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G = C6.532+ 1+4  order 192 = 26·3

53rd non-split extension by C6 of 2+ 1+4 acting via 2+ 1+4/C2xD4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C6.532+ 1+4, C4:C4:12D6, (C2xQ8):9D6, C22:Q8:14S3, (C6xQ8):9C22, D6:C4:68C22, D6:3Q8:19C2, D6:D4.2C2, C22:C4.62D6, Dic3:5D4:28C2, D6.19(C4oD4), D6.D4:19C2, (C2xC6).181C24, (C2xC12).60C23, C4:Dic3:37C22, (C22xC4).259D6, C2.55(D4:6D6), C12.23D4:14C2, Dic3:C4:19C22, (C4xDic3):29C22, C23.16D6:8C2, C3:5(C22.45C24), (C2xD12).150C22, C23.21D6:16C2, (S3xC23).54C22, (C22xS3).74C23, C22.202(S3xC23), C23.204(C22xS3), (C22xC6).209C23, C22.9(Q8:3S3), (C2xDic3).92C23, (C22xC12).381C22, C6.D4.121C22, (C22xDic3).122C22, (C2xD6:C4):37C2, (C4xC3:D4):57C2, (S3xC22:C4):9C2, (S3xC2xC4):51C22, C4:C4:S3:17C2, C4:C4:7S3:26C2, C2.52(S3xC4oD4), (C3xC4:C4):21C22, C6.164(C2xC4oD4), (C3xC22:Q8):17C2, (C2xC6).26(C4oD4), (C2xC4).51(C22xS3), C2.18(C2xQ8:3S3), (C2xC3:D4).128C22, (C3xC22:C4).36C22, SmallGroup(192,1196)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2xC6 — C6.532+ 1+4
Chief series
C1 — C3 — C6 — C2xC6 — C22xS3 — S3xC23 — S3xC22:C4 — C6.532+ 1+4
Lower central
C3 — C2xC6 — C6.532+ 1+4
Upper central
C1 — C22 — C22:Q8

Generators and relations for C6.532+ 1+4
 G = < a,b,c,d,e | a6=b4=c2=1, d2=a3b2, e2=a3, ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc=a3b-1, dbd-1=ebe-1=a3b, cd=dc, ce=ec, ede-1=b2d >

Subgroups: 688 in 248 conjugacy classes, 97 normal (91 characteristic)
C1, C2, C2, C3, C4, C22, C22, C22, S3, C6, C6, C2xC4, C2xC4, D4, Q8, C23, C23, Dic3, C12, D6, D6, C2xC6, C2xC6, C2xC6, C42, C22:C4, C22:C4, C4:C4, C4:C4, C22xC4, C22xC4, C2xD4, C2xQ8, C24, C4xS3, D12, C2xDic3, C2xDic3, C3:D4, C2xC12, C2xC12, C3xQ8, C22xS3, C22xS3, C22xC6, C2xC22:C4, C42:C2, C4xD4, C22wrC2, C22:Q8, C22:Q8, C22.D4, C4.4D4, C42:2C2, C4xDic3, Dic3:C4, C4:Dic3, D6:C4, C6.D4, C3xC22:C4, C3xC4:C4, S3xC2xC4, C2xD12, C22xDic3, C2xC3:D4, C22xC12, C6xQ8, S3xC23, C22.45C24, C23.16D6, S3xC22:C4, D6:D4, C23.21D6, C4:C4:7S3, Dic3:5D4, D6.D4, C4:C4:S3, C2xD6:C4, C4xC3:D4, D6:3Q8, C12.23D4, C3xC22:Q8, C6.532+ 1+4
Quotients: C1, C2, C22, S3, C23, D6, C4oD4, C24, C22xS3, C2xC4oD4, 2+ 1+4, Q8:3S3, S3xC23, C22.45C24, D4:6D6, C2xQ8:3S3, S3xC4oD4, C6.532+ 1+4

Smallest permutation representation of C6.532+ 1+4
►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 28 10 31)(2 29 11 32)(3 30 12 33)(4 25 7 34)(5 26 8 35)(6 27 9 36)(13 40 22 43)(14 41 23 44)(15 42 24 45)(16 37 19 46)(17 38 20 47)(18 39 21 48)

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

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

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

39 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I 3 4A4B4C···4G4H···4M4N4O6A6B6C6D6E12A12B12C12D12E12F12G12H
order12222222223444···44···444666661212121212121212
size1111226612122224···46···612122224444448888

39 irreducible representations

dim1111111111111122222224444
type+++++++++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2C2C2C2S3D6D6D6D6C4oD4C4oD42+ 1+4Q8:3S3D4:6D6S3xC4oD4
kernelC6.532+ 1+4C23.16D6S3xC22:C4D6:D4C23.21D6C4:C4:7S3Dic3:5D4D6.D4C4:C4:S3C2xD6:C4C4xC3:D4D6:3Q8C12.23D4C3xC22:Q8C22:Q8C22:C4C4:C4C22xC4C2xQ8D6C2xC6C6C22C2C2
# reps1111111221111112311441222

Matrix representation of C6.532+ 1+4 ►in GL6(F13)

1200000
0120000
0001200
0011200
0000120
0000012
,
800000
050000
0012000
0001200
0000012
0000120
,
1200000
0120000
001000
000100
0000120
000001
,
010000
100000
000100
001000
000080
000005
,
010000
1200000
001000
000100
000080
000005

G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,12,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[8,0,0,0,0,0,0,5,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,12,0],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,8,0,0,0,0,0,0,5],[0,12,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,8,0,0,0,0,0,0,5] >;
 

C6.532+ 1+4 in GAP, Magma, Sage, TeX 

C_6._{53}2_+^{1+4}
 
% in TeX
 
G:=Group("C6.53ES+(2,2)");
 
// GroupNames label
 
G:=SmallGroup(192,1196);
 
// by ID
 
G=gap.SmallGroup(192,1196);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,219,184,1571,297,136,6278]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^6=b^4=c^2=1,d^2=a^3*b^2,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=a^3*b^-1,d*b*d^-1=e*b*e^-1=a^3*b,c*d=d*c,c*e=e*c,e*d*e^-1=b^2*d>;
 
// generators/relations
 

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