C2^2xA4:C4 - GroupNames

G = C₂²×A₄⋊C₄ order 192 = 2⁶·3

Direct product of C₂² and A₄⋊C₄

direct product, non-abelian, soluble, monomial

Aliases: C₂²×A₄⋊C₄, C₂⁵.₂S₃, C₂³.₂₂S₄, C₂⁴⋊₃Dic₃, C₂⁴.₁₁D₆, C₂³⋊(C₂×Dic₃), (C₂²×A₄)⋊₃C₄, A₄⋊₂(C₂²×C₄), C₂.₂(C₂²×S₄), (C₂×A₄).₉C₂³, (C₂³×A₄).₃C₂, C₂².₃₁(C₂×S₄), C₂²⋊(C₂²×Dic₃), C₂³.₉(C₂²×S₃), (C₂²×A₄).₁₂C₂², (C₂×A₄)⋊₂(C₂×C₄), SmallGroup(192,1487)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂² — A₄ — C₂²×A₄⋊C₄

Chief series

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

Lower central

A₄ — C₂²×A₄⋊C₄

Upper central

C₁ — C₂³

Generators and relations for C₂²×A₄⋊C₄
G = < a,b,c,d,e,f | a²=b²=c²=d²=e³=f⁴=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, ece^-1=fcf^-1=cd=dc, ede^-1=c, df=fd, fef^-1=e^-1 >

Subgroups: 902 in 277 conjugacy classes, 59 normal (10 characteristic)
C₁, C₂, C₂, C₂, C₃, C₄, C₂², C₂², C₂², C₆, C₂×C₄, C₂³, C₂³, C₂³, Dic₃, A₄, C₂×C₆, C₂²⋊C₄, C₂²×C₄, C₂⁴, C₂⁴, C₂×Dic₃, C₂×A₄, C₂×A₄, C₂²×C₆, C₂×C₂²⋊C₄, C₂³×C₄, C₂⁵, A₄⋊C₄, C₂²×Dic₃, C₂²×A₄, C₂²×C₂²⋊C₄, C₂×A₄⋊C₄, C₂³×A₄, C₂²×A₄⋊C₄
Quotients: C₁, C₂, C₄, C₂², S₃, C₂×C₄, C₂³, Dic₃, D₆, C₂²×C₄, C₂×Dic₃, S₄, C₂²×S₃, A₄⋊C₄, C₂²×Dic₃, C₂×S₄, C₂×A₄⋊C₄, C₂²×S₄, C₂²×A₄⋊C₄

Smallest permutation representation of C₂²×A₄⋊C₄
►On 48 points

Generators in S₄₈

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

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

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

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

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

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

40 conjugacy classes

class	1	2A	···	2G	2H	···	2O	3	4A	···	4P	6A	···	6G
order	1	2	···	2	2	···	2	3	4	···	4	6	···	6
size	1	1	···	1	3	···	3	8	6	···	6	8	···	8

40 irreducible representations

dim	1	1	1	1	2	2	2	3	3	3
type	+	+	+		+	-	+	+		+
image	C₁	C₂	C₂	C₄	S₃	Dic₃	D₆	S₄	A₄⋊C₄	C₂×S₄
kernel	C₂²×A₄⋊C₄	C₂×A₄⋊C₄	C₂³×A₄	C₂²×A₄	C₂⁵	C₂⁴	C₂⁴	C₂³	C₂²	C₂²
# reps	1	6	1	8	1	4	3	2	8	6

Matrix representation of C₂²×A₄⋊C₄ ►in GL₇(𝔽₁₃)

12	0	0	0	0	0	0
0	12	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	0	12	0
0	0	0	0	0	0	12

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

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	0	12	0
0	0	0	0	0	0	1

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

3	0	0	0	0	0	0
0	9	0	0	0	0	0
0	0	3	0	0	0	0
0	0	0	9	0	0	0
0	0	0	0	0	0	1
0	0	0	0	1	0	0
0	0	0	0	0	1	0

0	8	0	0	0	0	0
8	0	0	0	0	0	0
0	0	0	8	0	0	0
0	0	8	0	0	0	0
0	0	0	0	5	0	0
0	0	0	0	0	0	5
0	0	0	0	0	5	0

G:=sub<GL(7,GF(13))| [12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12],[12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12],[3,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0],[0,8,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,8,0,0,0,0,0,8,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,5,0,0,0,0,0,5,0] >;

C₂²×A₄⋊C₄ in GAP, Magma, Sage, TeX

C_2^2\times A_4\rtimes C_4

% in TeX

G:=Group("C2^2xA4:C4");

// GroupNames label

G:=SmallGroup(192,1487);

// by ID

G=gap.SmallGroup(192,1487);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-2,2,56,1124,4037,285,2358,475]);

// Polycyclic

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

// generators/relations

G = C22×A4⋊C4 order 192 = 26·3

Direct product of C22 and A4⋊C4

G = C₂²×A₄⋊C₄ order 192 = 2⁶·3

Direct product of C₂² and A₄⋊C₄