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G = C6x2+ 1+4  order 192 = 26·3

Direct product of C6 and 2+ 1+4

direct product, metabelian, nilpotent (class 2), monomial, 2-elementary

Aliases: C6x2+ 1+4, C6.24C25, C12.90C24, C24:8(C2xC6), D4:4(C22xC6), C2.4(C24xC6), Q8:5(C22xC6), (C2xC12):11C23, (C22xD4):15C6, (C6xD4):68C22, (C3xD4):15C23, C4.13(C23xC6), C23:2(C22xC6), (C22xC6):4C23, (C23xC6):6C22, (C3xQ8):14C23, (C6xQ8):60C22, (C2xC6).387C24, C22.2(C23xC6), (C22xC12):53C22, (D4xC2xC6):27C2, C4oD4:8(C2xC6), (C6xC4oD4):29C2, (C2xC4oD4):17C6, (C2xD4):17(C2xC6), (C2xC4):2(C22xC6), (C2xQ8):22(C2xC6), (C22xC4):14(C2xC6), (C3xC4oD4):26C22, SmallGroup(192,1534)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2 — C6x2+ 1+4
Chief series
C1 — C2 — C6 — C2xC6 — C3xD4 — C6xD4 — C3x2+ 1+4 — C6x2+ 1+4
Lower central
C1 — C2 — C6x2+ 1+4
Upper central
C1 — C2xC6 — C6x2+ 1+4

Generators and relations for C6x2+ 1+4
 G = < a,b,c,d,e | a6=b4=c2=e2=1, d2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=b2d >

Subgroups: 1186 in 898 conjugacy classes, 754 normal (8 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, C6, C6, C6, C2xC4, D4, Q8, C23, C23, C12, C2xC6, C2xC6, C2xC6, C22xC4, C2xD4, C2xQ8, C4oD4, C24, C2xC12, C3xD4, C3xQ8, C22xC6, C22xC6, C22xD4, C2xC4oD4, 2+ 1+4, C22xC12, C6xD4, C6xQ8, C3xC4oD4, C23xC6, C2x2+ 1+4, D4xC2xC6, C6xC4oD4, C3x2+ 1+4, C6x2+ 1+4
Quotients: C1, C2, C3, C22, C6, C23, C2xC6, C24, C22xC6, 2+ 1+4, C25, C23xC6, C2x2+ 1+4, C3x2+ 1+4, C24xC6, C6x2+ 1+4

Smallest permutation representation of C6x2+ 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 43 18 8)(2 44 13 9)(3 45 14 10)(4 46 15 11)(5 47 16 12)(6 48 17 7)(19 34 29 38)(20 35 30 39)(21 36 25 40)(22 31 26 41)(23 32 27 42)(24 33 28 37)

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

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

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

102 conjugacy classes

class 1 2A2B2C2D···2U3A3B4A···4L6A···6F6G···6AP12A···12X
order12222···2334···46···66···612···12
size11112···2112···21···12···22···2

102 irreducible representations

dim1111111144
type+++++
imageC1C2C2C2C3C6C6C62+ 1+4C3x2+ 1+4
kernelC6x2+ 1+4D4xC2xC6C6xC4oD4C3x2+ 1+4C2x2+ 1+4C22xD4C2xC4oD42+ 1+4C6C2
# reps19616218123224

Matrix representation of C6x2+ 1+4 ►in GL5(F13)

120000
010000
001000
000100
000010
,
120000
000012
00010
001200
01000
,
120000
00100
01000
00001
00010
,
120000
00001
00010
001200
012000
,
10000
00010
00001
01000
00100

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

C6x2+ 1+4 in GAP, Magma, Sage, TeX 

C_6\times 2_+^{1+4}
 
% in TeX
 
G:=Group("C6xES+(2,2)");
 
// GroupNames label
 
G:=SmallGroup(192,1534);
 
// by ID
 
G=gap.SmallGroup(192,1534);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-2,1373,1059,2915]);
 
// 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,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=b^2*d>;
 
// generators/relations
 

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