Copied to
clipboard

G = A4xC24order 192 = 26·3

Direct product of C24 and A4

direct product, metabelian, soluble, monomial, A-group

Aliases: A4xC24, C26:1C3, C25:3C6, C24:9(C2xC6), C22:(C23xC6), C23:3(C22xC6), SmallGroup(192,1539)

Series: Derived Chief Lower central Upper central

C1C22 — A4xC24
C1C22A4C2xA4C22xA4C23xA4 — A4xC24
C22 — A4xC24
C1C24

Generators and relations for A4xC24
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=e2=f2=g3=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, bd=db, be=eb, bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, cg=gc, de=ed, df=fd, dg=gd, geg-1=ef=fe, gfg-1=e >

Subgroups: 3160 in 1165 conjugacy classes, 201 normal (6 characteristic)
C1, C2, C2, C3, C22, C22, C22, C6, C23, C23, A4, C2xC6, C24, C24, C24, C2xA4, C22xC6, C25, C25, C22xA4, C23xC6, C26, C23xA4, A4xC24
Quotients: C1, C2, C3, C22, C6, C23, A4, C2xC6, C24, C2xA4, C22xC6, C22xA4, C23xC6, C23xA4, A4xC24

Smallest permutation representation of A4xC24
On 48 points
Generators in S48
(1 8)(2 9)(3 7)(4 18)(5 16)(6 17)(10 44)(11 45)(12 43)(13 47)(14 48)(15 46)(19 36)(20 34)(21 35)(22 32)(23 33)(24 31)(25 40)(26 41)(27 42)(28 37)(29 38)(30 39)
(1 20)(2 21)(3 19)(4 32)(5 33)(6 31)(7 36)(8 34)(9 35)(10 30)(11 28)(12 29)(13 25)(14 26)(15 27)(16 23)(17 24)(18 22)(37 45)(38 43)(39 44)(40 47)(41 48)(42 46)
(1 4)(2 5)(3 6)(7 17)(8 18)(9 16)(10 46)(11 47)(12 48)(13 45)(14 43)(15 44)(19 31)(20 32)(21 33)(22 34)(23 35)(24 36)(25 37)(26 38)(27 39)(28 40)(29 41)(30 42)
(1 27)(2 25)(3 26)(4 39)(5 37)(6 38)(7 41)(8 42)(9 40)(10 22)(11 23)(12 24)(13 21)(14 19)(15 20)(16 28)(17 29)(18 30)(31 43)(32 44)(33 45)(34 46)(35 47)(36 48)
(1 15)(2 16)(3 12)(4 44)(5 9)(6 48)(7 43)(8 46)(10 18)(11 13)(14 17)(19 29)(20 27)(21 23)(22 30)(24 26)(25 28)(31 41)(32 39)(33 35)(34 42)(36 38)(37 40)(45 47)
(1 10)(2 13)(3 17)(4 46)(5 45)(6 7)(8 44)(9 47)(11 16)(12 14)(15 18)(19 24)(20 30)(21 25)(22 27)(23 28)(26 29)(31 36)(32 42)(33 37)(34 39)(35 40)(38 41)(43 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)| (1,8)(2,9)(3,7)(4,18)(5,16)(6,17)(10,44)(11,45)(12,43)(13,47)(14,48)(15,46)(19,36)(20,34)(21,35)(22,32)(23,33)(24,31)(25,40)(26,41)(27,42)(28,37)(29,38)(30,39), (1,20)(2,21)(3,19)(4,32)(5,33)(6,31)(7,36)(8,34)(9,35)(10,30)(11,28)(12,29)(13,25)(14,26)(15,27)(16,23)(17,24)(18,22)(37,45)(38,43)(39,44)(40,47)(41,48)(42,46), (1,4)(2,5)(3,6)(7,17)(8,18)(9,16)(10,46)(11,47)(12,48)(13,45)(14,43)(15,44)(19,31)(20,32)(21,33)(22,34)(23,35)(24,36)(25,37)(26,38)(27,39)(28,40)(29,41)(30,42), (1,27)(2,25)(3,26)(4,39)(5,37)(6,38)(7,41)(8,42)(9,40)(10,22)(11,23)(12,24)(13,21)(14,19)(15,20)(16,28)(17,29)(18,30)(31,43)(32,44)(33,45)(34,46)(35,47)(36,48), (1,15)(2,16)(3,12)(4,44)(5,9)(6,48)(7,43)(8,46)(10,18)(11,13)(14,17)(19,29)(20,27)(21,23)(22,30)(24,26)(25,28)(31,41)(32,39)(33,35)(34,42)(36,38)(37,40)(45,47), (1,10)(2,13)(3,17)(4,46)(5,45)(6,7)(8,44)(9,47)(11,16)(12,14)(15,18)(19,24)(20,30)(21,25)(22,27)(23,28)(26,29)(31,36)(32,42)(33,37)(34,39)(35,40)(38,41)(43,48), (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( (1,8)(2,9)(3,7)(4,18)(5,16)(6,17)(10,44)(11,45)(12,43)(13,47)(14,48)(15,46)(19,36)(20,34)(21,35)(22,32)(23,33)(24,31)(25,40)(26,41)(27,42)(28,37)(29,38)(30,39), (1,20)(2,21)(3,19)(4,32)(5,33)(6,31)(7,36)(8,34)(9,35)(10,30)(11,28)(12,29)(13,25)(14,26)(15,27)(16,23)(17,24)(18,22)(37,45)(38,43)(39,44)(40,47)(41,48)(42,46), (1,4)(2,5)(3,6)(7,17)(8,18)(9,16)(10,46)(11,47)(12,48)(13,45)(14,43)(15,44)(19,31)(20,32)(21,33)(22,34)(23,35)(24,36)(25,37)(26,38)(27,39)(28,40)(29,41)(30,42), (1,27)(2,25)(3,26)(4,39)(5,37)(6,38)(7,41)(8,42)(9,40)(10,22)(11,23)(12,24)(13,21)(14,19)(15,20)(16,28)(17,29)(18,30)(31,43)(32,44)(33,45)(34,46)(35,47)(36,48), (1,15)(2,16)(3,12)(4,44)(5,9)(6,48)(7,43)(8,46)(10,18)(11,13)(14,17)(19,29)(20,27)(21,23)(22,30)(24,26)(25,28)(31,41)(32,39)(33,35)(34,42)(36,38)(37,40)(45,47), (1,10)(2,13)(3,17)(4,46)(5,45)(6,7)(8,44)(9,47)(11,16)(12,14)(15,18)(19,24)(20,30)(21,25)(22,27)(23,28)(26,29)(31,36)(32,42)(33,37)(34,39)(35,40)(38,41)(43,48), (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([[(1,8),(2,9),(3,7),(4,18),(5,16),(6,17),(10,44),(11,45),(12,43),(13,47),(14,48),(15,46),(19,36),(20,34),(21,35),(22,32),(23,33),(24,31),(25,40),(26,41),(27,42),(28,37),(29,38),(30,39)], [(1,20),(2,21),(3,19),(4,32),(5,33),(6,31),(7,36),(8,34),(9,35),(10,30),(11,28),(12,29),(13,25),(14,26),(15,27),(16,23),(17,24),(18,22),(37,45),(38,43),(39,44),(40,47),(41,48),(42,46)], [(1,4),(2,5),(3,6),(7,17),(8,18),(9,16),(10,46),(11,47),(12,48),(13,45),(14,43),(15,44),(19,31),(20,32),(21,33),(22,34),(23,35),(24,36),(25,37),(26,38),(27,39),(28,40),(29,41),(30,42)], [(1,27),(2,25),(3,26),(4,39),(5,37),(6,38),(7,41),(8,42),(9,40),(10,22),(11,23),(12,24),(13,21),(14,19),(15,20),(16,28),(17,29),(18,30),(31,43),(32,44),(33,45),(34,46),(35,47),(36,48)], [(1,15),(2,16),(3,12),(4,44),(5,9),(6,48),(7,43),(8,46),(10,18),(11,13),(14,17),(19,29),(20,27),(21,23),(22,30),(24,26),(25,28),(31,41),(32,39),(33,35),(34,42),(36,38),(37,40),(45,47)], [(1,10),(2,13),(3,17),(4,46),(5,45),(6,7),(8,44),(9,47),(11,16),(12,14),(15,18),(19,24),(20,30),(21,25),(22,27),(23,28),(26,29),(31,36),(32,42),(33,37),(34,39),(35,40),(38,41),(43,48)], [(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)]])

64 conjugacy classes

class 1 2A···2O2P···2AE3A3B6A···6AD
order12···22···2336···6
size11···13···3444···4

64 irreducible representations

dim111133
type++++
imageC1C2C3C6A4C2xA4
kernelA4xC24C23xA4C26C25C24C23
# reps115230115

Matrix representation of A4xC24 in GL6(Z)

100000
0-10000
00-1000
000100
000010
000001
,
100000
0-10000
001000
000100
000010
000001
,
100000
010000
001000
000-100
0000-10
00000-1
,
-100000
010000
00-1000
000-100
0000-10
00000-1
,
100000
010000
001000
000-100
000010
00000-1
,
100000
010000
001000
000100
0000-10
00000-1
,
100000
010000
001000
000010
000001
000100

G:=sub<GL(6,Integers())| [1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1],[-1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0] >;

A4xC24 in GAP, Magma, Sage, TeX

A_4\times C_2^4
% in TeX

G:=Group("A4xC2^4");
// GroupNames label

G:=SmallGroup(192,1539);
// by ID

G=gap.SmallGroup(192,1539);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-2,2,285,475]);
// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=e^2=f^2=g^3=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,a*g=g*a,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,b*g=g*b,c*d=d*c,c*e=e*c,c*f=f*c,c*g=g*c,d*e=e*d,d*f=f*d,d*g=g*d,g*e*g^-1=e*f=f*e,g*f*g^-1=e>;
// generators/relations

׿
x
:
Z
F
o
wr
Q
<