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G = C6.482+ 1+4  order 192 = 26·3

48th non-split extension by C6 of 2+ 1+4 acting via 2+ 1+4/C2xD4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C6.482+ 1+4, C4:C4:9D6, (C2xD4):11D6, C4:D4:23S3, C22:C4:13D6, (C22xC4):24D6, C23:2D6:14C2, D6:3D4:28C2, Dic3:D4:23C2, D6:D4:14C2, C12:7D4:45C2, C12:3D4:20C2, D6:C4:32C22, (C6xD4):16C22, D6.D4:14C2, C2.32(D4oD12), (C2xC12).46C23, (C2xC6).164C24, C4:Dic3:13C22, C23.14D6:18C2, C2.50(D4:6D6), Dic3:C4:35C22, (C22xC12):31C22, (C4xDic3):26C22, C23.8D6:21C2, C3:2(C22.54C24), (C2xD12).146C22, C23.21D6:14C2, C6.D4:27C22, C23.28D6:12C2, (S3xC23).51C22, (C22xS3).71C23, C23.124(C22xS3), C22.185(S3xC23), (C22xC6).192C23, (C2xDic3).81C23, (C22xDic3):23C22, (S3xC2xC4):16C22, C4:C4:S3:15C2, (C3xC4:D4):26C2, (C3xC4:C4):16C22, (C2xC3:D4):17C22, (C2xC4).42(C22xS3), (C3xC22:C4):18C22, SmallGroup(192,1179)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2xC6 — C6.482+ 1+4
Chief series
C1 — C3 — C6 — C2xC6 — C22xS3 — S3xC23 — C23:2D6 — C6.482+ 1+4
Lower central
C3 — C2xC6 — C6.482+ 1+4
Upper central
C1 — C22 — C4:D4

Generators and relations for C6.482+ 1+4
 G = < a,b,c,d,e | a6=b4=c2=e2=1, d2=a3b2, ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc=b-1, dbd-1=a3b, be=eb, dcd-1=ece=a3c, ede=a3b2d >

Subgroups: 800 in 252 conjugacy classes, 91 normal (all characteristic)
C1, C2, C2, C3, C4, C22, C22, S3, C6, C6, C2xC4, C2xC4, D4, C23, C23, Dic3, C12, D6, C2xC6, C2xC6, C42, C22:C4, C22:C4, C4:C4, C4:C4, C22xC4, C22xC4, C2xD4, C2xD4, C24, C4xS3, D12, C2xDic3, C2xDic3, C3:D4, C2xC12, C2xC12, C3xD4, C22xS3, C22xS3, C22xC6, C22wrC2, C4:D4, C4:D4, C22.D4, C42:2C2, C4:1D4, C4xDic3, Dic3:C4, C4:Dic3, D6:C4, C6.D4, C3xC22:C4, C3xC4:C4, S3xC2xC4, C2xD12, C22xDic3, C2xC3:D4, C22xC12, C6xD4, S3xC23, C22.54C24, C23.8D6, D6:D4, Dic3:D4, C23.21D6, D6.D4, C4:C4:S3, C23.28D6, C12:7D4, C23:2D6, D6:3D4, C23.14D6, C12:3D4, C3xC4:D4, C6.482+ 1+4
Quotients: C1, C2, C22, S3, C23, D6, C24, C22xS3, 2+ 1+4, S3xC23, C22.54C24, D4:6D6, D4oD12, C6.482+ 1+4

Smallest permutation representation of C6.482+ 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 7 34)(2 29 8 35)(3 30 9 36)(4 25 10 31)(5 26 11 32)(6 27 12 33)(13 37 19 43)(14 38 20 44)(15 39 21 45)(16 40 22 46)(17 41 23 47)(18 42 24 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 22 10 13)(2 21 11 18)(3 20 12 17)(4 19 7 16)(5 24 8 15)(6 23 9 14)(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 40)(26 41)(27 42)(28 37)(29 38)(30 39)(31 46)(32 47)(33 48)(34 43)(35 44)(36 45)
 
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), (1,28,7,34)(2,29,8,35)(3,30,9,36)(4,25,10,31)(5,26,11,32)(6,27,12,33)(13,37,19,43)(14,38,20,44)(15,39,21,45)(16,40,22,46)(17,41,23,47)(18,42,24,48), (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), (1,22,10,13)(2,21,11,18)(3,20,12,17)(4,19,7,16)(5,24,8,15)(6,23,9,14)(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)(25,40)(26,41)(27,42)(28,37)(29,38)(30,39)(31,46)(32,47)(33,48)(34,43)(35,44)(36,45)>;
 
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), (1,28,7,34)(2,29,8,35)(3,30,9,36)(4,25,10,31)(5,26,11,32)(6,27,12,33)(13,37,19,43)(14,38,20,44)(15,39,21,45)(16,40,22,46)(17,41,23,47)(18,42,24,48), (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), (1,22,10,13)(2,21,11,18)(3,20,12,17)(4,19,7,16)(5,24,8,15)(6,23,9,14)(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)(25,40)(26,41)(27,42)(28,37)(29,38)(30,39)(31,46)(32,47)(33,48)(34,43)(35,44)(36,45) );
 
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)], [(1,28,7,34),(2,29,8,35),(3,30,9,36),(4,25,10,31),(5,26,11,32),(6,27,12,33),(13,37,19,43),(14,38,20,44),(15,39,21,45),(16,40,22,46),(17,41,23,47),(18,42,24,48)], [(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)], [(1,22,10,13),(2,21,11,18),(3,20,12,17),(4,19,7,16),(5,24,8,15),(6,23,9,14),(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),(25,40),(26,41),(27,42),(28,37),(29,38),(30,39),(31,46),(32,47),(33,48),(34,43),(35,44),(36,45)]])
 

33 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I 3 4A4B4C4D4E···4I6A6B6C6D6E6F6G12A12B12C12D12E12F
order1222222222344444···46666666121212121212
size11114441212122444412···122224488444488

33 irreducible representations

dim1111111111111122222444
type+++++++++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2C2C2C2S3D6D6D6D62+ 1+4D4:6D6D4oD12
kernelC6.482+ 1+4C23.8D6D6:D4Dic3:D4C23.21D6D6.D4C4:C4:S3C23.28D6C12:7D4C23:2D6D6:3D4C23.14D6C12:3D4C3xC4:D4C4:D4C22:C4C4:C4C22xC4C2xD4C6C2C2
# reps1111111112121112113342

Matrix representation of C6.482+ 1+4 ►in GL8(F13)

1212000000
10000000
000120000
001120000
00001100
000012000
00000001
000000121
,
120880000
0121080000
112100000
21010000
000011900
00004200
00000029
000000411
,
10000000
01000000
1211200000
11120120000
00001000
00000100
0000121120
00001112012
,
24000000
211000000
00920000
001140000
00001055
0000121253
00001112012
0000121120
,
24000000
911000000
001140000
00920000
00001055
00000135
000000120
000000012

G:=sub<GL(8,GF(13))| [12,1,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1,1],[12,0,1,2,0,0,0,0,0,12,12,1,0,0,0,0,8,10,1,0,0,0,0,0,8,8,0,1,0,0,0,0,0,0,0,0,11,4,0,0,0,0,0,0,9,2,0,0,0,0,0,0,0,0,2,4,0,0,0,0,0,0,9,11],[1,0,12,11,0,0,0,0,0,1,1,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,12,11,0,0,0,0,0,1,1,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12],[2,2,0,0,0,0,0,0,4,11,0,0,0,0,0,0,0,0,9,11,0,0,0,0,0,0,2,4,0,0,0,0,0,0,0,0,1,12,11,12,0,0,0,0,0,12,12,1,0,0,0,0,5,5,0,12,0,0,0,0,5,3,12,0],[2,9,0,0,0,0,0,0,4,11,0,0,0,0,0,0,0,0,11,9,0,0,0,0,0,0,4,2,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,5,3,12,0,0,0,0,0,5,5,0,12] >;
 

C6.482+ 1+4 in GAP, Magma, Sage, TeX 

C_6._{48}2_+^{1+4}
 
% in TeX
 
G:=Group("C6.48ES+(2,2)");
 
// GroupNames label
 
G:=SmallGroup(192,1179);
 
// by ID
 
G=gap.SmallGroup(192,1179);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,758,219,1571,570,297,6278]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^6=b^4=c^2=e^2=1,d^2=a^3*b^2,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,a*e=e*a,c*b*c=b^-1,d*b*d^-1=a^3*b,b*e=e*b,d*c*d^-1=e*c*e=a^3*c,e*d*e=a^3*b^2*d>;
 
// generators/relations
 

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