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G = C4×C12order 48 = 24·3

Abelian group of type [4,12]

Aliases: C4×C12, SmallGroup(48,20)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C4×C12
 Chief series C1 — C2 — C22 — C2×C6 — C2×C12 — C4×C12
 Lower central C1 — C4×C12
 Upper central C1 — C4×C12

Generators and relations for C4×C12
G = < a,b | a4=b12=1, ab=ba >

Smallest permutation representation of C4×C12
Regular action on 48 points
Generators in S48
(1 42 17 25)(2 43 18 26)(3 44 19 27)(4 45 20 28)(5 46 21 29)(6 47 22 30)(7 48 23 31)(8 37 24 32)(9 38 13 33)(10 39 14 34)(11 40 15 35)(12 41 16 36)
(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)

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

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

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

C4×C12 is a maximal subgroup of
C42.S3  C12⋊C8  C424S3  C122Q8  C12.6Q8  C422S3  C4⋊D12  C427S3  C423S3  C42⋊C9

48 conjugacy classes

 class 1 2A 2B 2C 3A 3B 4A ··· 4L 6A ··· 6F 12A ··· 12X order 1 2 2 2 3 3 4 ··· 4 6 ··· 6 12 ··· 12 size 1 1 1 1 1 1 1 ··· 1 1 ··· 1 1 ··· 1

48 irreducible representations

 dim 1 1 1 1 1 1 type + + image C1 C2 C3 C4 C6 C12 kernel C4×C12 C2×C12 C42 C12 C2×C4 C4 # reps 1 3 2 12 6 24

Matrix representation of C4×C12 in GL2(𝔽13) generated by

 12 0 0 5
,
 5 0 0 4
G:=sub<GL(2,GF(13))| [12,0,0,5],[5,0,0,4] >;

C4×C12 in GAP, Magma, Sage, TeX

C_4\times C_{12}
% in TeX

G:=Group("C4xC12");
// GroupNames label

G:=SmallGroup(48,20);
// by ID

G=gap.SmallGroup(48,20);
# by ID

G:=PCGroup([5,-2,-2,-3,-2,-2,60,126]);
// Polycyclic

G:=Group<a,b|a^4=b^12=1,a*b=b*a>;
// generators/relations

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