C3xC2^3.C2^3 - GroupNames

G = C₃×C₂³.C₂³ order 192 = 2⁶·3

Direct product of C₃ and C₂³.C₂³

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Aliases: C₃×C₂³.C₂³, (C₂×D₄)⋊₆C₁₂, (C₆×D₄)⋊₁₈C₄, C₂³⋊C₄⋊₅C₆, (C₆×Q₈)⋊₁₄C₄, (C₂×Q₈)⋊₆C₁₂, (C₂²×C₁₂)⋊₈C₄, (C₂²×C₄)⋊₅C₁₂, C₄²⋊C₂⋊₁C₆, (C₂×C₁₂).₅₁₄D₄, C₂³.₃(C₂×C₁₂), C₂².₁₁(C₆×D₄), C₂³.₂(C₂²×C₆), C₁₂.₇₇(C₂²⋊C₄), (C₆×D₄).₂₈₃C₂², C₂².₇(C₂²×C₁₂), (C₂²×C₆).₈₁C₂³, (C₂²×C₁₂).₄₀₆C₂², (C₂×C₄○D₄).₈C₆, C₄.₉(C₃×C₂²⋊C₄), (C₃×C₂³⋊C₄)⋊₁₁C₂, (C₂×C₄).₁₈(C₂×C₁₂), (C₂×C₁₂).₁₇(C₂×C₄), (C₆×C₄○D₄).₁₆C₂, (C₂×D₄).₄₁(C₂×C₆), (C₂×C₆).₄₀₆(C₂×D₄), (C₂×C₄).₁₂₀(C₃×D₄), C₂²⋊C₄.₉(C₂×C₆), C₂.₁₃(C₆×C₂²⋊C₄), C₆.₁₀₁(C₂×C₂²⋊C₄), (C₂²×C₄).₃₀(C₂×C₆), (C₂²×C₆).₁₀(C₂×C₄), (C₃×C₄²⋊C₂)⋊₂₂C₂, (C₂×C₆).₁₆₀(C₂²×C₄), (C₃×C₂²⋊C₄).₉₅C₂², SmallGroup(192,843)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂² — C₃×C₂³.C₂³

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₆ — C₃×C₂²⋊C₄ — C₃×C₂³⋊C₄ — C₃×C₂³.C₂³

Lower central

C₁ — C₂ — C₂² — C₃×C₂³.C₂³

Upper central

C₁ — C₁₂ — C₂²×C₁₂ — C₃×C₂³.C₂³

Generators and relations for C₃×C₂³.C₂³
G = < a,b,c,d,e,f,g | a³=b²=c²=d²=f²=1, e²=c, g²=d, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, ebe^-1=bd=db, bf=fb, bg=gb, fcf=cd=dc, ce=ec, cg=gc, de=ed, df=fd, dg=gd, fef=bde, eg=ge, fg=gf >

Subgroups: 274 in 158 conjugacy classes, 78 normal (26 characteristic)
C₁, C₂, C₂, C₃, C₄, C₄, C₄, C₂², C₂², C₂², C₆, C₆, C₂×C₄, C₂×C₄, C₂×C₄, D₄, Q₈, C₂³, C₂³, C₁₂, C₁₂, C₁₂, C₂×C₆, C₂×C₆, C₂×C₆, C₄², C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂×Q₈, C₄○D₄, C₂×C₁₂, C₂×C₁₂, C₂×C₁₂, C₃×D₄, C₃×Q₈, C₂²×C₆, C₂²×C₆, C₂³⋊C₄, C₄²⋊C₂, C₂×C₄○D₄, C₄×C₁₂, C₃×C₂²⋊C₄, C₃×C₄⋊C₄, C₂²×C₁₂, C₂²×C₁₂, C₆×D₄, C₆×D₄, C₆×Q₈, C₃×C₄○D₄, C₂³.C₂³, C₃×C₂³⋊C₄, C₃×C₄²⋊C₂, C₆×C₄○D₄, C₃×C₂³.C₂³
Quotients: C₁, C₂, C₃, C₄, C₂², C₆, C₂×C₄, D₄, C₂³, C₁₂, C₂×C₆, C₂²⋊C₄, C₂²×C₄, C₂×D₄, C₂×C₁₂, C₃×D₄, C₂²×C₆, C₂×C₂²⋊C₄, C₃×C₂²⋊C₄, C₂²×C₁₂, C₆×D₄, C₂³.C₂³, C₆×C₂²⋊C₄, C₃×C₂³.C₂³

Smallest permutation representation of C₃×C₂³.C₂³
►On 48 points

Generators in S₄₈

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

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

(25 27)(26 28)(29 31)(30 32)(33 35)(34 36)(37 39)(38 40)(41 43)(42 44)(45 47)(46 48)

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

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

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

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

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

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

66 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	3A	3B	4A	4B	4C	4D	4E	4F	···	4O	6A	6B	6C	···	6H	6I	6J	6K	6L	12A	12B	12C	12D	12E	···	12J	12K	···	12AD
order	1	2	2	2	2	2	2	3	3	4	4	4	4	4	4	···	4	6	6	6	···	6	6	6	6	6	12	12	12	12	12	···	12	12	···	12
size	1	1	2	2	2	4	4	1	1	1	1	2	2	2	4	···	4	1	1	2	···	2	4	4	4	4	1	1	1	1	2	···	2	4	···	4

66 irreducible representations

dim	1	1	1	1	1	1	1	1	1	1	1	1	1	1	2	2	4	4
type	+	+	+	+											+
image	C₁	C₂	C₂	C₂	C₃	C₄	C₄	C₄	C₆	C₆	C₆	C₁₂	C₁₂	C₁₂	D₄	C₃×D₄	C₂³.C₂³	C₃×C₂³.C₂³
kernel	C₃×C₂³.C₂³	C₃×C₂³⋊C₄	C₃×C₄²⋊C₂	C₆×C₄○D₄	C₂³.C₂³	C₂²×C₁₂	C₆×D₄	C₆×Q₈	C₂³⋊C₄	C₄²⋊C₂	C₂×C₄○D₄	C₂²×C₄	C₂×D₄	C₂×Q₈	C₂×C₁₂	C₂×C₄	C₃	C₁
# reps	1	4	2	1	2	4	2	2	8	4	2	8	4	4	4	8	2	4

Matrix representation of C₃×C₂³.C₂³ ►in GL₆(𝔽₁₃)

3	0	0	0	0	0
0	3	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

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

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

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

8	0	0	0	0	0
0	5	0	0	0	0
0	0	1	0	0	0
0	0	0	12	0	0
0	0	0	0	0	12
0	0	0	0	1	0

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

1	0	0	0	0	0
0	1	0	0	0	0
0	0	5	0	0	0
0	0	0	5	0	0
0	0	0	0	5	0
0	0	0	0	0	5

G:=sub<GL(6,GF(13))| [3,0,0,0,0,0,0,3,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[8,0,0,0,0,0,0,5,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,12,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,5,0,0,0,0,0,0,5,0,0,0,0,0,0,5,0,0,0,0,0,0,5] >;

C₃×C₂³.C₂³ in GAP, Magma, Sage, TeX

C_3\times C_2^3.C_2^3

% in TeX

G:=Group("C3xC2^3.C2^3");

// GroupNames label

G:=SmallGroup(192,843);

// by ID

G=gap.SmallGroup(192,843);

# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,336,365,520,4204,3036]);

// Polycyclic

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

// generators/relations

G = C3×C23.C23 order 192 = 26·3

Direct product of C3 and C23.C23

G = C₃×C₂³.C₂³ order 192 = 2⁶·3

Direct product of C₃ and C₂³.C₂³