C3xQ8.14D6 - GroupNames

G = C₃×Q₈.₁₄D₆ order 288 = 2⁵·3²

Direct product of C₃ and Q₈.₁₄D₆

direct product, metabelian, supersoluble, monomial

Aliases: C₃×Q₈.₁₄D₆, C₆².₆₈D₄, D₄.S₃⋊₆C₆, C₃⋊Q₁₆⋊₆C₆, D₄.₉(S₃×C₆), C₆.₆₁(C₆×D₄), (C₃×D₄).₄₉D₆, C₁₂.₅₉(C₃×D₄), Q₈.₁₉(S₃×C₆), (C₃×Q₈).₇₄D₆, (C₂×Dic₆)⋊₁₁C₆, (C₆×Dic₆)⋊₁₆C₂, (C₂×C₁₂).₂₄₈D₆, (C₃×C₁₂).₁₆₁D₄, C₄.Dic₃⋊₁₀C₆, (C₃×C₁₂).₉₀C₂³, C₁₂.₁₉(C₂²×C₆), Dic₆.₁₂(C₂×C₆), C₁₂.₁₄₂(C₃⋊D₄), C₁₂.₁₇₀(C₂²×S₃), (C₆×C₁₂).₁₃₃C₂², C₃²⋊₂₄(C₈.C₂²), (C₃×Dic₆).₄₂C₂², (D₄×C₃²).₂₅C₂², (Q₈×C₃²).₂₆C₂², C₃⋊C₈.₄(C₂×C₆), C₄.₁₉(S₃×C₂×C₆), (C₂×C₄).₂₁(S₃×C₆), C₄○D₄.₆(C₃×S₃), (C₃×D₄).₉(C₂×C₆), (C₂×C₆).₁₁(C₃×D₄), C₄.₂₅(C₃×C₃⋊D₄), C₂.₂₅(C₆×C₃⋊D₄), C₃⋊₅(C₃×C₈.C₂²), (C₂×C₁₂).₄₄(C₂×C₆), (C₃×D₄.S₃)⋊₁₂C₂, (C₃×C₄○D₄).₂₁S₃, (C₃×C₄○D₄).₁₁C₆, (C₃×C₃⋊Q₁₆)⋊₁₄C₂, (C₃×C₆).₂₆₉(C₂×D₄), C₆.₁₆₂(C₂×C₃⋊D₄), (C₃×C₃⋊C₈).₂₃C₂², (C₃×C₄.Dic₃)⋊₈C₂, (C₃×Q₈).₂₁(C₂×C₆), C₂².₆(C₃×C₃⋊D₄), (C₂×C₆).₄₇(C₃⋊D₄), (C₃²×C₄○D₄).₄C₂, SmallGroup(288,722)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₁₂ — C₃×Q₈.₁₄D₆

Chief series

C₁ — C₃ — C₆ — C₁₂ — C₃×C₁₂ — C₃×Dic₆ — C₆×Dic₆ — C₃×Q₈.₁₄D₆

Lower central

C₃ — C₆ — C₁₂ — C₃×Q₈.₁₄D₆

Upper central

C₁ — C₆ — C₂×C₁₂ — C₃×C₄○D₄

Generators and relations for C₃×Q₈.₁₄D₆
G = < a,b,c,d,e | a³=b⁴=d⁶=1, c²=e²=b², ab=ba, ac=ca, ad=da, ae=ea, cbc^-1=ebe^-1=b^-1, bd=db, dcd^-1=b²c, ece^-1=bc, ede^-1=d^-1 >

Subgroups: 290 in 140 conjugacy classes, 58 normal (42 characteristic)
C₁, C₂, C₂, C₃, C₃, C₄, C₄, C₂², C₂², C₆, C₆, C₈, C₂×C₄, C₂×C₄, D₄, D₄, Q₈, Q₈, C₃², Dic₃, C₁₂, C₁₂, C₂×C₆, C₂×C₆, M₄(2), SD₁₆, Q₁₆, C₂×Q₈, C₄○D₄, C₃×C₆, C₃×C₆, C₃⋊C₈, C₂₄, Dic₆, Dic₆, C₂×Dic₃, C₂×C₁₂, C₂×C₁₂, C₃×D₄, C₃×D₄, C₃×Q₈, C₃×Q₈, C₈.C₂², C₃×Dic₃, C₃×C₁₂, C₃×C₁₂, C₆², C₆², C₄.Dic₃, D₄.S₃, C₃⋊Q₁₆, C₃×M₄(2), C₃×SD₁₆, C₃×Q₁₆, C₂×Dic₆, C₆×Q₈, C₃×C₄○D₄, C₃×C₄○D₄, C₃×C₃⋊C₈, C₃×Dic₆, C₃×Dic₆, C₆×Dic₃, C₆×C₁₂, C₆×C₁₂, D₄×C₃², D₄×C₃², Q₈×C₃², Q₈.₁₄D₆, C₃×C₈.C₂², C₃×C₄.Dic₃, C₃×D₄.S₃, C₃×C₃⋊Q₁₆, C₆×Dic₆, C₃²×C₄○D₄, C₃×Q₈.₁₄D₆
Quotients: C₁, C₂, C₃, C₂², S₃, C₆, D₄, C₂³, D₆, C₂×C₆, C₂×D₄, C₃×S₃, C₃⋊D₄, C₃×D₄, C₂²×S₃, C₂²×C₆, C₈.C₂², S₃×C₆, C₂×C₃⋊D₄, C₆×D₄, C₃×C₃⋊D₄, S₃×C₂×C₆, Q₈.₁₄D₆, C₃×C₈.C₂², C₆×C₃⋊D₄, C₃×Q₈.₁₄D₆

Smallest permutation representation of C₃×Q₈.₁₄D₆
►On 48 points

Generators in S₄₈

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

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

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

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

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

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

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

63 conjugacy classes

class	1	2A	2B	2C	3A	3B	3C	3D	3E	4A	4B	4C	4D	4E	6A	6B	6C	···	6G	6H	···	6R	8A	8B	12A	···	12J	12K	···	12U	12V	12W	12X	12Y	24A	24B	24C	24D
order	1	2	2	2	3	3	3	3	3	4	4	4	4	4	6	6	6	···	6	6	···	6	8	8	12	···	12	12	···	12	12	12	12	12	24	24	24	24
size	1	1	2	4	1	1	2	2	2	2	2	4	12	12	1	1	2	···	2	4	···	4	12	12	2	···	2	4	···	4	12	12	12	12	12	12	12	12

63 irreducible representations

dim	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	4	4	4	4
type	+	+	+	+	+	+							+	+	+	+	+	+											-	-
image	C₁	C₂	C₂	C₂	C₂	C₂	C₃	C₆	C₆	C₆	C₆	C₆	S₃	D₄	D₄	D₆	D₆	D₆	C₃×S₃	C₃⋊D₄	C₃×D₄	C₃⋊D₄	C₃×D₄	S₃×C₆	S₃×C₆	S₃×C₆	C₃×C₃⋊D₄	C₃×C₃⋊D₄	C₈.C₂²	Q₈.₁₄D₆	C₃×C₈.C₂²	C₃×Q₈.₁₄D₆
kernel	C₃×Q₈.₁₄D₆	C₃×C₄.Dic₃	C₃×D₄.S₃	C₃×C₃⋊Q₁₆	C₆×Dic₆	C₃²×C₄○D₄	Q₈.₁₄D₆	C₄.Dic₃	D₄.S₃	C₃⋊Q₁₆	C₂×Dic₆	C₃×C₄○D₄	C₃×C₄○D₄	C₃×C₁₂	C₆²	C₂×C₁₂	C₃×D₄	C₃×Q₈	C₄○D₄	C₁₂	C₁₂	C₂×C₆	C₂×C₆	C₂×C₄	D₄	Q₈	C₄	C₂²	C₃²	C₃	C₃	C₁
# reps	1	1	2	2	1	1	2	2	4	4	2	2	1	1	1	1	1	1	2	2	2	2	2	2	2	2	4	4	1	2	2	4

Matrix representation of C₃×Q₈.₁₄D₆ ►in GL₆(𝔽₇₃)

8	0	0	0	0	0
0	8	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

1	0	0	0	0	0
0	1	0	0	0	0
0	0	1	48	0	0
0	0	3	72	0	0
0	0	0	60	72	2
0	0	56	60	72	1

1	0	0	0	0	0
0	1	0	0	0	0
0	0	0	0	72	0
0	0	0	56	0	70
0	0	1	0	0	0
0	0	0	48	0	17

9	0	0	0	0	0
64	65	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	72	0
0	0	0	13	0	72

65	66	0	0	0	0
9	8	0	0	0	0
0	0	0	4	0	0
0	0	18	0	0	0
0	0	0	68	61	12
0	0	44	68	67	12

G:=sub<GL(6,GF(73))| [8,0,0,0,0,0,0,8,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,3,0,56,0,0,48,72,60,60,0,0,0,0,72,72,0,0,0,0,2,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,56,0,48,0,0,72,0,0,0,0,0,0,70,0,17],[9,64,0,0,0,0,0,65,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,13,0,0,0,0,72,0,0,0,0,0,0,72],[65,9,0,0,0,0,66,8,0,0,0,0,0,0,0,18,0,44,0,0,4,0,68,68,0,0,0,0,61,67,0,0,0,0,12,12] >;

C₃×Q₈.₁₄D₆ in GAP, Magma, Sage, TeX

C_3\times Q_8._{14}D_6

% in TeX

G:=Group("C3xQ8.14D6");

// GroupNames label

G:=SmallGroup(288,722);

// by ID

G=gap.SmallGroup(288,722);

# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,336,590,555,2524,648,102,9414]);

// Polycyclic

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

// generators/relations

G = C3×Q8.14D6 order 288 = 25·32

Direct product of C3 and Q8.14D6

G = C₃×Q₈.₁₄D₆ order 288 = 2⁵·3²

Direct product of C₃ and Q₈.₁₄D₆