C3xD4.3D4 - GroupNames

G = C₃×D₄.₃D₄ order 192 = 2⁶·3

Direct product of C₃ and D₄.₃D₄

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

Aliases: C₃×D₄.₃D₄, C₂₄.₁₀₃D₄, C₈○D₄⋊₅C₆, C₈⋊C₂².C₆, D₄.₃(C₃×D₄), C₈.₂₃(C₃×D₄), C₄.₃₇(C₆×D₄), Q₈.₈(C₃×D₄), C₈.C₄⋊₅C₆, (C₂×SD₁₆)⋊₂C₆, (C₃×D₄).₂₈D₄, C₄.D₄⋊₃C₆, C₈.C₂²⋊₃C₆, (C₃×Q₈).₂₈D₄, (C₆×SD₁₆)⋊₁₃C₂, C₁₂.₃₉₈(C₂×D₄), C₄.₁₀D₄⋊₃C₆, M₄(2).₃(C₂×C₆), C₆.₁₅₄(C₄⋊D₄), (C₂×C₂₄).₂₀₀C₂², (C₂×C₁₂).₆₁₃C₂³, (C₆×D₄).₁₉₃C₂², (C₆×Q₈).₁₆₇C₂², (C₃×M₄(2)).₄₇C₂², (C₃×C₈○D₄)⋊₆C₂, (C₂×C₈).₂₄(C₂×C₆), C₄○D₄.₁₇(C₂×C₆), (C₂×D₄).₁₆(C₂×C₆), (C₃×C₄.D₄)⋊₇C₂, C₂.₂₃(C₃×C₄⋊D₄), (C₃×C₈⋊C₂²).₂C₂, (C₂×C₄).₈(C₂²×C₆), (C₂×Q₈).₁₂(C₂×C₆), (C₃×C₈.C₄)⋊₁₄C₂, C₂².₆(C₃×C₄○D₄), (C₃×C₄.₁₀D₄)⋊₇C₂, (C₃×C₈.C₂²)⋊₁₀C₂, (C₂×C₆).₁₁₅(C₄○D₄), (C₃×C₄○D₄).₅₅C₂², SmallGroup(192,904)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₄ — C₃×D₄.₃D₄

Chief series

C₁ — C₂ — C₄ — C₂×C₄ — C₂×C₁₂ — C₆×D₄ — C₆×SD₁₆ — C₃×D₄.₃D₄

Lower central

C₁ — C₂ — C₂×C₄ — C₃×D₄.₃D₄

Upper central

C₁ — C₆ — C₂×C₁₂ — C₃×D₄.₃D₄

Generators and relations for C₃×D₄.₃D₄
G = < a,b,c,d,e | a³=b⁴=c²=1, d⁴=e²=b², ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe^-1=b^-1, bd=db, cd=dc, ece^-1=bc, ede^-1=d³ >

Subgroups: 194 in 104 conjugacy classes, 50 normal (all characteristic)
C₁, C₂, C₂, C₃, C₄, C₄, C₂², C₂², C₆, C₆, C₈, C₈, C₂×C₄, C₂×C₄, D₄, D₄, Q₈, Q₈, C₂³, C₁₂, C₁₂, C₂×C₆, C₂×C₆, C₂×C₈, C₂×C₈, M₄(2), M₄(2), D₈, SD₁₆, Q₁₆, C₂×D₄, C₂×Q₈, C₄○D₄, C₂₄, C₂₄, C₂×C₁₂, C₂×C₁₂, C₃×D₄, C₃×D₄, C₃×Q₈, C₃×Q₈, C₂²×C₆, C₄.D₄, C₄.₁₀D₄, C₈.C₄, C₈○D₄, C₂×SD₁₆, C₈⋊C₂², C₈.C₂², C₂×C₂₄, C₂×C₂₄, C₃×M₄(2), C₃×M₄(2), C₃×D₈, C₃×SD₁₆, C₃×Q₁₆, C₆×D₄, C₆×Q₈, C₃×C₄○D₄, D₄.₃D₄, C₃×C₄.D₄, C₃×C₄.₁₀D₄, C₃×C₈.C₄, C₃×C₈○D₄, C₆×SD₁₆, C₃×C₈⋊C₂², C₃×C₈.C₂², C₃×D₄.₃D₄
Quotients: C₁, C₂, C₃, C₂², C₆, D₄, C₂³, C₂×C₆, C₂×D₄, C₄○D₄, C₃×D₄, C₂²×C₆, C₄⋊D₄, C₆×D₄, C₃×C₄○D₄, D₄.₃D₄, C₃×C₄⋊D₄, C₃×D₄.₃D₄

Smallest permutation representation of C₃×D₄.₃D₄
►On 48 points

Generators in S₄₈

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

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

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

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

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

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

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

48 conjugacy classes

class	1	2A	2B	2C	2D	3A	3B	4A	4B	4C	4D	6A	6B	6C	6D	6E	6F	6G	6H	8A	8B	8C	8D	8E	8F	8G	12A	12B	12C	12D	12E	12F	12G	12H	24A	24B	24C	24D	24E	···	24J	24K	24L	24M	24N
order	1	2	2	2	2	3	3	4	4	4	4	6	6	6	6	6	6	6	6	8	8	8	8	8	8	8	12	12	12	12	12	12	12	12	24	24	24	24	24	···	24	24	24	24	24
size	1	1	2	4	8	1	1	2	2	4	8	1	1	2	2	4	4	8	8	2	2	4	4	4	8	8	2	2	2	2	4	4	8	8	2	2	2	2	4	···	4	8	8	8	8

48 irreducible representations

dim	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	2	2	2	2	2	2	2	2	4	4
type	+	+	+	+	+	+	+	+									+	+	+
image	C₁	C₂	C₂	C₂	C₂	C₂	C₂	C₂	C₃	C₆	C₆	C₆	C₆	C₆	C₆	C₆	D₄	D₄	D₄	C₄○D₄	C₃×D₄	C₃×D₄	C₃×D₄	C₃×C₄○D₄	D₄.₃D₄	C₃×D₄.₃D₄
kernel	C₃×D₄.₃D₄	C₃×C₄.D₄	C₃×C₄.₁₀D₄	C₃×C₈.C₄	C₃×C₈○D₄	C₆×SD₁₆	C₃×C₈⋊C₂²	C₃×C₈.C₂²	D₄.₃D₄	C₄.D₄	C₄.₁₀D₄	C₈.C₄	C₈○D₄	C₂×SD₁₆	C₈⋊C₂²	C₈.C₂²	C₂₄	C₃×D₄	C₃×Q₈	C₂×C₆	C₈	D₄	Q₈	C₂²	C₃	C₁
# reps	1	1	1	1	1	1	1	1	2	2	2	2	2	2	2	2	2	1	1	2	4	2	2	4	2	4

Matrix representation of C₃×D₄.₃D₄ ►in GL₄(𝔽₇₃) generated by

8	0	0	0
0	8	0	0
0	0	8	0
0	0	0	8

0	1	0	0
72	0	0	0
0	0	0	72
0	0	1	0

0	0	0	72
0	0	1	0
0	1	0	0
72	0	0	0

6	6	0	0
67	6	0	0
0	0	6	6
0	0	67	6

6	67	0	0
67	67	0	0
0	0	6	6
0	0	6	67

G:=sub<GL(4,GF(73))| [8,0,0,0,0,8,0,0,0,0,8,0,0,0,0,8],[0,72,0,0,1,0,0,0,0,0,0,1,0,0,72,0],[0,0,0,72,0,0,1,0,0,1,0,0,72,0,0,0],[6,67,0,0,6,6,0,0,0,0,6,67,0,0,6,6],[6,67,0,0,67,67,0,0,0,0,6,6,0,0,6,67] >;

C₃×D₄.₃D₄ in GAP, Magma, Sage, TeX

C_3\times D_4._3D_4

% in TeX

G:=Group("C3xD4.3D4");

// GroupNames label

G:=SmallGroup(192,904);

// by ID

G=gap.SmallGroup(192,904);

# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,365,176,1094,4204,172,6053,1531,124]);

// Polycyclic

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

// generators/relations

G = C3×D4.3D4 order 192 = 26·3

Direct product of C3 and D4.3D4

G = C₃×D₄.₃D₄ order 192 = 2⁶·3

Direct product of C₃ and D₄.₃D₄