Copied to
clipboard

## G = C22×C12order 48 = 24·3

### Abelian group of type [2,2,12]

Aliases: C22×C12, SmallGroup(48,44)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22×C12
 Chief series C1 — C2 — C6 — C12 — C2×C12 — C22×C12
 Lower central C1 — C22×C12
 Upper central 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, C3, C4, C22, C6, C6, C2×C4, C23, C12, C2×C6, C22×C4, C2×C12, C22×C6, C22×C12
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, C23, C12, C2×C6, C22×C4, C2×C12, C22×C6, C22×C12

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

C22×C12 is a maximal subgroup of   C12.55D4  C6.C42  C12.48D4  C23.26D6  C23.28D6  C127D4

48 conjugacy classes

 class 1 2A ··· 2G 3A 3B 4A ··· 4H 6A ··· 6N 12A ··· 12P order 1 2 ··· 2 3 3 4 ··· 4 6 ··· 6 12 ··· 12 size 1 1 ··· 1 1 1 1 ··· 1 1 ··· 1 1 ··· 1

48 irreducible representations

 dim 1 1 1 1 1 1 1 1 type + + + image C1 C2 C2 C3 C4 C6 C6 C12 kernel C22×C12 C2×C12 C22×C6 C22×C4 C2×C6 C2×C4 C23 C22 # reps 1 6 1 2 8 12 2 16

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

 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
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

׿
×
𝔽