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G = C62.121C23  order 288 = 25·32

116th non-split extension by C62 of C23 acting via C23/C2=C22

metabelian, supersoluble, monomial

Aliases: C62.121C23, Dic32⋊7C2, C3⋊Dic3⋊3D4, C23.18S32, C6.72(S3×D4), (C3×Dic3)⋊5D4, C3⋊3(C12⋊3D4), C32⋊4(C4⋊1D4), (C22×C6).79D6, Dic3⋊1(C3⋊D4), (C2×Dic3).85D6, (C22×S3).27D6, C2.31(Dic3⋊D6), (C2×C62).40C22, (C6×Dic3).42C22, (C2×C3⋊D4)⋊6S3, (C6×C3⋊D4)⋊3C2, C6.67(C2×C3⋊D4), C2.45(S3×C3⋊D4), (C2×D6⋊S3)⋊9C2, C22.144(C2×S32), (C3×C6).167(C2×D4), (C2×C32⋊7D4)⋊5C2, (S3×C2×C6).49C22, (C2×C3⋊D12)⋊17C2, (C2×C6).140(C22×S3), (C22×C3⋊S3).35C22, (C2×C3⋊Dic3).73C22, SmallGroup(288,627)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C62 — C62.121C23
Chief series
C1 — C3 — C32 — C3×C6 — C62 — S3×C2×C6 — C2×D6⋊S3 — C62.121C23
Lower central
C32 — C62 — C62.121C23
Upper central
C1 — C22 — C23

Generators and relations for C62.121C23
 G = < a,b,c,d,e | a6=b6=c2=d2=e2=1, ab=ba, ac=ca, dad=a-1, ae=ea, cbc=b-1, bd=db, be=eb, dcd=b3c, ece=a3c, ede=a3b3d >

Subgroups: 1090 in 243 conjugacy classes, 54 normal (14 characteristic)
C1, C2, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C2×C4, D4, C23, C23, C32, Dic3, Dic3, C12, D6, C2×C6, C2×C6, C42, C2×D4, C3×S3, C3⋊S3, C3×C6, C3×C6, C3×C6, D12, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C22×S3, C22×S3, C22×C6, C22×C6, C4⋊1D4, C3×Dic3, C3⋊Dic3, S3×C6, C2×C3⋊S3, C62, C62, C4×Dic3, C2×D12, C2×C3⋊D4, C2×C3⋊D4, C6×D4, D6⋊S3, C3⋊D12, C6×Dic3, C3×C3⋊D4, C2×C3⋊Dic3, C32⋊7D4, S3×C2×C6, C22×C3⋊S3, C2×C62, C12⋊3D4, Dic32, C2×D6⋊S3, C2×C3⋊D12, C6×C3⋊D4, C2×C32⋊7D4, C62.121C23
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C3⋊D4, C22×S3, C4⋊1D4, S32, S3×D4, C2×C3⋊D4, C2×S32, C12⋊3D4, S3×C3⋊D4, Dic3⋊D6, C62.121C23

Smallest permutation representation of C62.121C23
►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 48)(2 43)(3 44)(4 45)(5 46)(6 47)(7 13)(8 14)(9 15)(10 16)(11 17)(12 18)(19 42)(20 37)(21 38)(22 39)(23 40)(24 41)(25 34)(26 35)(27 36)(28 31)(29 32)(30 33)

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

42 conjugacy classes

class 1 2A2B2C2D2E2F2G3A3B3C4A4B4C4D4E4F6A···6F6G···6Q6R6S6T6U12A12B12C12D
order122222223334444446···66···6666612121212
size11114121236224666618182···24···41212121212121212

42 irreducible representations

dim111111222222244444
type++++++++++++++++
imageC1C2C2C2C2C2S3D4D4D6D6D6C3⋊D4S32S3×D4C2×S32S3×C3⋊D4Dic3⋊D6
kernelC62.121C23Dic32C2×D6⋊S3C2×C3⋊D12C6×C3⋊D4C2×C32⋊7D4C2×C3⋊D4C3×Dic3C3⋊Dic3C2×Dic3C22×S3C22×C6Dic3C23C6C22C2C2
# reps111221242222814142

Matrix representation of C62.121C23 ►in GL8(ℤ)

10000000
01000000
00-100000
000-10000
00001000
00000100
000000-11
000000-10
,
-10000000
0-1000000
00-100000
000-10000
0000-1100
0000-1000
00000010
00000001
,
-1-2000000
01000000
00100000
00-1-10000
00000-100
0000-1000
00000010
00000001
,
-10000000
11000000
00120000
000-10000
0000-1000
00000-100
00000001
00000010
,
-1-2000000
01000000
00-1-20000
00010000
00001000
00000100
00000010
00000001

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,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,-1,0,0,0,0,0,0,1,0],[-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,-2,1,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,-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],[-1,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,2,-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],[-1,0,0,0,0,0,0,0,-2,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,-2,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1] >;
 

C62.121C23 in GAP, Magma, Sage, TeX 

C_6^2._{121}C_2^3
 
% in TeX
 
G:=Group("C6^2.121C2^3");
 
// GroupNames label
 
G:=SmallGroup(288,627);
 
// by ID
 
G=gap.SmallGroup(288,627);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,253,254,303,1356,9414]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^6=b^6=c^2=d^2=e^2=1,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=b^3*c,e*c*e=a^3*c,e*d*e=a^3*b^3*d>;
 
// generators/relations
 

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