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G = C23×C6order 48 = 24·3

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

Aliases: C23×C6, SmallGroup(48,52)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C23×C6
 Chief series C1 — C3 — C6 — C2×C6 — C22×C6 — C23×C6
 Lower central C1 — C23×C6
 Upper central 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, C3, C22, C6, C23, C2×C6, C24, C22×C6, C23×C6
Quotients: C1, C2, C3, C22, C6, C23, C2×C6, C24, C22×C6, 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)]])

C23×C6 is a maximal subgroup of   C244S3  C24⋊C9

48 conjugacy classes

 class 1 2A ··· 2O 3A 3B 6A ··· 6AD order 1 2 ··· 2 3 3 6 ··· 6 size 1 1 ··· 1 1 1 1 ··· 1

48 irreducible representations

 dim 1 1 1 1 type + + image C1 C2 C3 C6 kernel C23×C6 C22×C6 C24 C23 # reps 1 15 2 30

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

 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
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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