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G = C6×2+ 1+4  order 192 = 26·3

Direct product of C6 and 2+ 1+4

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

Aliases: C6×2+ 1+4, C6.24C25, C12.90C24, C24⋊8(C2×C6), D4⋊4(C22×C6), C2.4(C24×C6), Q8⋊5(C22×C6), (C2×C12)⋊11C23, (C22×D4)⋊15C6, (C6×D4)⋊68C22, (C3×D4)⋊15C23, C4.13(C23×C6), C23⋊2(C22×C6), (C22×C6)⋊4C23, (C23×C6)⋊6C22, (C3×Q8)⋊14C23, (C6×Q8)⋊60C22, (C2×C6).387C24, C22.2(C23×C6), (C22×C12)⋊53C22, (D4×C2×C6)⋊27C2, C4○D4⋊8(C2×C6), (C6×C4○D4)⋊29C2, (C2×C4○D4)⋊17C6, (C2×D4)⋊17(C2×C6), (C2×C4)⋊2(C22×C6), (C2×Q8)⋊22(C2×C6), (C22×C4)⋊14(C2×C6), (C3×C4○D4)⋊26C22, SmallGroup(192,1534)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2 — C6×2+ 1+4
Chief series
C1 — C2 — C6 — C2×C6 — C3×D4 — C6×D4 — C3×2+ 1+4 — C6×2+ 1+4
Lower central
C1 — C2 — C6×2+ 1+4
Upper central
C1 — C2×C6 — C6×2+ 1+4

Generators and relations for C6×2+ 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, C2×C4, D4, Q8, C23, C23, C12, C2×C6, C2×C6, C2×C6, C22×C4, C2×D4, C2×Q8, C4○D4, C24, C2×C12, C3×D4, C3×Q8, C22×C6, C22×C6, C22×D4, C2×C4○D4, 2+ 1+4, C22×C12, C6×D4, C6×Q8, C3×C4○D4, C23×C6, C2×2+ 1+4, D4×C2×C6, C6×C4○D4, C3×2+ 1+4, C6×2+ 1+4
Quotients: C1, C2, C3, C22, C6, C23, C2×C6, C24, C22×C6, 2+ 1+4, C25, C23×C6, C2×2+ 1+4, C3×2+ 1+4, C24×C6, C6×2+ 1+4

Smallest permutation representation of C6×2+ 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+4C3×2+ 1+4
kernelC6×2+ 1+4D4×C2×C6C6×C4○D4C3×2+ 1+4C2×2+ 1+4C22×D4C2×C4○D42+ 1+4C6C2
# reps19616218123224

Matrix representation of C6×2+ 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] >;
 

C6×2+ 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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