C3xC2wrC2^2 - GroupNames

G = C₃×C₂≀C₂² order 192 = 2⁶·3

Direct product of C₃ and C₂≀C₂²

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

Aliases: C₃×C₂≀C₂², 2₊¹⁺⁴⋊₇C₆, C₂³⋊(C₃×D₄), (C₂×C₁₂)⋊₄D₄, C₂³⋊C₄⋊₃C₆, C₂⁴⋊₄(C₂×C₆), (C₂²×C₆)⋊₁D₄, C₂²≀C₂⋊₃C₆, (C₂³×C₆)⋊₁C₂², C₂².₁₈(C₆×D₄), C₆.₁₀₄C₂²≀C₂, C₂³.₃(C₂²×C₆), (C₆×D₄).₁₈₃C₂², (C₂²×C₆).₈₂C₂³, (C₃×2₊¹⁺⁴)⋊₈C₂, (C₂×C₄)⋊(C₃×D₄), (C₃×C₂³⋊C₄)⋊₉C₂, C₂²⋊C₄⋊₁(C₂×C₆), (C₂×D₄).₈(C₂×C₆), (C₂×C₆).₄₁₃(C₂×D₄), (C₃×C₂²≀C₂)⋊₁₁C₂, C₂.₁₈(C₃×C₂²≀C₂), (C₃×C₂²⋊C₄)⋊₃₆C₂², SmallGroup(192,890)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂³ — C₃×C₂≀C₂²

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₆ — C₂³×C₆ — C₃×C₂²≀C₂ — C₃×C₂≀C₂²

Lower central

C₁ — C₂ — C₂³ — C₃×C₂≀C₂²

Upper central

C₁ — C₆ — C₂²×C₆ — C₃×C₂≀C₂²

Generators and relations for C₃×C₂≀C₂²
G = < a,b,c,d,e,f | a³=b²=c²=d²=e⁴=f²=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, ebe^-1=fbf=bcd, ece^-1=cd=dc, cf=fc, de=ed, df=fd, fef=e^-1 >

Subgroups: 450 in 198 conjugacy classes, 54 normal (12 characteristic)
C₁, C₂, C₂, C₃, C₄, C₂², C₂², C₆, C₆, C₂×C₄, C₂×C₄, D₄, Q₈, C₂³, C₂³, C₂³, C₁₂, C₂×C₆, C₂×C₆, C₂²⋊C₄, C₂²⋊C₄, C₂×D₄, C₂×D₄, C₄○D₄, C₂⁴, C₂×C₁₂, C₂×C₁₂, C₃×D₄, C₃×Q₈, C₂²×C₆, C₂²×C₆, C₂²×C₆, C₂³⋊C₄, C₂²≀C₂, 2₊¹⁺⁴, C₃×C₂²⋊C₄, C₃×C₂²⋊C₄, C₆×D₄, C₆×D₄, C₃×C₄○D₄, C₂³×C₆, C₂≀C₂², C₃×C₂³⋊C₄, C₃×C₂²≀C₂, C₃×2₊¹⁺⁴, C₃×C₂≀C₂²
Quotients: C₁, C₂, C₃, C₂², C₆, D₄, C₂³, C₂×C₆, C₂×D₄, C₃×D₄, C₂²×C₆, C₂²≀C₂, C₆×D₄, C₂≀C₂², C₃×C₂²≀C₂, C₃×C₂≀C₂²

Permutation representations of C₃×C₂≀C₂²
►On 24 points - transitive group 24T286

Generators in S₂₄

(1 7 3)(2 8 4)(5 10 11)(6 9 12)(13 17 22)(14 18 23)(15 19 24)(16 20 21)

(1 19)(2 18)(3 15)(4 14)(5 17)(6 20)(7 24)(8 23)(9 21)(10 22)(11 13)(12 16)

(2 6)(4 12)(8 9)(14 16)(18 20)(21 23)

(1 5)(2 6)(3 11)(4 12)(7 10)(8 9)(13 15)(14 16)(17 19)(18 20)(21 23)(22 24)

(1 2)(3 4)(5 6)(7 8)(9 10)(11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)

(13 15)(17 19)(22 24)

G:=sub<Sym(24)| (1,7,3)(2,8,4)(5,10,11)(6,9,12)(13,17,22)(14,18,23)(15,19,24)(16,20,21), (1,19)(2,18)(3,15)(4,14)(5,17)(6,20)(7,24)(8,23)(9,21)(10,22)(11,13)(12,16), (2,6)(4,12)(8,9)(14,16)(18,20)(21,23), (1,5)(2,6)(3,11)(4,12)(7,10)(8,9)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24), (1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (13,15)(17,19)(22,24)>;

G:=Group( (1,7,3)(2,8,4)(5,10,11)(6,9,12)(13,17,22)(14,18,23)(15,19,24)(16,20,21), (1,19)(2,18)(3,15)(4,14)(5,17)(6,20)(7,24)(8,23)(9,21)(10,22)(11,13)(12,16), (2,6)(4,12)(8,9)(14,16)(18,20)(21,23), (1,5)(2,6)(3,11)(4,12)(7,10)(8,9)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24), (1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (13,15)(17,19)(22,24) );

G=PermutationGroup([[(1,7,3),(2,8,4),(5,10,11),(6,9,12),(13,17,22),(14,18,23),(15,19,24),(16,20,21)], [(1,19),(2,18),(3,15),(4,14),(5,17),(6,20),(7,24),(8,23),(9,21),(10,22),(11,13),(12,16)], [(2,6),(4,12),(8,9),(14,16),(18,20),(21,23)], [(1,5),(2,6),(3,11),(4,12),(7,10),(8,9),(13,15),(14,16),(17,19),(18,20),(21,23),(22,24)], [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24)], [(13,15),(17,19),(22,24)]])

G:=TransitiveGroup(24,286);

►On 24 points - transitive group 24T352

Generators in S₂₄

(1 15 21)(2 16 22)(3 13 23)(4 14 24)(5 9 20)(6 10 17)(7 11 18)(8 12 19)

(1 5)(2 8)(3 4)(6 7)(9 15)(10 11)(12 16)(13 14)(17 18)(19 22)(20 21)(23 24)

(1 3)(2 7)(4 5)(6 8)(9 14)(10 12)(11 16)(13 15)(17 19)(18 22)(20 24)(21 23)

(1 8)(2 5)(3 6)(4 7)(9 16)(10 13)(11 14)(12 15)(17 23)(18 24)(19 21)(20 22)

(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)

(1 8)(2 7)(3 6)(4 5)(9 14)(10 13)(11 16)(12 15)(17 23)(18 22)(19 21)(20 24)

G:=sub<Sym(24)| (1,15,21)(2,16,22)(3,13,23)(4,14,24)(5,9,20)(6,10,17)(7,11,18)(8,12,19), (1,5)(2,8)(3,4)(6,7)(9,15)(10,11)(12,16)(13,14)(17,18)(19,22)(20,21)(23,24), (1,3)(2,7)(4,5)(6,8)(9,14)(10,12)(11,16)(13,15)(17,19)(18,22)(20,24)(21,23), (1,8)(2,5)(3,6)(4,7)(9,16)(10,13)(11,14)(12,15)(17,23)(18,24)(19,21)(20,22), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,8)(2,7)(3,6)(4,5)(9,14)(10,13)(11,16)(12,15)(17,23)(18,22)(19,21)(20,24)>;

G:=Group( (1,15,21)(2,16,22)(3,13,23)(4,14,24)(5,9,20)(6,10,17)(7,11,18)(8,12,19), (1,5)(2,8)(3,4)(6,7)(9,15)(10,11)(12,16)(13,14)(17,18)(19,22)(20,21)(23,24), (1,3)(2,7)(4,5)(6,8)(9,14)(10,12)(11,16)(13,15)(17,19)(18,22)(20,24)(21,23), (1,8)(2,5)(3,6)(4,7)(9,16)(10,13)(11,14)(12,15)(17,23)(18,24)(19,21)(20,22), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,8)(2,7)(3,6)(4,5)(9,14)(10,13)(11,16)(12,15)(17,23)(18,22)(19,21)(20,24) );

G=PermutationGroup([[(1,15,21),(2,16,22),(3,13,23),(4,14,24),(5,9,20),(6,10,17),(7,11,18),(8,12,19)], [(1,5),(2,8),(3,4),(6,7),(9,15),(10,11),(12,16),(13,14),(17,18),(19,22),(20,21),(23,24)], [(1,3),(2,7),(4,5),(6,8),(9,14),(10,12),(11,16),(13,15),(17,19),(18,22),(20,24),(21,23)], [(1,8),(2,5),(3,6),(4,7),(9,16),(10,13),(11,14),(12,15),(17,23),(18,24),(19,21),(20,22)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24)], [(1,8),(2,7),(3,6),(4,5),(9,14),(10,13),(11,16),(12,15),(17,23),(18,22),(19,21),(20,24)]])

G:=TransitiveGroup(24,352);

48 conjugacy classes

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

48 irreducible representations

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

Matrix representation of C₃×C₂≀C₂² ►in GL₄(𝔽₇) generated by

2	0	0	0
0	2	0	0
0	0	2	0
0	0	0	2

4	3	2	0
5	0	3	1
6	1	2	2
4	4	5	1

4	3	1	0
4	1	4	3
1	6	5	5
6	2	0	4

6	0	0	0
0	6	0	0
0	0	6	0
0	0	0	6

1	6	4	2
3	6	0	1
6	5	2	6
6	1	4	5

6	5	2	1
0	4	2	1
0	5	1	1
0	3	4	1

G:=sub<GL(4,GF(7))| [2,0,0,0,0,2,0,0,0,0,2,0,0,0,0,2],[4,5,6,4,3,0,1,4,2,3,2,5,0,1,2,1],[4,4,1,6,3,1,6,2,1,4,5,0,0,3,5,4],[6,0,0,0,0,6,0,0,0,0,6,0,0,0,0,6],[1,3,6,6,6,6,5,1,4,0,2,4,2,1,6,5],[6,0,0,0,5,4,5,3,2,2,1,4,1,1,1,1] >;

C₃×C₂≀C₂² in GAP, Magma, Sage, TeX

C_3\times C_2\wr C_2^2

% in TeX

G:=Group("C3xC2wrC2^2");

// GroupNames label

G:=SmallGroup(192,890);

// by ID

G=gap.SmallGroup(192,890);

# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,365,1094,1068,3036]);

// Polycyclic

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

// generators/relations

G = C3×C2≀C22 order 192 = 26·3

Direct product of C3 and C2≀C22

G = C₃×C₂≀C₂² order 192 = 2⁶·3

Direct product of C₃ and C₂≀C₂²