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G = C62.19C23  order 288 = 25·32

14th non-split extension by C62 of C23 acting via C23/C2=C22

metabelian, supersoluble, monomial

Aliases: C62.19C23, Dic32:12C2, C12.36(C4xS3), C4:Dic3:11S3, (C2xC12).129D6, C6.2(D4:2S3), (C6xC12).89C22, (C2xDic3).56D6, C2.2(D12:S3), C6.22(Q8:3S3), C32:4(C42:C2), C6.D12.6C2, C4.10(C6.D6), (C6xDic3).54C22, (C4xC3:S3):1C4, C6.30(S3xC2xC4), (C2xC4).112S32, (C3xC4:Dic3):6C2, C22.22(C2xS32), C3:2(C4:C4:7S3), (C3xC12).64(C2xC4), (C3xC6).8(C4oD4), C2.8(C2xC6.D6), C3:Dic3.43(C2xC4), (C2xC6).38(C22xS3), (C3xC6).50(C22xC4), (C22xC3:S3).63C22, (C2xC3:Dic3).115C22, (C2xC4xC3:S3).2C2, (C2xC3:S3).37(C2xC4), SmallGroup(288,497)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3xC6 — C62.19C23
Chief series
C1 — C3 — C32 — C3xC6 — C62 — C6xDic3 — Dic32 — C62.19C23
Lower central
C32 — C3xC6 — C62.19C23
Upper central
C1 — C22 — C2xC4

Generators and relations for C62.19C23
 G = < a,b,c,d,e | a6=b6=1, c2=a3, d2=a3b3, e2=b3, ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc-1=b-1, bd=db, be=eb, cd=dc, ece-1=b3c, ede-1=b3d >

Subgroups: 658 in 179 conjugacy classes, 60 normal (12 characteristic)
C1, C2, C2, C2, C3, C3, C4, C4, C22, C22, S3, C6, C6, C2xC4, C2xC4, C23, C32, Dic3, C12, C12, D6, C2xC6, C2xC6, C42, C22:C4, C4:C4, C22xC4, C3:S3, C3xC6, C3xC6, C4xS3, C2xDic3, C2xDic3, C2xC12, C2xC12, C22xS3, C42:C2, C3xDic3, C3:Dic3, C3xC12, C2xC3:S3, C2xC3:S3, C62, C4xDic3, C4:Dic3, D6:C4, C3xC4:C4, S3xC2xC4, C6xDic3, C4xC3:S3, C2xC3:Dic3, C6xC12, C22xC3:S3, C4:C4:7S3, Dic32, C6.D12, C3xC4:Dic3, C2xC4xC3:S3, C62.19C23
Quotients: C1, C2, C4, C22, S3, C2xC4, C23, D6, C22xC4, C4oD4, C4xS3, C22xS3, C42:C2, S32, S3xC2xC4, D4:2S3, Q8:3S3, C6.D6, C2xS32, C4:C4:7S3, D12:S3, C2xC6.D6, C62.19C23

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

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

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

48 conjugacy classes

class 1 2A2B2C2D2E3A3B3C4A4B4C···4J4K4L4M4N6A···6F6G6H6I12A···12H12I···12P
order122222333444···444446···666612···1212···12
size11111818224226···699992···24444···412···12

48 irreducible representations

dim11111122222444444
type+++++++++-+++
imageC1C2C2C2C2C4S3D6D6C4oD4C4xS3S32D4:2S3Q8:3S3C6.D6C2xS32D12:S3
kernelC62.19C23Dic32C6.D12C3xC4:Dic3C2xC4xC3:S3C4xC3:S3C4:Dic3C2xDic3C2xC12C3xC6C12C2xC4C6C6C4C22C2
# reps12221824248122214

Matrix representation of C62.19C23 ►in GL8(F13)

10000000
01000000
00100000
00010000
000012000
000001200
00000001
0000001212
,
121000000
120000000
001200000
000120000
000012000
000001200
00000010
00000001
,
012000000
120000000
00830000
00550000
00000800
00008000
000000120
000000012
,
120000000
012000000
0012110000
00110000
00000100
00001000
00000010
0000001212
,
10000000
01000000
00500000
00880000
00008000
00000500
000000120
000000012

G:=sub<GL(8,GF(13))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1,12],[12,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,8,5,0,0,0,0,0,0,3,5,0,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12],[12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,1,0,0,0,0,0,0,11,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,5,8,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12] >;
 

C62.19C23 in GAP, Magma, Sage, TeX 

C_6^2._{19}C_2^3
 
% in TeX
 
G:=Group("C6^2.19C2^3");
 
// GroupNames label
 
G:=SmallGroup(288,497);
 
// by ID
 
G=gap.SmallGroup(288,497);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,56,176,422,219,100,1356,9414]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^6=b^6=1,c^2=a^3,d^2=a^3*b^3,e^2=b^3,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,a*e=e*a,c*b*c^-1=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e^-1=b^3*c,e*d*e^-1=b^3*d>;
 
// generators/relations
 

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