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G = C3×D42  order 192 = 26·3

Direct product of C3, D4 and D4

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

Aliases: C3×D42, C6.1602+ (1+4), C4⋊2(C6×D4), (C4×D4)⋊14C6, C4⋊1D4⋊9C6, C12⋊14(C2×D4), C24⋊6(C2×C6), C22⋊3(C6×D4), C22≀C2⋊7C6, (D4×C12)⋊43C2, C4⋊D4⋊10C6, C42⋊11(C2×C6), (C4×C12)⋊42C22, (C6×D4)⋊64C22, (C22×D4)⋊11C6, (C23×C6)⋊4C22, (C2×C6).365C24, C6.193(C22×D4), (C2×C12).674C23, (C22×C12)⋊50C22, C22.39(C23×C6), C23.15(C22×C6), (C22×C6).264C23, C2.12(C3×2+ (1+4)), (D4×C2×C6)⋊23C2, C4⋊C4⋊16(C2×C6), C2.17(D4×C2×C6), (C2×C6)⋊13(C2×D4), (C2×D4)⋊12(C2×C6), C22⋊C4⋊5(C2×C6), (C3×C4⋊D4)⋊37C2, (C3×C4⋊1D4)⋊18C2, (C3×C4⋊C4)⋊72C22, (C22×C4)⋊11(C2×C6), (C3×C22≀C2)⋊15C2, (C2×C4).32(C22×C6), (C3×C22⋊C4)⋊40C22, SmallGroup(192,1434)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C22 — C3×D42
Chief series
C1 — C2 — C22 — C2×C6 — C22×C6 — C6×D4 — C3×C22≀C2 — C3×D42
Lower central
C1 — C22 — C3×D42
Upper central
C1 — C2×C6 — C3×D42

Subgroups: 778 in 428 conjugacy classes, 182 normal (14 characteristic)
C1, C2, C2 [×2], C2 [×12], C3, C4 [×4], C4 [×5], C22, C22 [×8], C22 [×36], C6, C6 [×2], C6 [×12], C2×C4, C2×C4 [×6], C2×C4 [×8], D4 [×8], D4 [×26], C23 [×8], C23 [×20], C12 [×4], C12 [×5], C2×C6, C2×C6 [×8], C2×C6 [×36], C42, C22⋊C4 [×8], C4⋊C4 [×2], C22×C4 [×4], C2×D4 [×16], C2×D4 [×16], C24 [×4], C2×C12, C2×C12 [×6], C2×C12 [×8], C3×D4 [×8], C3×D4 [×26], C22×C6 [×8], C22×C6 [×20], C4×D4 [×2], C22≀C2 [×4], C4⋊D4 [×4], C4⋊1D4, C22×D4 [×4], C4×C12, C3×C22⋊C4 [×8], C3×C4⋊C4 [×2], C22×C12 [×4], C6×D4 [×16], C6×D4 [×16], C23×C6 [×4], D42, D4×C12 [×2], C3×C22≀C2 [×4], C3×C4⋊D4 [×4], C3×C4⋊1D4, D4×C2×C6 [×4], C3×D42

Quotients:
C1, C2 [×15], C3, C22 [×35], C6 [×15], D4 [×8], C23 [×15], C2×C6 [×35], C2×D4 [×12], C24, C3×D4 [×8], C22×C6 [×15], C22×D4 [×2], 2+ (1+4), C6×D4 [×12], C23×C6, D42, D4×C2×C6 [×2], C3×2+ (1+4), C3×D42

Generators and relations
 G = < a,b,c,d,e | a3=b4=c2=d4=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

Smallest permutation representation
►On 48 points
Generators in S48
(1 39 47)(2 40 48)(3 37 45)(4 38 46)(5 35 43)(6 36 44)(7 33 41)(8 34 42)(9 13 26)(10 14 27)(11 15 28)(12 16 25)(17 21 31)(18 22 32)(19 23 29)(20 24 30)

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

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

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

Matrix representation ►G ⊆ GL4(𝔽13) generated by

3000
0300
0010
0001
,
12000
01200
00811
0005
,
1000
0100
00811
00125
,
1200
121200
00120
00012
,
1000
121200
00120
00012
G:=sub<GL(4,GF(13))| [3,0,0,0,0,3,0,0,0,0,1,0,0,0,0,1],[12,0,0,0,0,12,0,0,0,0,8,0,0,0,11,5],[1,0,0,0,0,1,0,0,0,0,8,12,0,0,11,5],[1,12,0,0,2,12,0,0,0,0,12,0,0,0,0,12],[1,12,0,0,0,12,0,0,0,0,12,0,0,0,0,12] >;
 

75 conjugacy classes

class 1 2A2B2C2D···2K2L2M2N2O3A3B4A4B4C4D4E···4I6A···6F6G···6V6W···6AD12A···12H12I···12R
order12222···222223344444···46···66···66···612···1212···12
size11112···244441122224···41···12···24···42···24···4

75 irreducible representations

dim1111111111112244
type++++++++
imageC1C2C2C2C2C2C3C6C6C6C6C6D4C3×D42+ (1+4)C3×2+ (1+4)
kernelC3×D42D4×C12C3×C22≀C2C3×C4⋊D4C3×C4⋊1D4D4×C2×C6D42C4×D4C22≀C2C4⋊D4C4⋊1D4C22×D4C3×D4D4C6C2
# reps12441424882881612

In GAP, Magma, Sage, TeX 

C_3\times D_4^2
 
% in TeX
 
G:=Group("C3xD4^2");
 
// GroupNames label
 
G:=SmallGroup(192,1434);
 
// by ID
 
G=gap.SmallGroup(192,1434);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-2,701,2102,794]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^3=b^4=c^2=d^4=e^2=1,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=d^-1>;
 
// generators/relations
 

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