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G = C24.59D6order 192 = 26·3

6th non-split extension by C24 of D6 acting via D6/S3=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C24.59D6, C23.49D12, (C22xC4):4D6, (S3xC23):5C4, C3:1(C24:3C4), D6:3(C22:C4), C6.37C22wrC2, C22:4(D6:C4), (S3xC24).1C2, C23.56(C4xS3), (C22xC6).67D4, C2.2(C23:2D6), C2.4(D6:D4), (C22xC12):1C22, (C22xS3).87D4, C22.100(S3xD4), C22.43(C2xD12), C23.59(C3:D4), (C23xC6).38C22, (S3xC23).87C22, C23.292(C22xS3), (C22xC6).329C23, (C22xDic3):2C22, (C2xD6:C4):3C2, C2.9(C2xD6:C4), (C2xC22:C4):2S3, (C6xC22:C4):2C2, (C2xC6):1(C22:C4), (C2xC6).321(C2xD4), C2.28(S3xC22:C4), C6.36(C2xC22:C4), C22.126(S3xC2xC4), (C2xC6.D4):2C2, (C22xC6).53(C2xC4), C22.50(C2xC3:D4), (C22xS3).60(C2xC4), (C2xC6).108(C22xC4), SmallGroup(192,514)

Series: Derived Chief Lower central Upper central

C1C2xC6 — C24.59D6
C1C3C6C2xC6C22xC6S3xC23S3xC24 — C24.59D6
C3C2xC6 — C24.59D6
C1C23C2xC22:C4

Generators and relations for C24.59D6
 G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e6=b, f2=cb=bc, ab=ba, ac=ca, eae-1=faf-1=ad=da, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=ce5 >

Subgroups: 1608 in 506 conjugacy classes, 95 normal (19 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, S3, C6, C6, C6, C2xC4, C23, C23, C23, Dic3, C12, D6, D6, C2xC6, C2xC6, C2xC6, C22:C4, C22xC4, C22xC4, C24, C24, C2xDic3, C2xC12, C22xS3, C22xS3, C22xC6, C22xC6, C22xC6, C2xC22:C4, C2xC22:C4, C25, D6:C4, C6.D4, C3xC22:C4, C22xDic3, C22xC12, S3xC23, S3xC23, C23xC6, C24:3C4, C2xD6:C4, C2xC6.D4, C6xC22:C4, S3xC24, C24.59D6
Quotients: C1, C2, C4, C22, S3, C2xC4, D4, C23, D6, C22:C4, C22xC4, C2xD4, C4xS3, D12, C3:D4, C22xS3, C2xC22:C4, C22wrC2, D6:C4, S3xC2xC4, C2xD12, S3xD4, C2xC3:D4, C24:3C4, S3xC22:C4, D6:D4, C2xD6:C4, C23:2D6, C24.59D6

Smallest permutation representation of C24.59D6
On 48 points
Generators in S48
(1 14)(2 38)(3 16)(4 40)(5 18)(6 42)(7 20)(8 44)(9 22)(10 46)(11 24)(12 48)(13 30)(15 32)(17 34)(19 36)(21 26)(23 28)(25 43)(27 45)(29 47)(31 37)(33 39)(35 41)
(1 7)(2 8)(3 9)(4 10)(5 11)(6 12)(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)(25 31)(26 32)(27 33)(28 34)(29 35)(30 36)(37 43)(38 44)(39 45)(40 46)(41 47)(42 48)
(1 37)(2 38)(3 39)(4 40)(5 41)(6 42)(7 43)(8 44)(9 45)(10 46)(11 47)(12 48)(13 30)(14 31)(15 32)(16 33)(17 34)(18 35)(19 36)(20 25)(21 26)(22 27)(23 28)(24 29)
(1 31)(2 32)(3 33)(4 34)(5 35)(6 36)(7 25)(8 26)(9 27)(10 28)(11 29)(12 30)(13 48)(14 37)(15 38)(16 39)(17 40)(18 41)(19 42)(20 43)(21 44)(22 45)(23 46)(24 47)
(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 36 43 13)(2 24 44 35)(3 34 45 23)(4 22 46 33)(5 32 47 21)(6 20 48 31)(7 30 37 19)(8 18 38 29)(9 28 39 17)(10 16 40 27)(11 26 41 15)(12 14 42 25)

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

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

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

48 conjugacy classes

class 1 2A···2G2H2I2J2K2L···2S 3 4A4B4C4D4E4F4G4H6A···6G6H6I6J6K12A···12H
order12···222222···23444444446···6666612···12
size11···122226···624444121212122···244444···4

48 irreducible representations

dim111111222222224
type++++++++++++
imageC1C2C2C2C2C4S3D4D4D6D6C4xS3D12C3:D4S3xD4
kernelC24.59D6C2xD6:C4C2xC6.D4C6xC22:C4S3xC24S3xC23C2xC22:C4C22xS3C22xC6C22xC4C24C23C23C23C22
# reps141118184214444

Matrix representation of C24.59D6 in GL5(F13)

10000
012000
001200
00010
000012
,
120000
01000
00100
000120
000012
,
10000
012000
001200
000120
000012
,
10000
01000
00100
000120
000012
,
50000
09200
0111100
00001
000120
,
80000
02900
0111100
00001
00010

G:=sub<GL(5,GF(13))| [1,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,1,0,0,0,0,0,12],[12,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,12,0,0,0,0,0,12],[1,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,12,0,0,0,0,0,12],[5,0,0,0,0,0,9,11,0,0,0,2,11,0,0,0,0,0,0,12,0,0,0,1,0],[8,0,0,0,0,0,2,11,0,0,0,9,11,0,0,0,0,0,0,1,0,0,0,1,0] >;

C24.59D6 in GAP, Magma, Sage, TeX

C_2^4._{59}D_6
% in TeX

G:=Group("C2^4.59D6");
// GroupNames label

G:=SmallGroup(192,514);
// by ID

G=gap.SmallGroup(192,514);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,422,387,58,6278]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^6=b,f^2=c*b=b*c,a*b=b*a,a*c=c*a,e*a*e^-1=f*a*f^-1=a*d=d*a,b*d=d*b,b*e=e*b,b*f=f*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=c*e^5>;
// generators/relations

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