C3xC2^3:C4 - GroupNames

G = C₃xC₂³:C₄ order 96 = 2⁵·3

Direct product of C₃ and C₂³:C₄

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

Aliases: C₃xC₂³:C₄, C₂³:C₁₂, (C₂xC₄):C₁₂, (C₂xC₁₂):₂C₄, C₂²:C₄:₁C₆, (C₂²xC₆):₁C₄, (C₆xD₄).₇C₂, (C₂xD₄).₁C₆, (C₂xC₆).₂₁D₄, C₂³.₁(C₂xC₆), C₂².₂(C₃xD₄), C₂².₂(C₂xC₁₂), C₆.₂₁(C₂²:C₄), (C₂²xC₆).₁C₂², (C₃xC₂²:C₄):₂C₂, (C₂xC₆).₁₉(C₂xC₄), C₂.₃(C₃xC₂²:C₄), SmallGroup(96,49)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂² — C₃xC₂³:C₄

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²xC₆ — C₃xC₂²:C₄ — C₃xC₂³:C₄

Lower central

C₁ — C₂ — C₂² — C₃xC₂³:C₄

Upper central

C₁ — C₆ — C₂²xC₆ — C₃xC₂³:C₄

Generators and relations for C₃xC₂³:C₄
G = < a,b,c,d,e | a³=b²=c²=d²=e⁴=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, ebe^-1=bcd, ece^-1=cd=dc, de=ed >

Subgroups: 100 in 52 conjugacy classes, 24 normal (16 characteristic)
Quotients: C₁, C₂, C₃, C₄, C₂², C₆, C₂xC₄, D₄, C₁₂, C₂xC₆, C₂²:C₄, C₂xC₁₂, C₃xD₄, C₂³:C₄, C₃xC₂²:C₄, C₃xC₂³:C₄

C₁

C₂

₂C₂

₄C₂

C₃

C₂²

₂C₂²

₂C₄

₄C₂²

₄C₄

₄C₂²

₄C₄

C₆

₂C₆

₄C₆

C₂³

C₂xC₄

₂D₄

₂C₂xC₄

₂D₄

₂C₂xC₄

C₂xC₆

₂C₁₂

₂C₂xC₆

₄C₂xC₆

₄C₁₂

₄C₂xC₆

C₂xD₄

C₂²:C₄

C₂xC₁₂

C₂²xC₆

₂C₃xD₄

₂C₂xC₁₂

₂C₃xD₄

C₂³:C₄

C₃xC₂²:C₄

C₆xD₄

C₃xC₂³:C₄

Permutation representations of C₃xC₂³:C₄
►On 24 points - transitive group 24T91

Generators in S₂₄

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

(2 12)(3 9)(5 16)(8 15)(17 21)(20 24)

(2 12)(4 10)(6 13)(8 15)(18 22)(20 24)

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

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

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

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

G:=TransitiveGroup(24,91);

►On 24 points - transitive group 24T93

Generators in S₂₄

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

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

(1 6)(3 10)(8 12)(13 15)(17 19)(22 24)

(1 6)(2 5)(3 10)(4 9)(7 11)(8 12)(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)

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

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

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

G:=TransitiveGroup(24,93);

►On 24 points - transitive group 24T115

Generators in S₂₄

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

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

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

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

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

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

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

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

G:=TransitiveGroup(24,115);

C₃xC₂³:C₄ is a maximal subgroup of C₃:C₂wrC₄ (C₂xD₄).D₆ C₂³.D₁₂ C₂³.₂D₁₂ C₂³:C₄:₅S₃ C₂³:D₁₂ C₂³.₅D₁₂

33 conjugacy classes

class	1	2A	2B	2C	2D	2E	3A	3B	4A	···	4E	6A	6B	6C	···	6H	6I	6J	12A	···	12J
order	1	2	2	2	2	2	3	3	4	···	4	6	6	6	···	6	6	6	12	···	12
size	1	1	2	2	2	4	1	1	4	···	4	1	1	2	···	2	4	4	4	···	4

33 irreducible representations

dim	1	1	1	1	1	1	1	1	1	1	2	2	4	4
type	+	+	+								+		+
image	C₁	C₂	C₂	C₃	C₄	C₄	C₆	C₆	C₁₂	C₁₂	D₄	C₃xD₄	C₂³:C₄	C₃xC₂³:C₄
kernel	C₃xC₂³:C₄	C₃xC₂²:C₄	C₆xD₄	C₂³:C₄	C₂xC₁₂	C₂²xC₆	C₂²:C₄	C₂xD₄	C₂xC₄	C₂³	C₂xC₆	C₂²	C₃	C₁
# reps	1	2	1	2	2	2	4	2	4	4	2	4	1	2

Matrix representation of C₃xC₂³:C₄ ►in GL₄(F₇) generated by

4	0	0	0
0	4	0	0
0	0	4	0
0	0	0	4

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

1	1	2	2
2	0	2	2
5	2	1	0
1	1	4	5

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

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

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

C₃xC₂³:C₄ in GAP, Magma, Sage, TeX

C_3\times C_2^3\rtimes C_4

% in TeX

G:=Group("C3xC2^3:C4");

// GroupNames label

G:=SmallGroup(96,49);

// by ID

G=gap.SmallGroup(96,49);

# by ID

G:=PCGroup([6,-2,-2,-3,-2,-2,-2,144,169,1443,1090]);

// Polycyclic

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

// generators/relations

Export

Subgroup lattice of C₃xC₂³:C₄ in TeX

G = C3xC23:C4 order 96 = 25·3

Direct product of C3 and C23:C4

G = C₃xC₂³:C₄ order 96 = 2⁵·3

Direct product of C₃ and C₂³:C₄