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G = C22×C12order 48 = 24·3

Abelian group of type [2,2,12]

direct product, abelian, monomial, 2-elementary

Aliases: C22×C12, SmallGroup(48,44)

Series: Derived Chief Lower central Upper central

C1 — C22×C12
C1C2C6C12C2×C12 — C22×C12
C1 — C22×C12
C1 — C22×C12

Generators and relations for C22×C12
 G = < a,b,c | a2=b2=c12=1, ab=ba, ac=ca, bc=cb >

Subgroups: 54, all normal (8 characteristic)
C1, C2, C2 [×6], C3, C4 [×4], C22 [×7], C6, C6 [×6], C2×C4 [×6], C23, C12 [×4], C2×C6 [×7], C22×C4, C2×C12 [×6], C22×C6, C22×C12
Quotients: C1, C2 [×7], C3, C4 [×4], C22 [×7], C6 [×7], C2×C4 [×6], C23, C12 [×4], C2×C6 [×7], C22×C4, C2×C12 [×6], C22×C6, C22×C12

Smallest permutation representation of C22×C12
Regular action on 48 points
Generators in S48
(1 24)(2 13)(3 14)(4 15)(5 16)(6 17)(7 18)(8 19)(9 20)(10 21)(11 22)(12 23)(25 38)(26 39)(27 40)(28 41)(29 42)(30 43)(31 44)(32 45)(33 46)(34 47)(35 48)(36 37)
(1 48)(2 37)(3 38)(4 39)(5 40)(6 41)(7 42)(8 43)(9 44)(10 45)(11 46)(12 47)(13 36)(14 25)(15 26)(16 27)(17 28)(18 29)(19 30)(20 31)(21 32)(22 33)(23 34)(24 35)
(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)| (1,24)(2,13)(3,14)(4,15)(5,16)(6,17)(7,18)(8,19)(9,20)(10,21)(11,22)(12,23)(25,38)(26,39)(27,40)(28,41)(29,42)(30,43)(31,44)(32,45)(33,46)(34,47)(35,48)(36,37), (1,48)(2,37)(3,38)(4,39)(5,40)(6,41)(7,42)(8,43)(9,44)(10,45)(11,46)(12,47)(13,36)(14,25)(15,26)(16,27)(17,28)(18,29)(19,30)(20,31)(21,32)(22,33)(23,34)(24,35), (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( (1,24)(2,13)(3,14)(4,15)(5,16)(6,17)(7,18)(8,19)(9,20)(10,21)(11,22)(12,23)(25,38)(26,39)(27,40)(28,41)(29,42)(30,43)(31,44)(32,45)(33,46)(34,47)(35,48)(36,37), (1,48)(2,37)(3,38)(4,39)(5,40)(6,41)(7,42)(8,43)(9,44)(10,45)(11,46)(12,47)(13,36)(14,25)(15,26)(16,27)(17,28)(18,29)(19,30)(20,31)(21,32)(22,33)(23,34)(24,35), (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([(1,24),(2,13),(3,14),(4,15),(5,16),(6,17),(7,18),(8,19),(9,20),(10,21),(11,22),(12,23),(25,38),(26,39),(27,40),(28,41),(29,42),(30,43),(31,44),(32,45),(33,46),(34,47),(35,48),(36,37)], [(1,48),(2,37),(3,38),(4,39),(5,40),(6,41),(7,42),(8,43),(9,44),(10,45),(11,46),(12,47),(13,36),(14,25),(15,26),(16,27),(17,28),(18,29),(19,30),(20,31),(21,32),(22,33),(23,34),(24,35)], [(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)])

48 conjugacy classes

class 1 2A···2G3A3B4A···4H6A···6N12A···12P
order12···2334···46···612···12
size11···1111···11···11···1

48 irreducible representations

dim11111111
type+++
imageC1C2C2C3C4C6C6C12
kernelC22×C12C2×C12C22×C6C22×C4C2×C6C2×C4C23C22
# reps1612812216

Matrix representation of C22×C12 in GL3(𝔽13) generated by

1200
0120
001
,
100
010
0012
,
1200
0110
008
G:=sub<GL(3,GF(13))| [12,0,0,0,12,0,0,0,1],[1,0,0,0,1,0,0,0,12],[12,0,0,0,11,0,0,0,8] >;

C22×C12 in GAP, Magma, Sage, TeX

C_2^2\times C_{12}
% in TeX

G:=Group("C2^2xC12");
// GroupNames label

G:=SmallGroup(48,44);
// by ID

G=gap.SmallGroup(48,44);
# by ID

G:=PCGroup([5,-2,-2,-2,-3,-2,120]);
// Polycyclic

G:=Group<a,b,c|a^2=b^2=c^12=1,a*b=b*a,a*c=c*a,b*c=c*b>;
// generators/relations

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