A4xC2xC8 - GroupNames

G = A₄xC₂xC₈ order 192 = 2⁶·3

Direct product of C₂xC₈ and A₄

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

Aliases: A₄xC₂xC₈, C₂³:₂C₂₄, C₂⁴.₄C₁₂, (C₂³xC₈):C₃, C₂²:(C₂xC₂₄), C₄.₇(C₄xA₄), (C₄xA₄).₆C₄, (C₂²xC₈):₄C₆, (C₂³xC₄).₅C₆, (C₂²xA₄).₄C₄, (C₂²xC₄).₉C₁₂, C₂².₁₀(C₄xA₄), C₄.₁₂(C₂²xA₄), (C₄xA₄).₂₂C₂², C₂³.₁₇(C₂xC₁₂), C₂.₂(C₂xC₄xA₄), (C₂xC₄xA₄).₁₁C₂, (C₂xC₄).₁₉(C₂xA₄), (C₂xA₄).₁₃(C₂xC₄), (C₂²xC₄).₈₅(C₂xC₆), SmallGroup(192,1010)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂² — A₄xC₂xC₈

Chief series

C₁ — C₂² — C₂³ — C₂²xC₄ — C₄xA₄ — C₂xC₄xA₄ — A₄xC₂xC₈

Lower central

C₂² — A₄xC₂xC₈

Upper central

C₁ — C₂xC₈

Generators and relations for A₄xC₂xC₈
G = < a,b,c,d,e | a²=b⁸=c²=d²=e³=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, ece^-1=cd=dc, ede^-1=c >

Subgroups: 224 in 93 conjugacy classes, 33 normal (21 characteristic)
C₁, C₂, C₂, C₂, C₃, C₄, C₄, C₂², C₂², C₆, C₈, C₈, C₂xC₄, C₂xC₄, C₂³, C₂³, C₂³, C₁₂, A₄, C₂xC₆, C₂xC₈, C₂xC₈, C₂²xC₄, C₂²xC₄, C₂⁴, C₂₄, C₂xC₁₂, C₂xA₄, C₂xA₄, C₂²xC₈, C₂²xC₈, C₂³xC₄, C₂xC₂₄, C₄xA₄, C₂²xA₄, C₂³xC₈, C₈xA₄, C₂xC₄xA₄, A₄xC₂xC₈
Quotients: C₁, C₂, C₃, C₄, C₂², C₆, C₈, C₂xC₄, C₁₂, A₄, C₂xC₆, C₂xC₈, C₂₄, C₂xC₁₂, C₂xA₄, C₂xC₂₄, C₄xA₄, C₂²xA₄, C₈xA₄, C₂xC₄xA₄, A₄xC₂xC₈

Smallest permutation representation of A₄xC₂xC₈
►On 48 points

Generators in S₄₈

(1 23)(2 24)(3 17)(4 18)(5 19)(6 20)(7 21)(8 22)(9 32)(10 25)(11 26)(12 27)(13 28)(14 29)(15 30)(16 31)(33 41)(34 42)(35 43)(36 44)(37 45)(38 46)(39 47)(40 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)

(1 5)(2 6)(3 7)(4 8)(9 28)(10 29)(11 30)(12 31)(13 32)(14 25)(15 26)(16 27)(17 21)(18 22)(19 23)(20 24)(33 41)(34 42)(35 43)(36 44)(37 45)(38 46)(39 47)(40 48)

(1 23)(2 24)(3 17)(4 18)(5 19)(6 20)(7 21)(8 22)(9 13)(10 14)(11 15)(12 16)(25 29)(26 30)(27 31)(28 32)(33 45)(34 46)(35 47)(36 48)(37 41)(38 42)(39 43)(40 44)

(1 15 39)(2 16 40)(3 9 33)(4 10 34)(5 11 35)(6 12 36)(7 13 37)(8 14 38)(17 32 41)(18 25 42)(19 26 43)(20 27 44)(21 28 45)(22 29 46)(23 30 47)(24 31 48)

G:=sub<Sym(48)| (1,23)(2,24)(3,17)(4,18)(5,19)(6,20)(7,21)(8,22)(9,32)(10,25)(11,26)(12,27)(13,28)(14,29)(15,30)(16,31)(33,41)(34,42)(35,43)(36,44)(37,45)(38,46)(39,47)(40,48), (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), (1,5)(2,6)(3,7)(4,8)(9,28)(10,29)(11,30)(12,31)(13,32)(14,25)(15,26)(16,27)(17,21)(18,22)(19,23)(20,24)(33,41)(34,42)(35,43)(36,44)(37,45)(38,46)(39,47)(40,48), (1,23)(2,24)(3,17)(4,18)(5,19)(6,20)(7,21)(8,22)(9,13)(10,14)(11,15)(12,16)(25,29)(26,30)(27,31)(28,32)(33,45)(34,46)(35,47)(36,48)(37,41)(38,42)(39,43)(40,44), (1,15,39)(2,16,40)(3,9,33)(4,10,34)(5,11,35)(6,12,36)(7,13,37)(8,14,38)(17,32,41)(18,25,42)(19,26,43)(20,27,44)(21,28,45)(22,29,46)(23,30,47)(24,31,48)>;

G:=Group( (1,23)(2,24)(3,17)(4,18)(5,19)(6,20)(7,21)(8,22)(9,32)(10,25)(11,26)(12,27)(13,28)(14,29)(15,30)(16,31)(33,41)(34,42)(35,43)(36,44)(37,45)(38,46)(39,47)(40,48), (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), (1,5)(2,6)(3,7)(4,8)(9,28)(10,29)(11,30)(12,31)(13,32)(14,25)(15,26)(16,27)(17,21)(18,22)(19,23)(20,24)(33,41)(34,42)(35,43)(36,44)(37,45)(38,46)(39,47)(40,48), (1,23)(2,24)(3,17)(4,18)(5,19)(6,20)(7,21)(8,22)(9,13)(10,14)(11,15)(12,16)(25,29)(26,30)(27,31)(28,32)(33,45)(34,46)(35,47)(36,48)(37,41)(38,42)(39,43)(40,44), (1,15,39)(2,16,40)(3,9,33)(4,10,34)(5,11,35)(6,12,36)(7,13,37)(8,14,38)(17,32,41)(18,25,42)(19,26,43)(20,27,44)(21,28,45)(22,29,46)(23,30,47)(24,31,48) );

G=PermutationGroup([[(1,23),(2,24),(3,17),(4,18),(5,19),(6,20),(7,21),(8,22),(9,32),(10,25),(11,26),(12,27),(13,28),(14,29),(15,30),(16,31),(33,41),(34,42),(35,43),(36,44),(37,45),(38,46),(39,47),(40,48)], [(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)], [(1,5),(2,6),(3,7),(4,8),(9,28),(10,29),(11,30),(12,31),(13,32),(14,25),(15,26),(16,27),(17,21),(18,22),(19,23),(20,24),(33,41),(34,42),(35,43),(36,44),(37,45),(38,46),(39,47),(40,48)], [(1,23),(2,24),(3,17),(4,18),(5,19),(6,20),(7,21),(8,22),(9,13),(10,14),(11,15),(12,16),(25,29),(26,30),(27,31),(28,32),(33,45),(34,46),(35,47),(36,48),(37,41),(38,42),(39,43),(40,44)], [(1,15,39),(2,16,40),(3,9,33),(4,10,34),(5,11,35),(6,12,36),(7,13,37),(8,14,38),(17,32,41),(18,25,42),(19,26,43),(20,27,44),(21,28,45),(22,29,46),(23,30,47),(24,31,48)]])

64 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	3A	3B	4A	4B	4C	4D	4E	4F	4G	4H	6A	···	6F	8A	···	8H	8I	···	8P	12A	···	12H	24A	···	24P
order	1	2	2	2	2	2	2	2	3	3	4	4	4	4	4	4	4	4	6	···	6	8	···	8	8	···	8	12	···	12	24	···	24
size	1	1	1	1	3	3	3	3	4	4	1	1	1	1	3	3	3	3	4	···	4	1	···	1	3	···	3	4	···	4	4	···	4

64 irreducible representations

dim	1	1	1	1	1	1	1	1	1	1	1	1	3	3	3	3	3	3
type	+	+	+										+	+	+
image	C₁	C₂	C₂	C₃	C₄	C₄	C₆	C₆	C₈	C₁₂	C₁₂	C₂₄	A₄	C₂xA₄	C₂xA₄	C₄xA₄	C₄xA₄	C₈xA₄
kernel	A₄xC₂xC₈	C₈xA₄	C₂xC₄xA₄	C₂³xC₈	C₄xA₄	C₂²xA₄	C₂²xC₈	C₂³xC₄	C₂xA₄	C₂²xC₄	C₂⁴	C₂³	C₂xC₈	C₈	C₂xC₄	C₄	C₂²	C₂
# reps	1	2	1	2	2	2	4	2	8	4	4	16	1	2	1	2	2	8

Matrix representation of A₄xC₂xC₈ ►in GL₄(F₇₃) generated by

1	0	0	0
0	72	0	0
0	0	72	0
0	0	0	72

10	0	0	0
0	10	0	0
0	0	10	0
0	0	0	10

1	0	0	0
0	72	0	0
0	0	72	0
0	0	0	1

1	0	0	0
0	72	0	0
0	0	1	0
0	0	0	72

1	0	0	0
0	0	1	0
0	0	0	1
0	1	0	0

G:=sub<GL(4,GF(73))| [1,0,0,0,0,72,0,0,0,0,72,0,0,0,0,72],[10,0,0,0,0,10,0,0,0,0,10,0,0,0,0,10],[1,0,0,0,0,72,0,0,0,0,72,0,0,0,0,1],[1,0,0,0,0,72,0,0,0,0,1,0,0,0,0,72],[1,0,0,0,0,0,0,1,0,1,0,0,0,0,1,0] >;

A₄xC₂xC₈ in GAP, Magma, Sage, TeX

A_4\times C_2\times C_8

% in TeX

G:=Group("A4xC2xC8");

// GroupNames label

G:=SmallGroup(192,1010);

// by ID

G=gap.SmallGroup(192,1010);

# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,2,92,80,1027,1784]);

// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^8=c^2=d^2=e^3=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,e*c*e^-1=c*d=d*c,e*d*e^-1=c>;

// generators/relations

G = A4xC2xC8 order 192 = 26·3

Direct product of C2xC8 and A4

G = A₄xC₂xC₈ order 192 = 2⁶·3

Direct product of C₂xC₈ and A₄