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## G = C3×D42order 192 = 26·3

### Direct product of C3, D4 and D4

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

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

Generators and relations for C3×D42
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 >

Subgroups: 778 in 428 conjugacy classes, 182 normal (14 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C22, C6, C6, C6, C2×C4, C2×C4, C2×C4, D4, D4, C23, C23, C12, C12, C2×C6, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×D4, C24, C2×C12, C2×C12, C2×C12, C3×D4, C3×D4, C22×C6, C22×C6, C4×D4, C22≀C2, C4⋊D4, C41D4, C22×D4, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C22×C12, C6×D4, C6×D4, C23×C6, D42, D4×C12, C3×C22≀C2, C3×C4⋊D4, C3×C41D4, D4×C2×C6, C3×D42
Quotients: C1, C2, C3, C22, C6, D4, C23, C2×C6, C2×D4, C24, C3×D4, C22×C6, C22×D4, 2+ 1+4, C6×D4, C23×C6, D42, D4×C2×C6, C3×2+ 1+4, C3×D42

Smallest permutation representation of C3×D42
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)]])

75 conjugacy classes

 class 1 2A 2B 2C 2D ··· 2K 2L 2M 2N 2O 3A 3B 4A 4B 4C 4D 4E ··· 4I 6A ··· 6F 6G ··· 6V 6W ··· 6AD 12A ··· 12H 12I ··· 12R order 1 2 2 2 2 ··· 2 2 2 2 2 3 3 4 4 4 4 4 ··· 4 6 ··· 6 6 ··· 6 6 ··· 6 12 ··· 12 12 ··· 12 size 1 1 1 1 2 ··· 2 4 4 4 4 1 1 2 2 2 2 4 ··· 4 1 ··· 1 2 ··· 2 4 ··· 4 2 ··· 2 4 ··· 4

75 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 4 4 type + + + + + + + + image C1 C2 C2 C2 C2 C2 C3 C6 C6 C6 C6 C6 D4 C3×D4 2+ 1+4 C3×2+ 1+4 kernel C3×D42 D4×C12 C3×C22≀C2 C3×C4⋊D4 C3×C4⋊1D4 D4×C2×C6 D42 C4×D4 C22≀C2 C4⋊D4 C4⋊1D4 C22×D4 C3×D4 D4 C6 C2 # reps 1 2 4 4 1 4 2 4 8 8 2 8 8 16 1 2

Matrix representation of C3×D42 in GL4(𝔽13) generated by

 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 11 0 0 0 5
,
 1 0 0 0 0 1 0 0 0 0 8 11 0 0 12 5
,
 1 2 0 0 12 12 0 0 0 0 12 0 0 0 0 12
,
 1 0 0 0 12 12 0 0 0 0 12 0 0 0 0 12
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] >;

C3×D42 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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