C3xC4^2:3C4 - GroupNames

G = C₃xC₄²:₃C₄ order 192 = 2⁶·3

Direct product of C₃ and C₄²:₃C₄

direct product, metabelian, nilpotent (class 4), monomial, 2-elementary

Aliases: C₃xC₄²:₃C₄, C₄²:₅C₁₂, (C₄xC₁₂):₆C₄, (C₆xQ₈):₄C₄, (C₂xQ₈):₃C₁₂, C₂³:C₄.₂C₆, (C₂²xC₆).₄D₄, C₂³.₄(C₃xD₄), C₄.₄D₄.₂C₆, C₆.₃₅(C₂³:C₄), (C₆xD₄).₁₇₇C₂², (C₂xC₄).₂(C₂xC₁₂), (C₂xD₄).₄(C₂xC₆), C₂.₉(C₃xC₂³:C₄), (C₂xC₁₂).₁₃(C₂xC₄), (C₃xC₂³:C₄).₄C₂, (C₂xC₆).₇₆(C₂²:C₄), (C₃xC₄.₄D₄).₁₁C₂, C₂².₁₃(C₃xC₂²:C₄), SmallGroup(192,160)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂xC₄ — C₃xC₄²:₃C₄

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂xD₄ — C₆xD₄ — C₃xC₂³:C₄ — C₃xC₄²:₃C₄

Lower central

C₁ — C₂ — C₂² — C₂xC₄ — C₃xC₄²:₃C₄

Upper central

C₁ — C₆ — C₂xC₆ — C₆xD₄ — C₃xC₄²:₃C₄

Generators and relations for C₃xC₄²:₃C₄
G = < a,b,c,d | a³=b⁴=c⁴=d⁴=1, ab=ba, ac=ca, ad=da, bc=cb, dbd^-1=b^-1c^-1, dcd^-1=b²c^-1 >

Subgroups: 178 in 70 conjugacy classes, 26 normal (18 characteristic)
C₁, C₂, C₂, C₃, C₄, C₂², C₂², C₆, C₆, C₂xC₄, C₂xC₄, D₄, Q₈, C₂³, C₁₂, C₂xC₆, C₂xC₆, C₄², C₂²:C₄, C₂xD₄, C₂xQ₈, C₂xC₁₂, C₂xC₁₂, C₃xD₄, C₃xQ₈, C₂²xC₆, C₂³:C₄, C₄.₄D₄, C₄xC₁₂, C₃xC₂²:C₄, C₆xD₄, C₆xQ₈, C₄²:₃C₄, C₃xC₂³:C₄, C₃xC₄.₄D₄, C₃xC₄²:₃C₄
Quotients: C₁, C₂, C₃, C₄, C₂², C₆, C₂xC₄, D₄, C₁₂, C₂xC₆, C₂²:C₄, C₂xC₁₂, C₃xD₄, C₂³:C₄, C₃xC₂²:C₄, C₄²:₃C₄, C₃xC₂³:C₄, C₃xC₄²:₃C₄

Smallest permutation representation of C₃xC₄²:₃C₄
►On 48 points

Generators in S₄₈

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

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

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

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

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

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

39 conjugacy classes

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

39 irreducible representations

dim	1	1	1	1	1	1	1	1	1	1	2	2	4	4	4	4
type	+	+	+								+		+
image	C₁	C₂	C₂	C₃	C₄	C₄	C₆	C₆	C₁₂	C₁₂	D₄	C₃xD₄	C₂³:C₄	C₄²:₃C₄	C₃xC₂³:C₄	C₃xC₄²:₃C₄
kernel	C₃xC₄²:₃C₄	C₃xC₂³:C₄	C₃xC₄.₄D₄	C₄²:₃C₄	C₄xC₁₂	C₆xQ₈	C₂³:C₄	C₄.₄D₄	C₄²	C₂xQ₈	C₂²xC₆	C₂³	C₆	C₃	C₂	C₁
# reps	1	2	1	2	2	2	4	2	4	4	2	4	1	2	2	4

Matrix representation of C₃xC₄²:₃C₄ ►in GL₆(F₁₃)

9	0	0	0	0	0
0	9	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

0	12	0	0	0	0
12	0	0	0	0	0
0	0	4	9	4	4
0	0	9	4	4	4
0	0	9	9	4	9
0	0	9	9	9	4

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

0	5	0	0	0	0
8	0	0	0	0	0
0	0	4	9	9	9
0	0	4	9	4	4
0	0	9	9	4	9
0	0	4	4	4	9

G:=sub<GL(6,GF(13))| [9,0,0,0,0,0,0,9,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,12,0,0,0,0,12,0,0,0,0,0,0,0,4,9,9,9,0,0,9,4,9,9,0,0,4,4,4,9,0,0,4,4,9,4],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,1,0,0,0,0,0,0,1,0,0],[0,8,0,0,0,0,5,0,0,0,0,0,0,0,4,4,9,4,0,0,9,9,9,4,0,0,9,4,4,4,0,0,9,4,9,9] >;

C₃xC₄²:₃C₄ in GAP, Magma, Sage, TeX

C_3\times C_4^2\rtimes_3C_4

% in TeX

G:=Group("C3xC4^2:3C4");

// GroupNames label

G:=SmallGroup(192,160);

// by ID

G=gap.SmallGroup(192,160);

# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-2,168,197,680,1683,1522,248,2951,375,6053]);

// Polycyclic

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

// generators/relations

G = C3xC42:3C4 order 192 = 26·3

Direct product of C3 and C42:3C4

G = C₃xC₄²:₃C₄ order 192 = 2⁶·3

Direct product of C₃ and C₄²:₃C₄