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G = C62.77C23  order 288 = 25·32

72nd non-split extension by C62 of C23 acting via C23/C2=C22

metabelian, supersoluble, monomial

Aliases: C62.77C23, Dic32⋊3C2, D6⋊C4⋊17S3, (C2×Dic6)⋊6S3, C6.144(S3×D4), D6⋊Dic3⋊33C2, (C6×Dic6)⋊11C2, (C2×C12).229D6, C6.15(C4○D12), (C2×Dic3).31D6, (C3×Dic3).10D4, (C22×S3).13D6, C32⋊6(C4.4D4), C6.D12⋊18C2, C6.11D12⋊11C2, C6.11(D4⋊2S3), C3⋊2(C12.23D4), (C6×C12).187C22, C6.16(Q8⋊3S3), Dic3.2(C3⋊D4), C2.18(D12⋊S3), C3⋊6(C23.11D6), C2.17(D6.6D6), (C6×Dic3).83C22, (C2×C4).32S32, (C3×D6⋊C4)⋊13C2, C2.18(S3×C3⋊D4), C6.39(C2×C3⋊D4), C22.115(C2×S32), (C3×C6).104(C2×D4), (S3×C2×C6).28C22, (C2×C3⋊D12).9C2, (C3×C6).47(C4○D4), (C2×C6).96(C22×S3), (C22×C3⋊S3).22C22, (C2×C3⋊Dic3).54C22, SmallGroup(288,555)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C62 — C62.77C23
Chief series
C1 — C3 — C32 — C3×C6 — C62 — C6×Dic3 — C2×C3⋊D12 — C62.77C23
Lower central
C32 — C62 — C62.77C23
Upper central
C1 — C22 — C2×C4

Generators and relations for C62.77C23
 G = < a,b,c,d,e | a6=b6=d2=1, c2=e2=b3, ab=ba, ac=ca, dad=a-1, ae=ea, cbc-1=b-1, bd=db, be=eb, dcd=ece-1=b3c, ede-1=a3d >

Subgroups: 730 in 169 conjugacy classes, 48 normal (44 characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, Q8, C23, C32, Dic3, Dic3, C12, D6, C2×C6, C2×C6, C42, C22⋊C4, C2×D4, C2×Q8, C3×S3, C3⋊S3, C3×C6, Dic6, D12, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×Q8, C22×S3, C22×S3, C22×C6, C4.4D4, C3×Dic3, C3×Dic3, C3⋊Dic3, C3×C12, S3×C6, C2×C3⋊S3, C62, C4×Dic3, D6⋊C4, D6⋊C4, C6.D4, C3×C22⋊C4, C2×Dic6, C2×D12, C2×C3⋊D4, C6×Q8, C3⋊D12, C3×Dic6, C6×Dic3, C2×C3⋊Dic3, C6×C12, S3×C2×C6, C22×C3⋊S3, C23.11D6, C12.23D4, Dic32, D6⋊Dic3, C6.D12, C3×D6⋊C4, C6.11D12, C2×C3⋊D12, C6×Dic6, C62.77C23
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, C3⋊D4, C22×S3, C4.4D4, S32, C4○D12, S3×D4, D4⋊2S3, Q8⋊3S3, C2×C3⋊D4, C2×S32, C23.11D6, C12.23D4, D12⋊S3, D6.6D6, S3×C3⋊D4, C62.77C23

Smallest permutation representation of C62.77C23
►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 15 3 17 5 13)(2 16 4 18 6 14)(7 44 11 48 9 46)(8 45 12 43 10 47)(19 29 21 25 23 27)(20 30 22 26 24 28)(31 37 35 41 33 39)(32 38 36 42 34 40)

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

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

42 conjugacy classes

class 1 2A2B2C2D2E3A3B3C4A4B4C4D4E4F4G4H6A···6F6G6H6I6J6K12A···12H12I···12N
order122222333444444446···66666612···1212···12
size11111236224466661218182···244412124···412···12

42 irreducible representations

dim1111111122222222244444444
type++++++++++++++++-+++
imageC1C2C2C2C2C2C2C2S3S3D4D6D6D6C4○D4C3⋊D4C4○D12S32S3×D4D4⋊2S3Q8⋊3S3C2×S32D12⋊S3D6.6D6S3×C3⋊D4
kernelC62.77C23Dic32D6⋊Dic3C6.D12C3×D6⋊C4C6.11D12C2×C3⋊D12C6×Dic6D6⋊C4C2×Dic6C3×Dic3C2×Dic3C2×C12C22×S3C3×C6Dic3C6C2×C4C6C6C6C22C2C2C2
# reps1111111111232144411121222

Matrix representation of C62.77C23 ►in GL8(𝔽13)

120000000
012000000
00100000
00010000
00001000
00000100
000000121
000000120
,
10000000
01000000
001200000
000120000
000012100
000012000
00000010
00000001
,
120000000
012000000
00050000
00500000
000001200
000012000
00000010
00000001
,
120000000
121000000
00080000
00500000
000012000
000001200
00000001
00000010
,
111000000
012000000
00010000
001200000
000012000
000001200
00000010
00000001

G:=sub<GL(8,GF(13))| [12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,1,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[12,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[1,0,0,0,0,0,0,0,11,12,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1] >;
 

C62.77C23 in GAP, Magma, Sage, TeX 

C_6^2._{77}C_2^3
 
% in TeX
 
G:=Group("C6^2.77C2^3");
 
// GroupNames label
 
G:=SmallGroup(288,555);
 
// by ID
 
G=gap.SmallGroup(288,555);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,253,422,135,100,1356,9414]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^6=b^6=d^2=1,c^2=e^2=b^3,a*b=b*a,a*c=c*a,d*a*d=a^-1,a*e=e*a,c*b*c^-1=b^-1,b*d=d*b,b*e=e*b,d*c*d=e*c*e^-1=b^3*c,e*d*e^-1=a^3*d>;
 
// generators/relations
 

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