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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, (S3xC6).16D4, C6.167(S3xD4), D6:Dic3:16C2, C62:5C4:7C2, C6.D4:8S3, (C22xC6).71D6, C6.66(C4oD12), C3:7(C23.9D6), D6.10(C3:D4), (C2xDic3).44D6, (C22xS3).47D6, Dic3:Dic3:36C2, C6.53(D4:2S3), C62.C22:20C2, (C2xC62).30C22, C2.16(D6.4D6), C2.27(D6.3D6), C3:3(C23.23D6), (C6xDic3).84C22, C32:13(C22.D4), (C2xS3xDic3):19C2, (C2xC3:D4).5S3, (C6xC3:D4).8C2, C2.39(S3xC3:D4), C6.63(C2xC3:D4), C22.137(C2xS32), (C3xC6).157(C2xD4), (S3xC2xC6).43C22, (C3xC6).82(C4oD4), (C3xC6.D4):14C2, (C2xC6).130(C22xS3), (C2xC3:Dic3).67C22, SmallGroup(288,617)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C62 — C62.111C23
Chief series
C1 — C3 — C32 — C3xC6 — C62 — S3xC2xC6 — C2xS3xDic3 — 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, C2xC4, D4, C23, C23, C32, Dic3, C12, D6, D6, C2xC6, C2xC6, C22:C4, C4:C4, C22xC4, C2xD4, C3xS3, C3xC6, C3xC6, C4xS3, C2xDic3, C2xDic3, C3:D4, C2xC12, C3xD4, C22xS3, C22xC6, C22xC6, C22.D4, C3xDic3, C3:Dic3, S3xC6, S3xC6, C62, C62, Dic3:C4, C4:Dic3, D6:C4, C6.D4, C6.D4, C3xC22:C4, S3xC2xC4, C22xDic3, C2xC3:D4, C6xD4, S3xDic3, C6xDic3, C3xC3:D4, C2xC3:Dic3, S3xC2xC6, C2xC62, C23.9D6, C23.23D6, D6:Dic3, Dic3:Dic3, C62.C22, C3xC6.D4, C62:5C4, C2xS3xDic3, C6xC3:D4, C62.111C23
Quotients: C1, C2, C22, S3, D4, C23, D6, C2xD4, C4oD4, C3:D4, C22xS3, C22.D4, S32, C4oD12, S3xD4, D4:2S3, C2xC3:D4, C2xS32, C23.9D6, C23.23D6, D6.3D6, D6.4D6, S3xC3: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++++++++++++++++-+-
imageC1C2C2C2C2C2C2C2S3S3D4D6D6D6C4oD4C3:D4C4oD12S32S3xD4D4:2S3C2xS32D6.3D6D6.4D6S3xC3:D4
kernelC62.111C23D6:Dic3Dic3:Dic3C62.C22C3xC6.D4C62:5C4C2xS3xDic3C6xC3:D4C6.D4C2xC3:D4S3xC6C2xDic3C22xS3C22xC6C3xC6D6C6C23C6C6C22C2C2C2
# 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
 

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