A4xC4:C4 - GroupNames

G = A₄×C₄⋊C₄ order 192 = 2⁶·3

Direct product of A₄ and C₄⋊C₄

direct product, metabelian, soluble, monomial

Aliases: A₄×C₄⋊C₄, C₄⋊(C₄×A₄), (C₄×A₄)⋊₃C₄, (C₂²×C₄)⋊C₁₂, C₂.₂(D₄×A₄), C₂.₁(Q₈×A₄), (C₂×A₄).₃Q₈, (C₂×A₄).₁₃D₄, (C₂³×C₄).₁C₆, C₂³.₆(C₃×Q₈), C₂⁴.₂₅(C₂×C₆), C₂³.₂₂(C₃×D₄), C₂³.₁₆(C₂×C₁₂), C₂².₁₂(C₂²×A₄), (C₂²×A₄).₁₅C₂², (C₂²×C₄⋊C₄)⋊C₃, C₂²⋊(C₃×C₄⋊C₄), C₂.₄(C₂×C₄×A₄), (C₂×C₄×A₄).₇C₂, (C₂×C₄).₁(C₂×A₄), (C₂×A₄).₁₂(C₂×C₄), SmallGroup(192,995)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂³ — A₄×C₄⋊C₄

Chief series

C₁ — C₂² — C₂³ — C₂⁴ — C₂²×A₄ — C₂×C₄×A₄ — A₄×C₄⋊C₄

Lower central

C₂² — C₂³ — A₄×C₄⋊C₄

Upper central

C₁ — C₂² — C₄⋊C₄

Generators and relations for A₄×C₄⋊C₄
G = < a,b,c,d,e | a²=b²=c³=d⁴=e⁴=1, cac^-1=ab=ba, ad=da, ae=ea, cbc^-1=a, bd=db, be=eb, cd=dc, ce=ec, ede^-1=d^-1 >

Subgroups: 324 in 113 conjugacy classes, 33 normal (21 characteristic)
C₁, C₂, C₂, C₃, C₄, C₄, C₂², C₂², C₆, C₂×C₄, C₂×C₄, C₂×C₄, C₂³, C₂³, C₁₂, A₄, C₂×C₆, C₄⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂⁴, C₂×C₁₂, C₂×A₄, C₂×C₄⋊C₄, C₂³×C₄, C₂³×C₄, C₃×C₄⋊C₄, C₄×A₄, C₄×A₄, C₂²×A₄, C₂²×C₄⋊C₄, C₂×C₄×A₄, C₂×C₄×A₄, A₄×C₄⋊C₄
Quotients: C₁, C₂, C₃, C₄, C₂², C₆, C₂×C₄, D₄, Q₈, C₁₂, A₄, C₂×C₆, C₄⋊C₄, C₂×C₁₂, C₃×D₄, C₃×Q₈, C₂×A₄, C₃×C₄⋊C₄, C₄×A₄, C₂²×A₄, C₂×C₄×A₄, D₄×A₄, Q₈×A₄, A₄×C₄⋊C₄

Smallest permutation representation of A₄×C₄⋊C₄
►On 48 points

Generators in S₄₈

(5 7)(6 8)(13 15)(14 16)(17 19)(18 20)(25 27)(26 28)(33 35)(34 36)(37 39)(38 40)(41 43)(42 44)(45 47)(46 48)

(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(17 19)(18 20)(21 23)(22 24)(29 31)(30 32)(37 39)(38 40)(45 47)(46 48)

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

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

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

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

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

40 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	3A	3B	4A	···	4F	4G	···	4L	6A	···	6F	12A	···	12L
order	1	2	2	2	2	2	2	2	3	3	4	···	4	4	···	4	6	···	6	12	···	12
size	1	1	1	1	3	3	3	3	4	4	2	···	2	6	···	6	4	···	4	8	···	8

40 irreducible representations

dim	1	1	1	1	1	1	2	2	2	2	3	3	3	6	6
type	+	+					+	-			+	+		+	-
image	C₁	C₂	C₃	C₄	C₆	C₁₂	D₄	Q₈	C₃×D₄	C₃×Q₈	A₄	C₂×A₄	C₄×A₄	D₄×A₄	Q₈×A₄
kernel	A₄×C₄⋊C₄	C₂×C₄×A₄	C₂²×C₄⋊C₄	C₄×A₄	C₂³×C₄	C₂²×C₄	C₂×A₄	C₂×A₄	C₂³	C₂³	C₄⋊C₄	C₂×C₄	C₄	C₂	C₂
# reps	1	3	2	4	6	8	1	1	2	2	1	3	4	1	1

Matrix representation of A₄×C₄⋊C₄ ►in GL₇(𝔽₁₃)

1	0	0	0	0	0	0
0	1	0	0	0	0	0
0	0	1	0	0	0	0
0	0	0	1	0	0	0
0	0	0	0	12	0	0
0	0	0	0	12	0	1
0	0	0	0	12	1	0

1	0	0	0	0	0	0
0	1	0	0	0	0	0
0	0	1	0	0	0	0
0	0	0	1	0	0	0
0	0	0	0	0	12	1
0	0	0	0	0	12	0
0	0	0	0	1	12	0

1	0	0	0	0	0	0
0	1	0	0	0	0	0
0	0	3	0	0	0	0
0	0	0	3	0	0	0
0	0	0	0	0	0	1
0	0	0	0	1	0	0
0	0	0	0	0	1	0

12	12	0	0	0	0	0
2	1	0	0	0	0	0
0	0	1	1	0	0	0
0	0	11	12	0	0	0
0	0	0	0	1	0	0
0	0	0	0	0	1	0
0	0	0	0	0	0	1

8	0	0	0	0	0	0
10	5	0	0	0	0	0
0	0	12	0	0	0	0
0	0	2	1	0	0	0
0	0	0	0	1	0	0
0	0	0	0	0	1	0
0	0	0	0	0	0	1

G:=sub<GL(7,GF(13))| [1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,12,12,0,0,0,0,0,0,1,0,0,0,0,0,1,0],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,12,12,12,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0],[12,2,0,0,0,0,0,12,1,0,0,0,0,0,0,0,1,11,0,0,0,0,0,1,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[8,10,0,0,0,0,0,0,5,0,0,0,0,0,0,0,12,2,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1] >;

A₄×C₄⋊C₄ in GAP, Magma, Sage, TeX

A_4\times C_4\rtimes C_4

% in TeX

G:=Group("A4xC4:C4");

// GroupNames label

G:=SmallGroup(192,995);

// by ID

G=gap.SmallGroup(192,995);

# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,2,168,365,92,1027,1784]);

// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^3=d^4=e^4=1,c*a*c^-1=a*b=b*a,a*d=d*a,a*e=e*a,c*b*c^-1=a,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^-1>;

// generators/relations

G = A4×C4⋊C4 order 192 = 26·3

Direct product of A4 and C4⋊C4

G = A₄×C₄⋊C₄ order 192 = 2⁶·3

Direct product of A₄ and C₄⋊C₄