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G = C6.682+ 1+4  order 192 = 26·3

68th non-split extension by C6 of 2+ 1+4 acting via 2+ 1+4/C2xD4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C6.682+ 1+4, C4:C4:16D6, D6:C4:5C22, C22:C4:19D6, (C22xC4):28D6, Dic3:D4:35C2, C12:D4:32C2, C23:2D6:18C2, D6:D4:22C2, C12:3D4:22C2, C12:7D4:14C2, (C2xD4).105D6, C23.9D6:38C2, C2.46(D4oD12), (C2xD12):27C22, (C2xC6).210C24, C4:Dic3:16C22, C2.70(D4:6D6), (C2xC12).184C23, Dic3:C4:25C22, (C22xC12):13C22, (C4xDic3):34C22, (C6xD4).148C22, C22.D4:15S3, C3:3(C22.54C24), (S3xC23).61C22, (C22xS3).91C23, C22.231(S3xC23), C23.141(C22xS3), (C22xC6).224C23, (C2xDic3).109C23, C6.D4.48C22, (S3xC2xC4):24C22, C4:C4:S3:32C2, (C3xC4:C4):30C22, (C2xC3:D4):21C22, (C2xC4).71(C22xS3), (C3xC22:C4):26C22, (C3xC22.D4):18C2, SmallGroup(192,1225)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2xC6 — C6.682+ 1+4
Chief series
C1 — C3 — C6 — C2xC6 — C22xS3 — S3xC23 — D6:D4 — C6.682+ 1+4
Lower central
C3 — C2xC6 — C6.682+ 1+4
Upper central
C1 — C22 — C22.D4

Generators and relations for C6.682+ 1+4
 G = < a,b,c,d,e | a6=b4=c2=e2=1, d2=b2, ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc=a3b-1, bd=db, ebe=a3b, dcd-1=ece=a3c, ede=b2d >

Subgroups: 832 in 252 conjugacy classes, 91 normal (27 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, S3, C6, C6, C6, C2xC4, C2xC4, C2xC4, D4, C23, C23, Dic3, C12, D6, C2xC6, C2xC6, C42, C22:C4, C22:C4, C22:C4, C4:C4, C4:C4, C22xC4, C22xC4, C2xD4, C2xD4, C24, C4xS3, D12, C2xDic3, C3:D4, C2xC12, C2xC12, C2xC12, C3xD4, C22xS3, C22xS3, C22xC6, C22wrC2, C4:D4, C22.D4, C22.D4, C42:2C2, C4:1D4, C4xDic3, Dic3:C4, C4:Dic3, D6:C4, C6.D4, C3xC22:C4, C3xC22:C4, C3xC4:C4, S3xC2xC4, C2xD12, C2xD12, C2xC3:D4, C22xC12, C6xD4, S3xC23, C22.54C24, D6:D4, C23.9D6, Dic3:D4, C12:D4, C4:C4:S3, C12:7D4, C23:2D6, C12:3D4, C3xC22.D4, C6.682+ 1+4
Quotients: C1, C2, C22, S3, C23, D6, C24, C22xS3, 2+ 1+4, S3xC23, C22.54C24, D4:6D6, D4oD12, C6.682+ 1+4

Smallest permutation representation of C6.682+ 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 25 10 34)(2 26 11 35)(3 27 12 36)(4 28 7 31)(5 29 8 32)(6 30 9 33)(13 40 22 43)(14 41 23 44)(15 42 24 45)(16 37 19 46)(17 38 20 47)(18 39 21 48)

(13 16)(14 17)(15 18)(19 22)(20 23)(21 24)(25 31)(26 32)(27 33)(28 34)(29 35)(30 36)(37 46)(38 47)(39 48)(40 43)(41 44)(42 45)

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

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

33 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I 3 4A···4E4F4G4H4I6A6B6C6D6E6F12A12B12C12D12E12F12G
order122222222234···4444466666612121212121212
size1111441212121224···4121212122224484444888

33 irreducible representations

dim111111111122222444
type+++++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2S3D6D6D6D62+ 1+4D4:6D6D4oD12
kernelC6.682+ 1+4D6:D4C23.9D6Dic3:D4C12:D4C4:C4:S3C12:7D4C23:2D6C12:3D4C3xC22.D4C22.D4C22:C4C4:C4C22xC4C2xD4C6C2C2
# reps122222211113211324

Matrix representation of C6.682+ 1+4 ►in GL8(F13)

11000000
120000000
00110000
001200000
00001000
00000100
00000010
00000001
,
001070000
00630000
36000000
710000000
000000120
0000112125
00001000
000031001
,
10000000
01000000
001200000
000120000
00001000
00000100
000000120
0000103012
,
001200000
00110000
120000000
11000000
00000100
000012000
000012118
0000100312
,
00100000
00010000
10000000
01000000
00000100
00001000
000012118
000000012

G:=sub<GL(8,GF(13))| [1,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,0,3,7,0,0,0,0,0,0,6,10,0,0,0,0,10,6,0,0,0,0,0,0,7,3,0,0,0,0,0,0,0,0,0,0,0,1,1,3,0,0,0,0,0,12,0,10,0,0,0,0,12,12,0,0,0,0,0,0,0,5,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,10,0,0,0,0,0,1,0,3,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12],[0,0,12,1,0,0,0,0,0,0,0,1,0,0,0,0,12,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,12,12,10,0,0,0,0,1,0,1,0,0,0,0,0,0,0,1,3,0,0,0,0,0,0,8,12],[0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,1,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,8,12] >;
 

C6.682+ 1+4 in GAP, Magma, Sage, TeX 

C_6._{68}2_+^{1+4}
 
% in TeX
 
G:=Group("C6.68ES+(2,2)");
 
// GroupNames label
 
G:=SmallGroup(192,1225);
 
// by ID
 
G=gap.SmallGroup(192,1225);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,758,219,184,1571,570,6278]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^6=b^4=c^2=e^2=1,d^2=b^2,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,b*d=d*b,e*b*e=a^3*b,d*c*d^-1=e*c*e=a^3*c,e*d*e=b^2*d>;
 
// generators/relations
 

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