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G = C62.82C23  order 288 = 25·32

77th non-split extension by C62 of C23 acting via C23/C2=C22

metabelian, supersoluble, monomial

Aliases: C62.82C23, D6⋊C4⋊11S3, C6.52(S3×D4), C3⋊Dic3⋊12D4, C3⋊2(Dic3⋊D4), (C2×C12).202D6, C32⋊9(C4⋊D4), C6.34(C4○D12), (C2×Dic3).33D6, C62.C22⋊4C2, (C22×S3).17D6, C2.15(D6⋊D6), C2.12(Dic3⋊D6), (C6×C12).238C22, C2.18(D6.D6), (C6×Dic3).17C22, (C2×C4).97S32, (C2×C3⋊S3)⋊9D4, (C3×D6⋊C4)⋊8C2, (C2×D6⋊S3)⋊4C2, (C2×C3⋊D12)⋊5C2, C22.120(C2×S32), (C3×C6).108(C2×D4), (S3×C2×C6).32C22, (C3×C6).51(C4○D4), (C2×C6).101(C22×S3), (C22×C3⋊S3).73C22, (C2×C3⋊Dic3).136C22, (C2×C4×C3⋊S3)⋊14C2, SmallGroup(288,560)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C62 — C62.82C23
Chief series
C1 — C3 — C32 — C3×C6 — C62 — S3×C2×C6 — C2×D6⋊S3 — C62.82C23
Lower central
C32 — C62 — C62.82C23
Upper central
C1 — C22 — C2×C4

Generators and relations for C62.82C23
 G = < a,b,c,d,e | a6=b6=c2=d2=1, e2=a3, ab=ba, ac=ca, dad=a-1, ae=ea, cbc=b-1, bd=db, be=eb, dcd=a3c, ece-1=b3c, ede-1=b3d >

Subgroups: 978 in 215 conjugacy classes, 48 normal (18 characteristic)
C1, 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⋊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×C12, S3×C6, C2×C3⋊S3, C2×C3⋊S3, C62, Dic3⋊C4, D6⋊C4, C3×C22⋊C4, S3×C2×C4, C2×D12, C2×C3⋊D4, D6⋊S3, C3⋊D12, C6×Dic3, C4×C3⋊S3, C2×C3⋊Dic3, C6×C12, S3×C2×C6, C22×C3⋊S3, Dic3⋊D4, C62.C22, C3×D6⋊C4, C2×D6⋊S3, C2×C3⋊D12, C2×C4×C3⋊S3, C62.82C23
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, C22×S3, C4⋊D4, S32, C4○D12, S3×D4, C2×S32, Dic3⋊D4, D6.D6, D6⋊D6, Dic3⋊D6, C62.82C23

Smallest permutation representation of C62.82C23
►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 10)(2 11)(3 12)(4 7)(5 8)(6 9)(13 43)(14 44)(15 45)(16 46)(17 47)(18 48)(19 37)(20 38)(21 39)(22 40)(23 41)(24 42)(25 35)(26 36)(27 31)(28 32)(29 33)(30 34)

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

(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), (1,10)(2,11)(3,12)(4,7)(5,8)(6,9)(13,43)(14,44)(15,45)(16,46)(17,47)(18,48)(19,37)(20,38)(21,39)(22,40)(23,41)(24,42)(25,35)(26,36)(27,31)(28,32)(29,33)(30,34), (1,34)(2,33)(3,32)(4,31)(5,36)(6,35)(7,30)(8,29)(9,28)(10,27)(11,26)(12,25)(13,38)(14,37)(15,42)(16,41)(17,40)(18,39)(19,47)(20,46)(21,45)(22,44)(23,43)(24,48), (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), (1,10)(2,11)(3,12)(4,7)(5,8)(6,9)(13,43)(14,44)(15,45)(16,46)(17,47)(18,48)(19,37)(20,38)(21,39)(22,40)(23,41)(24,42)(25,35)(26,36)(27,31)(28,32)(29,33)(30,34), (1,34)(2,33)(3,32)(4,31)(5,36)(6,35)(7,30)(8,29)(9,28)(10,27)(11,26)(12,25)(13,38)(14,37)(15,42)(16,41)(17,40)(18,39)(19,47)(20,46)(21,45)(22,44)(23,43)(24,48), (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)], [(1,10),(2,11),(3,12),(4,7),(5,8),(6,9),(13,43),(14,44),(15,45),(16,46),(17,47),(18,48),(19,37),(20,38),(21,39),(22,40),(23,41),(24,42),(25,35),(26,36),(27,31),(28,32),(29,33),(30,34)], [(1,34),(2,33),(3,32),(4,31),(5,36),(6,35),(7,30),(8,29),(9,28),(10,27),(11,26),(12,25),(13,38),(14,37),(15,42),(16,41),(17,40),(18,39),(19,47),(20,46),(21,45),(22,44),(23,43),(24,48)], [(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 2A2B2C2D2E2F2G3A3B3C4A4B4C4D4E4F6A···6F6G6H6I6J6K6L6M12A···12H12I12J12K12L
order122222223334444446···6666666612···1212121212
size11111212181822422121218182···2444121212124···412121212

42 irreducible representations

dim11111122222222444444
type++++++++++++++++
imageC1C2C2C2C2C2S3D4D4D6D6D6C4○D4C4○D12S32S3×D4C2×S32D6.D6D6⋊D6Dic3⋊D6
kernelC62.82C23C62.C22C3×D6⋊C4C2×D6⋊S3C2×C3⋊D12C2×C4×C3⋊S3D6⋊C4C3⋊Dic3C2×C3⋊S3C2×Dic3C2×C12C22×S3C3×C6C6C2×C4C6C22C2C2C2
# reps11212122222228141222

Matrix representation of C62.82C23 ►in GL8(ℤ)

-10000000
0-1000000
00100000
00010000
0000-1100
0000-1000
00000010
00000001
,
-10000000
0-1000000
00010000
00-1-10000
00001000
00000100
000000-10
0000000-1
,
0-1000000
-10000000
00-100000
00110000
0000-1000
00000-100
000000-12
00000001
,
-10000000
01000000
00100000
00010000
00000100
00001000
0000001-2
0000000-1
,
01000000
-10000000
00-100000
000-10000
0000-1000
00000-100
00000010
0000001-1

G:=sub<GL(8,Integers())| [-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,-1,-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],[-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,-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,-1],[0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,2,1],[-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,-2,-1],[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,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,-1] >;
 

C62.82C23 in GAP, Magma, Sage, TeX 

C_6^2._{82}C_2^3
 
% in TeX
 
G:=Group("C6^2.82C2^3");
 
// GroupNames label
 
G:=SmallGroup(288,560);
 
// by ID
 
G=gap.SmallGroup(288,560);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,141,422,219,58,1356,9414]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^6=b^6=c^2=d^2=1,e^2=a^3,a*b=b*a,a*c=c*a,d*a*d=a^-1,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,d*c*d=a^3*c,e*c*e^-1=b^3*c,e*d*e^-1=b^3*d>;
 
// generators/relations
 

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