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G = C23×C6order 48 = 24·3

Abelian group of type [2,2,2,6]

direct product, abelian, monomial, 2-elementary

Aliases: C23×C6, SmallGroup(48,52)

Series: Derived Chief Lower central Upper central

C1 — C23×C6
C1C3C6C2×C6C22×C6 — C23×C6
C1 — C23×C6
C1 — C23×C6

Generators and relations for C23×C6
 G = < a,b,c,d | a2=b2=c2=d6=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, cd=dc >

Subgroups: 134, all normal (4 characteristic)
C1, C2 [×15], C3, C22 [×35], C6 [×15], C23 [×15], C2×C6 [×35], C24, C22×C6 [×15], C23×C6
Quotients: C1, C2 [×15], C3, C22 [×35], C6 [×15], C23 [×15], C2×C6 [×35], C24, C22×C6 [×15], C23×C6

Smallest permutation representation of C23×C6
Regular action on 48 points
Generators in S48
(1 29)(2 30)(3 25)(4 26)(5 27)(6 28)(7 31)(8 32)(9 33)(10 34)(11 35)(12 36)(13 37)(14 38)(15 39)(16 40)(17 41)(18 42)(19 43)(20 44)(21 45)(22 46)(23 47)(24 48)
(1 17)(2 18)(3 13)(4 14)(5 15)(6 16)(7 19)(8 20)(9 21)(10 22)(11 23)(12 24)(25 37)(26 38)(27 39)(28 40)(29 41)(30 42)(31 43)(32 44)(33 45)(34 46)(35 47)(36 48)
(1 11)(2 12)(3 7)(4 8)(5 9)(6 10)(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)(25 31)(26 32)(27 33)(28 34)(29 35)(30 36)(37 43)(38 44)(39 45)(40 46)(41 47)(42 48)
(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)| (1,29)(2,30)(3,25)(4,26)(5,27)(6,28)(7,31)(8,32)(9,33)(10,34)(11,35)(12,36)(13,37)(14,38)(15,39)(16,40)(17,41)(18,42)(19,43)(20,44)(21,45)(22,46)(23,47)(24,48), (1,17)(2,18)(3,13)(4,14)(5,15)(6,16)(7,19)(8,20)(9,21)(10,22)(11,23)(12,24)(25,37)(26,38)(27,39)(28,40)(29,41)(30,42)(31,43)(32,44)(33,45)(34,46)(35,47)(36,48), (1,11)(2,12)(3,7)(4,8)(5,9)(6,10)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24)(25,31)(26,32)(27,33)(28,34)(29,35)(30,36)(37,43)(38,44)(39,45)(40,46)(41,47)(42,48), (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( (1,29)(2,30)(3,25)(4,26)(5,27)(6,28)(7,31)(8,32)(9,33)(10,34)(11,35)(12,36)(13,37)(14,38)(15,39)(16,40)(17,41)(18,42)(19,43)(20,44)(21,45)(22,46)(23,47)(24,48), (1,17)(2,18)(3,13)(4,14)(5,15)(6,16)(7,19)(8,20)(9,21)(10,22)(11,23)(12,24)(25,37)(26,38)(27,39)(28,40)(29,41)(30,42)(31,43)(32,44)(33,45)(34,46)(35,47)(36,48), (1,11)(2,12)(3,7)(4,8)(5,9)(6,10)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24)(25,31)(26,32)(27,33)(28,34)(29,35)(30,36)(37,43)(38,44)(39,45)(40,46)(41,47)(42,48), (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([(1,29),(2,30),(3,25),(4,26),(5,27),(6,28),(7,31),(8,32),(9,33),(10,34),(11,35),(12,36),(13,37),(14,38),(15,39),(16,40),(17,41),(18,42),(19,43),(20,44),(21,45),(22,46),(23,47),(24,48)], [(1,17),(2,18),(3,13),(4,14),(5,15),(6,16),(7,19),(8,20),(9,21),(10,22),(11,23),(12,24),(25,37),(26,38),(27,39),(28,40),(29,41),(30,42),(31,43),(32,44),(33,45),(34,46),(35,47),(36,48)], [(1,11),(2,12),(3,7),(4,8),(5,9),(6,10),(13,19),(14,20),(15,21),(16,22),(17,23),(18,24),(25,31),(26,32),(27,33),(28,34),(29,35),(30,36),(37,43),(38,44),(39,45),(40,46),(41,47),(42,48)], [(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···2O3A3B6A···6AD
order12···2336···6
size11···1111···1

48 irreducible representations

dim1111
type++
imageC1C2C3C6
kernelC23×C6C22×C6C24C23
# reps115230

Matrix representation of C23×C6 in GL4(𝔽7) generated by

6000
0600
0060
0006
,
6000
0600
0060
0001
,
6000
0600
0010
0006
,
5000
0200
0020
0005
G:=sub<GL(4,GF(7))| [6,0,0,0,0,6,0,0,0,0,6,0,0,0,0,6],[6,0,0,0,0,6,0,0,0,0,6,0,0,0,0,1],[6,0,0,0,0,6,0,0,0,0,1,0,0,0,0,6],[5,0,0,0,0,2,0,0,0,0,2,0,0,0,0,5] >;

C23×C6 in GAP, Magma, Sage, TeX

C_2^3\times C_6
% in TeX

G:=Group("C2^3xC6");
// GroupNames label

G:=SmallGroup(48,52);
// by ID

G=gap.SmallGroup(48,52);
# by ID

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

G:=Group<a,b,c,d|a^2=b^2=c^2=d^6=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,c*d=d*c>;
// generators/relations

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