Copied to
clipboard

G = C62.111C23  order 288 = 25·32

106th non-split extension by C62 of C23 acting via C23/C2=C22

metabelian, supersoluble, monomial

Aliases: C62.111C23, C23.14S32, (S3×C6).16D4, C6.167(S3×D4), D6⋊Dic3⋊16C2, C62⋊5C4⋊7C2, C6.D4⋊8S3, (C22×C6).71D6, C6.66(C4○D12), C3⋊7(C23.9D6), D6.10(C3⋊D4), (C2×Dic3).44D6, (C22×S3).47D6, Dic3⋊Dic3⋊36C2, C6.53(D4⋊2S3), C62.C22⋊20C2, (C2×C62).30C22, C2.16(D6.4D6), C2.27(D6.3D6), C3⋊3(C23.23D6), (C6×Dic3).84C22, C32⋊13(C22.D4), (C2×S3×Dic3)⋊19C2, (C2×C3⋊D4).5S3, (C6×C3⋊D4).8C2, C2.39(S3×C3⋊D4), C6.63(C2×C3⋊D4), C22.137(C2×S32), (C3×C6).157(C2×D4), (S3×C2×C6).43C22, (C3×C6).82(C4○D4), (C3×C6.D4)⋊14C2, (C2×C6).130(C22×S3), (C2×C3⋊Dic3).67C22, SmallGroup(288,617)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C62 — C62.111C23
Chief series
C1 — C3 — C32 — C3×C6 — C62 — S3×C2×C6 — C2×S3×Dic3 — C62.111C23
Lower central
C32 — C62 — C62.111C23
Upper central
C1 — C22 — C23

Generators and relations for C62.111C23
 G = < a,b,c,d,e | a6=b6=c2=e2=1, d2=a3, ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ece=a3c, ede=b3d >

Subgroups: 602 in 173 conjugacy classes, 48 normal (44 characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C2×C4, D4, C23, C23, C32, Dic3, C12, D6, D6, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C3×S3, C3×C6, C3×C6, C4×S3, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C22×S3, C22×C6, C22×C6, C22.D4, C3×Dic3, C3⋊Dic3, S3×C6, S3×C6, C62, C62, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C6.D4, C6.D4, C3×C22⋊C4, S3×C2×C4, C22×Dic3, C2×C3⋊D4, C6×D4, S3×Dic3, C6×Dic3, C3×C3⋊D4, C2×C3⋊Dic3, S3×C2×C6, C2×C62, C23.9D6, C23.23D6, D6⋊Dic3, Dic3⋊Dic3, C62.C22, C3×C6.D4, C62⋊5C4, C2×S3×Dic3, C6×C3⋊D4, C62.111C23
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, C3⋊D4, C22×S3, C22.D4, S32, C4○D12, S3×D4, D4⋊2S3, C2×C3⋊D4, C2×S32, C23.9D6, C23.23D6, D6.3D6, D6.4D6, S3×C3⋊D4, C62.111C23

Smallest permutation representation of C62.111C23
►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 16 3 18 5 14)(2 17 4 13 6 15)(7 48 9 44 11 46)(8 43 10 45 12 47)(19 28 23 26 21 30)(20 29 24 27 22 25)(31 41 35 39 33 37)(32 42 36 40 34 38)

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

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

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

42 conjugacy classes

class 1 2A2B2C2D2E2F3A3B3C4A4B4C4D4E4F4G6A···6F6G···6Q6R6S12A···12F
order122222233344444446···66···66612···12
size11114662246612121818362···24···4121212···12

42 irreducible representations

dim111111112222222224444444
type++++++++++++++++-+-
imageC1C2C2C2C2C2C2C2S3S3D4D6D6D6C4○D4C3⋊D4C4○D12S32S3×D4D4⋊2S3C2×S32D6.3D6D6.4D6S3×C3⋊D4
kernelC62.111C23D6⋊Dic3Dic3⋊Dic3C62.C22C3×C6.D4C62⋊5C4C2×S3×Dic3C6×C3⋊D4C6.D4C2×C3⋊D4S3×C6C2×Dic3C22×S3C22×C6C3×C6D6C6C23C6C6C22C2C2C2
# reps111111111123124441131222

Matrix representation of C62.111C23 ►in GL8(𝔽13)

10000000
01000000
001200000
000120000
00001000
00000100
00000001
0000001212
,
120000000
012000000
001200000
000120000
00000100
0000121200
00000010
00000001
,
120000000
012000000
00080000
00500000
00001000
0000121200
00000010
00000001
,
01000000
10000000
00010000
001200000
00001000
00000100
00000010
0000001212
,
10000000
012000000
00100000
000120000
000012000
000001200
00000010
00000001

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

C62.111C23 in GAP, Magma, Sage, TeX 

C_6^2._{111}C_2^3
 
% in TeX
 
G:=Group("C6^2.111C2^3");
 
// GroupNames label
 
G:=SmallGroup(288,617);
 
// by ID
 
G=gap.SmallGroup(288,617);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,64,254,219,1356,9414]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^6=b^6=c^2=e^2=1,d^2=a^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=a^3*c,e*d*e=b^3*d>;
 
// generators/relations
 

׿
×
⋊
ℤ
𝔽
○
≀
ℚ
◁