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G = C62.53C23order 288 = 25·32

48th non-split extension by C62 of C23 acting via C23/C2=C22

metabelian, supersoluble, monomial

Aliases: C62.53C23, C6.9(S3xQ8), C6.49(S3xD4), Dic3:6(C4xS3), C6.D6:3C4, Dic3:C4:14S3, (C2xC12).199D6, C2.2(Dic3:D6), (C2xDic3).65D6, Dic3:Dic3:24C2, (C6xC12).231C22, C2.3(Dic3.D6), (C6xDic3).34C22, C3:1(S3xC4:C4), C2.18(C4xS32), (C2xC4).95S32, C32:5(C2xC4:C4), C6.17(S3xC2xC4), C3:S3:3(C4:C4), (C2xC3:S3).8Q8, (C2xC3:S3).54D4, C22.33(C2xS32), (C3xC6).93(C2xD4), (C3xC6).26(C2xQ8), (C3xDic3):3(C2xC4), (C3xDic3:C4):13C2, (C3xC6).16(C22xC4), (C2xC6).72(C22xS3), (C2xC6.D6).2C2, (C22xC3:S3).70C22, (C2xC3:Dic3).130C22, (C2xC4xC3:S3).20C2, (C2xC3:S3).30(C2xC4), SmallGroup(288,531)

Series: Derived Chief Lower central Upper central

C1C3xC6 — C62.53C23
C1C3C32C3xC6C62C6xDic3C2xC6.D6 — C62.53C23
C32C3xC6 — C62.53C23
C1C22C2xC4

Generators and relations for C62.53C23
 G = < a,b,c,d,e | a6=b6=1, c2=d2=b3, e2=a3, 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: 786 in 211 conjugacy classes, 66 normal (18 characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C2xC4, C2xC4, C23, C32, Dic3, Dic3, C12, D6, C2xC6, C2xC6, C4:C4, C22xC4, C3:S3, C3xC6, C4xS3, C2xDic3, C2xDic3, C2xC12, C2xC12, C22xS3, C2xC4:C4, C3xDic3, C3xDic3, C3:Dic3, C3xC12, C2xC3:S3, C62, Dic3:C4, Dic3:C4, C4:Dic3, C3xC4:C4, S3xC2xC4, C6.D6, C6.D6, C6xDic3, C4xC3:S3, C2xC3:Dic3, C6xC12, C22xC3:S3, S3xC4:C4, Dic3:Dic3, C3xDic3:C4, C2xC6.D6, C2xC4xC3:S3, C62.53C23
Quotients: C1, C2, C4, C22, S3, C2xC4, D4, Q8, C23, D6, C4:C4, C22xC4, C2xD4, C2xQ8, C4xS3, C22xS3, C2xC4:C4, S32, S3xC2xC4, S3xD4, S3xQ8, C2xS32, S3xC4:C4, Dic3.D6, C4xS32, Dic3:D6, C62.53C23

Smallest permutation representation of C62.53C23
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 36 16 41)(2 31 17 42)(3 32 18 37)(4 33 13 38)(5 34 14 39)(6 35 15 40)(7 21 45 30)(8 22 46 25)(9 23 47 26)(10 24 48 27)(11 19 43 28)(12 20 44 29)
(1 41 16 36)(2 40 17 35)(3 39 18 34)(4 38 13 33)(5 37 14 32)(6 42 15 31)(7 30 45 21)(8 29 46 20)(9 28 47 19)(10 27 48 24)(11 26 43 23)(12 25 44 22)
(1 21 4 24)(2 22 5 19)(3 23 6 20)(7 33 10 36)(8 34 11 31)(9 35 12 32)(13 27 16 30)(14 28 17 25)(15 29 18 26)(37 47 40 44)(38 48 41 45)(39 43 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), (1,36,16,41)(2,31,17,42)(3,32,18,37)(4,33,13,38)(5,34,14,39)(6,35,15,40)(7,21,45,30)(8,22,46,25)(9,23,47,26)(10,24,48,27)(11,19,43,28)(12,20,44,29), (1,41,16,36)(2,40,17,35)(3,39,18,34)(4,38,13,33)(5,37,14,32)(6,42,15,31)(7,30,45,21)(8,29,46,20)(9,28,47,19)(10,27,48,24)(11,26,43,23)(12,25,44,22), (1,21,4,24)(2,22,5,19)(3,23,6,20)(7,33,10,36)(8,34,11,31)(9,35,12,32)(13,27,16,30)(14,28,17,25)(15,29,18,26)(37,47,40,44)(38,48,41,45)(39,43,42,46)>;

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

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

48 conjugacy classes

class 1 2A2B2C2D2E2F2G3A3B3C4A4B4C···4J4K4L6A···6F6G6H6I12A···12H12I···12P
order12222222333444···4446···666612···1212···12
size11119999224226···618182···24444···412···12

48 irreducible representations

dim1111112222224444444
type+++++++-++++-++
imageC1C2C2C2C2C4S3D4Q8D6D6C4xS3S32S3xD4S3xQ8C2xS32Dic3.D6C4xS32Dic3:D6
kernelC62.53C23Dic3:Dic3C3xDic3:C4C2xC6.D6C2xC4xC3:S3C6.D6Dic3:C4C2xC3:S3C2xC3:S3C2xDic3C2xC12Dic3C2xC4C6C6C22C2C2C2
# reps1222182224281221222

Matrix representation of C62.53C23 in GL8(F13)

10000000
01000000
00100000
00010000
000012000
000001200
000000121
000000120
,
01000000
1212000000
001200000
000120000
000012000
000001200
00000010
00000001
,
10000000
1212000000
000120000
00100000
00007100
00002600
00000010
00000001
,
10000000
01000000
00010000
001200000
000061200
000011700
00000001
00000010
,
10000000
01000000
00010000
00100000
000012300
00008100
00000010
00000001

G:=sub<GL(8,GF(13))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,1,0],[0,12,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,7,2,0,0,0,0,0,0,1,6,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,6,11,0,0,0,0,0,0,12,7,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,12,8,0,0,0,0,0,0,3,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1] >;

C62.53C23 in GAP, Magma, Sage, TeX

C_6^2._{53}C_2^3
% in TeX

G:=Group("C6^2.53C2^3");
// GroupNames label

G:=SmallGroup(288,531);
// by ID

G=gap.SmallGroup(288,531);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,120,422,219,58,1356,9414]);
// Polycyclic

G:=Group<a,b,c,d,e|a^6=b^6=1,c^2=d^2=b^3,e^2=a^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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