C3xQ8.13D6 - GroupNames

G = C₃xQ₈.₁₃D₆ order 288 = 2⁵·3²

Direct product of C₃ and Q₈.₁₃D₆

direct product, metabelian, supersoluble, monomial

Aliases: C₃xQ₈.₁₃D₆, C₆².₆₇D₄, D₄:S₃:₇C₆, C₄oD₁₂:₄C₆, D₄.S₃:₇C₆, D₄.₈(S₃xC₆), C₃:Q₁₆:₇C₆, C₆.₆₀(C₆xD₄), Q₈:₂S₃:₇C₆, (C₃xD₄).₄₈D₆, C₁₂.₆₆(C₃xD₄), (C₃xQ₈).₇₃D₆, Q₈.₁₈(S₃xC₆), D₁₂.₁₂(C₂xC₆), (C₂xC₁₂).₃₃₃D₆, (C₃xC₁₂).₁₇₄D₄, C₃²:₂₄(C₄oD₈), (C₃xC₁₂).₈₉C₂³, C₁₂.₁₈(C₂²xC₆), Dic₆.₁₁(C₂xC₆), C₁₂.₁₄₉(C₃:D₄), C₁₂.₁₆₉(C₂²xS₃), (C₆xC₁₂).₁₃₂C₂², (C₃xD₁₂).₄₁C₂², (C₃xDic₆).₄₁C₂², (D₄xC₃²).₂₄C₂², (Q₈xC₃²).₂₅C₂², (C₂xC₃:C₈):₈C₆, (C₆xC₃:C₈):₁₄C₂, C₃:₅(C₃xC₄oD₈), C₄.₁₈(S₃xC₂xC₆), C₄oD₄:₆(C₃xS₃), (C₃xC₄oD₄):₆C₆, C₃:C₈.₁₀(C₂xC₆), (C₃xD₄:S₃):₁₅C₂, (C₃xC₄oD₄):₁₁S₃, (C₃xC₄oD₁₂):₈C₂, (C₂xC₄).₅₉(S₃xC₆), (C₃xD₄).₈(C₂xC₆), (C₂xC₆).₁₀(C₃xD₄), C₄.₃₂(C₃xC₃:D₄), C₂.₂₄(C₆xC₃:D₄), (C₂xC₁₂).₄₃(C₂xC₆), (C₃xD₄.S₃):₁₅C₂, (C₃xC₃:Q₁₆):₁₅C₂, (C₃xC₆).₂₆₈(C₂xD₄), (C₃²xC₄oD₄):₂C₂, C₆.₁₆₁(C₂xC₃:D₄), (C₃xC₃:C₈).₄₁C₂², (C₃xQ₈).₂₀(C₂xC₆), C₂².₁(C₃xC₃:D₄), (C₃xQ₈:₂S₃):₁₅C₂, (C₂xC₆).₂₉(C₃:D₄), SmallGroup(288,721)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₁₂ — C₃xQ₈.₁₃D₆

Chief series

C₁ — C₃ — C₆ — C₁₂ — C₃xC₁₂ — C₃xD₁₂ — C₃xC₄oD₁₂ — C₃xQ₈.₁₃D₆

Lower central

C₃ — C₆ — C₁₂ — C₃xQ₈.₁₃D₆

Upper central

C₁ — C₁₂ — C₂xC₁₂ — C₃xC₄oD₄

Generators and relations for C₃xQ₈.₁₃D₆
G = < a,b,c,d,e | a³=b⁴=1, c²=d⁶=e²=b², ab=ba, ac=ca, ad=da, ae=ea, cbc^-1=ebe^-1=b^-1, bd=db, cd=dc, ece^-1=b^-1c, ede^-1=d⁵ >

Subgroups: 322 in 144 conjugacy classes, 58 normal (all characteristic)
C₁, C₂, C₂, C₃, C₃, C₄, C₄, C₂², C₂², S₃, C₆, C₆, C₈, C₂xC₄, C₂xC₄, D₄, D₄, Q₈, Q₈, C₃², Dic₃, C₁₂, C₁₂, D₆, C₂xC₆, C₂xC₆, C₂xC₈, D₈, SD₁₆, Q₁₆, C₄oD₄, C₄oD₄, C₃xS₃, C₃xC₆, C₃xC₆, C₃:C₈, C₂₄, Dic₆, C₄xS₃, D₁₂, C₃:D₄, C₂xC₁₂, C₂xC₁₂, C₃xD₄, C₃xD₄, C₃xQ₈, C₃xQ₈, C₄oD₈, C₃xDic₃, C₃xC₁₂, C₃xC₁₂, S₃xC₆, C₆², C₆², C₂xC₃:C₈, D₄:S₃, D₄.S₃, Q₈:₂S₃, C₃:Q₁₆, C₂xC₂₄, C₃xD₈, C₃xSD₁₆, C₃xQ₁₆, C₄oD₁₂, C₃xC₄oD₄, C₃xC₄oD₄, C₃xC₃:C₈, C₃xDic₆, S₃xC₁₂, C₃xD₁₂, C₃xC₃:D₄, C₆xC₁₂, C₆xC₁₂, D₄xC₃², D₄xC₃², Q₈xC₃², Q₈.₁₃D₆, C₃xC₄oD₈, C₆xC₃:C₈, C₃xD₄:S₃, C₃xD₄.S₃, C₃xQ₈:₂S₃, C₃xC₃:Q₁₆, C₃xC₄oD₁₂, C₃²xC₄oD₄, C₃xQ₈.₁₃D₆
Quotients: C₁, C₂, C₃, C₂², S₃, C₆, D₄, C₂³, D₆, C₂xC₆, C₂xD₄, C₃xS₃, C₃:D₄, C₃xD₄, C₂²xS₃, C₂²xC₆, C₄oD₈, S₃xC₆, C₂xC₃:D₄, C₆xD₄, C₃xC₃:D₄, S₃xC₂xC₆, Q₈.₁₃D₆, C₃xC₄oD₈, C₆xC₃:D₄, C₃xQ₈.₁₃D₆

Smallest permutation representation of C₃xQ₈.₁₃D₆
►On 48 points

Generators in S₄₈

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

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

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

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

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

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

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

72 conjugacy classes

class	1	2A	2B	2C	2D	3A	3B	3C	3D	3E	4A	4B	4C	4D	4E	6A	6B	6C	···	6G	6H	···	6R	6S	6T	8A	8B	8C	8D	12A	12B	12C	12D	12E	···	12L	12M	···	12W	12X	12Y	24A	···	24H
order	1	2	2	2	2	3	3	3	3	3	4	4	4	4	4	6	6	6	···	6	6	···	6	6	6	8	8	8	8	12	12	12	12	12	···	12	12	···	12	12	12	24	···	24
size	1	1	2	4	12	1	1	2	2	2	1	1	2	4	12	1	1	2	···	2	4	···	4	12	12	6	6	6	6	1	1	1	1	2	···	2	4	···	4	12	12	6	···	6

72 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	2	2	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	S₃	D₄	D₄	D₆	D₆	D₆	C₃xS₃	C₃:D₄	C₃xD₄	C₃:D₄	C₃xD₄	C₄oD₈	S₃xC₆	S₃xC₆	S₃xC₆	C₃xC₃:D₄	C₃xC₃:D₄	C₃xC₄oD₈	Q₈.₁₃D₆	C₃xQ₈.₁₃D₆
kernel	C₃xQ₈.₁₃D₆	C₆xC₃:C₈	C₃xD₄:S₃	C₃xD₄.S₃	C₃xQ₈:₂S₃	C₃xC₃:Q₁₆	C₃xC₄oD₁₂	C₃²xC₄oD₄	Q₈.₁₃D₆	C₂xC₃:C₈	D₄:S₃	D₄.S₃	Q₈:₂S₃	C₃:Q₁₆	C₄oD₁₂	C₃xC₄oD₄	C₃xC₄oD₄	C₃xC₁₂	C₆²	C₂xC₁₂	C₃xD₄	C₃xQ₈	C₄oD₄	C₁₂	C₁₂	C₂xC₆	C₂xC₆	C₃²	C₂xC₄	D₄	Q₈	C₄	C₂²	C₃	C₃	C₁
# reps	1	1	1	1	1	1	1	1	2	2	2	2	2	2	2	2	1	1	1	1	1	1	2	2	2	2	2	4	2	2	2	4	4	8	2	4

Matrix representation of C₃xQ₈.₁₃D₆ ►in GL₄(F₇₃) generated by

64	0	0	0
0	64	0	0
0	0	1	0
0	0	0	1

1	0	0	0
0	1	0	0
0	0	27	0
0	0	0	46

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

8	0	0	0
0	64	0	0
0	0	27	0
0	0	0	27

0	64	0	0
8	0	0	0
0	0	0	63
0	0	22	0

G:=sub<GL(4,GF(73))| [64,0,0,0,0,64,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,27,0,0,0,0,46],[1,0,0,0,0,1,0,0,0,0,0,72,0,0,1,0],[8,0,0,0,0,64,0,0,0,0,27,0,0,0,0,27],[0,8,0,0,64,0,0,0,0,0,0,22,0,0,63,0] >;

C₃xQ₈.₁₃D₆ in GAP, Magma, Sage, TeX

C_3\times Q_8._{13}D_6

% in TeX

G:=Group("C3xQ8.13D6");

// GroupNames label

G:=SmallGroup(288,721);

// by ID

G=gap.SmallGroup(288,721);

# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,344,590,2524,648,102,9414]);

// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^4=1,c^2=d^6=e^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=e*b*e^-1=b^-1,b*d=d*b,c*d=d*c,e*c*e^-1=b^-1*c,e*d*e^-1=d^5>;

// generators/relations

G = C3xQ8.13D6 order 288 = 25·32

Direct product of C3 and Q8.13D6

G = C₃xQ₈.₁₃D₆ order 288 = 2⁵·3²

Direct product of C₃ and Q₈.₁₃D₆