C3xD4.8D4 - GroupNames

G = C₃xD₄.₈D₄ order 192 = 2⁶·3

Direct product of C₃ and D₄.₈D₄

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

Aliases: C₃xD₄.₈D₄, 2_-¹⁺⁴:₄C₆, C₄wrC₂:₂C₆, C₈:C₂²:₂C₆, D₄.₈(C₃xD₄), C₄.₂₈(C₆xD₄), (C₃xD₄).₄₂D₄, (C₂xC₁₂).₂₄D₄, C₄.₄D₄:₁C₆, (C₃xQ₈).₄₂D₄, Q₈.₁₃(C₃xD₄), C₄.₁₀D₄:₁C₆, C₁₂.₃₈₉(C₂xD₄), C₄².₁₂(C₂xC₆), C₂².₁₅(C₆xD₄), C₆.₁₀₁C₂²wrC₂, M₄(2).₁(C₂xC₆), (C₄xC₁₂).₂₅₄C₂², (C₂xC₁₂).₆₁₀C₂³, (C₆xD₄).₁₈₁C₂², (C₃x2_-¹⁺⁴):₆C₂, (C₆xQ₈).₁₅₇C₂², (C₃xM₄(2)).₁₅C₂², (C₃xC₄wrC₂):₆C₂, (C₂xC₄).₅(C₃xD₄), (C₃xC₈:C₂²):₉C₂, C₄oD₄.₇(C₂xC₆), (C₂xD₄).₆(C₂xC₆), (C₂xQ₈).₄(C₂xC₆), (C₂xC₆).₄₁₀(C₂xD₄), (C₂xC₄).₅(C₂²xC₆), C₂.₁₅(C₃xC₂²wrC₂), (C₃xC₄.₁₀D₄):₅C₂, (C₃xC₄.₄D₄):₂₁C₂, (C₃xC₄oD₄).₃₂C₂², SmallGroup(192,887)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂xC₄ — C₃xD₄.₈D₄

Chief series

C₁ — C₂ — C₂² — C₂xC₄ — C₂xC₁₂ — C₆xD₄ — C₃xC₈:C₂² — C₃xD₄.₈D₄

Lower central

C₁ — C₂ — C₂xC₄ — C₃xD₄.₈D₄

Upper central

C₁ — C₆ — C₂xC₁₂ — C₃xD₄.₈D₄

Generators and relations for C₃xD₄.₈D₄
G = < a,b,c,d,e | a³=b⁴=c²=e²=1, d⁴=b², ab=ba, ac=ca, ad=da, ae=ea, cbc=dbd^-1=ebe=b^-1, dcd^-1=bc, ece=b^-1c, ede=b²d³ >

Subgroups: 274 in 146 conjugacy classes, 54 normal (22 characteristic)
C₁, C₂, C₂, C₃, C₄, C₄, C₂², C₂², C₆, C₆, C₈, C₂xC₄, C₂xC₄, C₂xC₄, D₄, D₄, Q₈, Q₈, C₂³, C₁₂, C₁₂, C₂xC₆, C₂xC₆, C₄², C₂²:C₄, M₄(2), D₈, SD₁₆, C₂xD₄, C₂xQ₈, C₂xQ₈, C₄oD₄, C₄oD₄, C₂₄, C₂xC₁₂, C₂xC₁₂, C₂xC₁₂, C₃xD₄, C₃xD₄, C₃xQ₈, C₃xQ₈, C₂²xC₆, C₄.₁₀D₄, C₄wrC₂, C₄.₄D₄, C₈:C₂², 2_-¹⁺⁴, C₄xC₁₂, C₃xC₂²:C₄, C₃xM₄(2), C₃xD₈, C₃xSD₁₆, C₆xD₄, C₆xQ₈, C₆xQ₈, C₃xC₄oD₄, C₃xC₄oD₄, D₄.₈D₄, C₃xC₄.₁₀D₄, C₃xC₄wrC₂, C₃xC₄.₄D₄, C₃xC₈:C₂², C₃x2_-¹⁺⁴, C₃xD₄.₈D₄
Quotients: C₁, C₂, C₃, C₂², C₆, D₄, C₂³, C₂xC₆, C₂xD₄, C₃xD₄, C₂²xC₆, C₂²wrC₂, C₆xD₄, D₄.₈D₄, C₃xC₂²wrC₂, C₃xD₄.₈D₄

Smallest permutation representation of C₃xD₄.₈D₄
►On 48 points

Generators in S₄₈

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

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

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

(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)

(2 8)(3 7)(4 6)(9 11)(12 16)(13 15)(17 19)(20 24)(21 23)(25 27)(28 32)(29 31)(34 40)(35 39)(36 38)(41 45)(42 44)(46 48)

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

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

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

48 conjugacy classes

class	1	2A	2B	2C	2D	2E	3A	3B	4A	4B	4C	···	4H	6A	6B	6C	6D	6E	6F	6G	6H	6I	6J	8A	8B	12A	12B	12C	12D	12E	···	12P	24A	24B	24C	24D
order	1	2	2	2	2	2	3	3	4	4	4	···	4	6	6	6	6	6	6	6	6	6	6	8	8	12	12	12	12	12	···	12	24	24	24	24
size	1	1	2	4	4	8	1	1	2	2	4	···	4	1	1	2	2	4	4	4	4	8	8	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	2	2	2	2	2	2	4	4
type	+	+	+	+	+	+							+	+	+
image	C₁	C₂	C₂	C₂	C₂	C₂	C₃	C₆	C₆	C₆	C₆	C₆	D₄	D₄	D₄	C₃xD₄	C₃xD₄	C₃xD₄	D₄.₈D₄	C₃xD₄.₈D₄
kernel	C₃xD₄.₈D₄	C₃xC₄.₁₀D₄	C₃xC₄wrC₂	C₃xC₄.₄D₄	C₃xC₈:C₂²	C₃x2_-¹⁺⁴	D₄.₈D₄	C₄.₁₀D₄	C₄wrC₂	C₄.₄D₄	C₈:C₂²	2_-¹⁺⁴	C₂xC₁₂	C₃xD₄	C₃xQ₈	C₂xC₄	D₄	Q₈	C₃	C₁
# reps	1	1	2	1	2	1	2	2	4	2	4	2	2	2	2	4	4	4	2	4

Matrix representation of C₃xD₄.₈D₄ ►in GL₄(F₇₃) generated by

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	72
0	0	1	0

0	0	0	27
0	0	27	0
0	46	0	0
46	0	0	0

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

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

G:=sub<GL(4,GF(73))| [8,0,0,0,0,8,0,0,0,0,8,0,0,0,0,8],[0,1,0,0,72,0,0,0,0,0,0,1,0,0,72,0],[0,0,0,46,0,0,46,0,0,27,0,0,27,0,0,0],[0,0,0,72,0,0,72,0,1,0,0,0,0,72,0,0],[1,0,0,0,0,72,0,0,0,0,0,72,0,0,72,0] >;

C₃xD₄.₈D₄ in GAP, Magma, Sage, TeX

C_3\times D_4._8D_4

% in TeX

G:=Group("C3xD4.8D4");

// GroupNames label

G:=SmallGroup(192,887);

// by ID

G=gap.SmallGroup(192,887);

# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,365,1094,520,4204,2111,1068,172,3036]);

// Polycyclic

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

// generators/relations

G = C3xD4.8D4 order 192 = 26·3

Direct product of C3 and D4.8D4

G = C₃xD₄.₈D₄ order 192 = 2⁶·3

Direct product of C₃ and D₄.₈D₄