators and relations for C62.51C23
 G = < a,b,c,d,e | a6=b6=c2=1, d2=e2=b3, ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ece-1=a3c, ede-1=a3d >

Subgroups: 850 in 215 conjugacy classes, 60 normal (44 characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, C23, C32, Dic3, Dic3, C12, D6, D6, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C3×S3, C3⋊S3, C3×C6, C4×S3, D12, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C22×S3, C22×S3, C22×C6, C4×D4, C3×Dic3, C3×Dic3, C3⋊Dic3, C3×C12, S3×C6, S3×C6, C2×C3⋊S3, C2×C3⋊S3, C62, C4×Dic3, Dic3⋊C4, D6⋊C4, D6⋊C4, C3×C22⋊C4, C3×C4⋊C4, S3×C2×C4, C2×D12, C22×Dic3, C2×C3⋊D4, S3×Dic3, C3⋊D12, C6×Dic3, C4×C3⋊S3, C2×C3⋊Dic3, C6×C12, S3×C2×C6, C22×C3⋊S3, Dic3⋊4D4, Dic3⋊5D4, Dic32, C6.D12, C3×Dic3⋊C4, C3×D6⋊C4, C2×S3×Dic3, C2×C3⋊D12, C2×C4×C3⋊S3, C62.51C23
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D6, C22×C4, C2×D4, C4○D4, C4×S3, C22×S3, C4×D4, S32, S3×C2×C4, S3×D4, D4⋊2S3, Q8⋊3S3, C2×S32, Dic3⋊4D4, Dic3⋊5D4, D12⋊S3, C4×S32, Dic3⋊D6, C62.51C23

Smallest permutation representation of C62.51C23
►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 10)(2 11)(3 12)(4 7)(5 8)(6 9)(13 45)(14 46)(15 47)(16 48)(17 43)(18 44)(19 31)(20 32)(21 33)(22 34)(23 35)(24 36)(25 39)(26 40)(27 41)(28 42)(29 37)(30 38)

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

48 conjugacy classes

class 1 2A2B2C2D2E2F2G3A3B3C4A4B4C···4H4I4J4K4L6A···6F6G6H6I6J6K12A···12H12I···12N
order12222222333444···444446···66666612···1212···12
size1111661818224226···699992···244412124···412···12

48 irreducible representations

dim11111111122222222244444444
type++++++++++++++++-+++
imageC1C2C2C2C2C2C2C2C4S3S3D4D6D6D6C4○D4C4×S3C4×S3S32S3×D4D4⋊2S3Q8⋊3S3C2×S32D12⋊S3C4×S32Dic3⋊D6
kernelC62.51C23Dic32C6.D12C3×Dic3⋊C4C3×D6⋊C4C2×S3×Dic3C2×C3⋊D12C2×C4×C3⋊S3C3⋊D12Dic3⋊C4D6⋊C4C3⋊Dic3C2×Dic3C2×C12C22×S3C3×C6Dic3D6C2×C4C6C6C6C22C2C2C2
# reps11111111811232124412111222

Matrix representation of C62.51C23 ►in GL8(𝔽13)

10000000
01000000
001200000
000120000
000012000
000001200
00000001
0000001212
,
121000000
120000000
00100000
00010000
000012000
000001200
00000010
00000001
,
012000000
120000000
00010000
00100000
00000800
00005000
00000010
00000001
,
10000000
01000000
00010000
00100000
00000100
000012000
000000120
00000011
,
120000000
012000000
00100000
000120000
00005000
00000800
000000120
000000012

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

C62.51C23 in GAP, Magma, Sage, TeX 

C_6^2._{51}C_2^3
 
% in TeX
 
G:=Group("C6^2.51C2^3");
 
// GroupNames label
 
G:=SmallGroup(288,529);
 
// by ID
 
G=gap.SmallGroup(288,529);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,120,254,135,100,1356,9414]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^6=b^6=c^2=1,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=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*d>;
 
// generators/relations
 

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