C3xC2^3.11D6 - GroupNames

G = C₃×C₂³.₁₁D₆ order 288 = 2⁵·3²

Direct product of C₃ and C₂³.₁₁D₆

direct product, metabelian, supersoluble, monomial

Aliases: C₃×C₂³.₁₁D₆, C₆².₁₇₆C₂³, D₆⋊C₄⋊₁₁C₆, C₆.₂₁(C₆×D₄), (C₂×Dic₆)⋊₃C₆, C₆.₁₈₀(S₃×D₄), (C₄×Dic₃)⋊₁₂C₆, (C₆×Dic₆)⋊₂₇C₂, (C₂×C₁₂).₂₆₇D₆, C₆.D₄⋊₅C₆, C₂³.₁₁(S₃×C₆), Dic₃.₁(C₃×D₄), (C₂²×C₆).₂₉D₆, (Dic₃×C₁₂)⋊₃₃C₂, (C₃×Dic₃).₂₈D₄, C₆.₁₁₉(C₄○D₁₂), (C₆×C₁₂).₂₄₄C₂², C₃²⋊₁₃(C₄.₄D₄), (C₂×C₆²).₅₂C₂², C₆.₁₁₄(D₄⋊₂S₃), (C₆×Dic₃).₁₂₂C₂², C₂.₁₀(C₃×S₃×D₄), C₆.₉(C₃×C₄○D₄), (C₃×D₆⋊C₄)⋊₃₀C₂, (C₃×C₂²⋊C₄)⋊₇C₆, C₂²⋊C₄⋊₅(C₃×S₃), (C₂×C₄).₂₈(S₃×C₆), (C₂×C₃⋊D₄).₄C₆, C₃⋊₂(C₃×C₄.₄D₄), C₂².₄₄(S₃×C₂×C₆), (C₃×C₂²⋊C₄)⋊₁₃S₃, (C₂×C₁₂).₅₄(C₂×C₆), C₂.₁₂(C₃×C₄○D₁₂), (C₃×C₆).₂₀₉(C₂×D₄), C₂.₉(C₃×D₄⋊₂S₃), (C₆×C₃⋊D₄).₁₁C₂, (S₃×C₂×C₆).₅₅C₂², (C₃×C₆).₉₉(C₄○D₄), (C₂²×S₃).₅(C₂×C₆), (C₂²×C₆).₂₆(C₂×C₆), (C₂×C₆).₃₁(C₂²×C₆), (C₃×C₆.D₄)⋊₂₂C₂, (C₃²×C₂²⋊C₄)⋊₁₁C₂, (C₂×C₆).₃₀₉(C₂²×S₃), (C₂×Dic₃).₂₂(C₂×C₆), SmallGroup(288,656)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₆ — C₃×C₂³.₁₁D₆

Chief series

C₁ — C₃ — C₆ — C₂×C₆ — C₆² — S₃×C₂×C₆ — C₃×D₆⋊C₄ — C₃×C₂³.₁₁D₆

Lower central

C₃ — C₂×C₆ — C₃×C₂³.₁₁D₆

Upper central

C₁ — C₂×C₆ — C₃×C₂²⋊C₄

Generators and relations for C₃×C₂³.₁₁D₆
G = < a,b,c,d,e,f | a³=b²=c²=d²=1, e⁶=c, f²=dc=cd, ab=ba, ac=ca, ad=da, ae=ea, af=fa, fbf^-1=bc=cb, ebe^-1=bd=db, ce=ec, cf=fc, de=ed, df=fd, fef^-1=de⁵ >

Subgroups: 418 in 169 conjugacy classes, 62 normal (58 characteristic)
C₁, C₂, C₂, C₃, C₃, C₄, C₂², C₂², S₃, C₆, C₆, C₂×C₄, C₂×C₄, D₄, Q₈, C₂³, C₂³, C₃², Dic₃, Dic₃, C₁₂, D₆, C₂×C₆, C₂×C₆, C₄², C₂²⋊C₄, C₂²⋊C₄, C₂×D₄, C₂×Q₈, C₃×S₃, C₃×C₆, C₃×C₆, Dic₆, C₂×Dic₃, C₃⋊D₄, C₂×C₁₂, C₂×C₁₂, C₃×D₄, C₃×Q₈, C₂²×S₃, C₂²×C₆, C₂²×C₆, C₄.₄D₄, C₃×Dic₃, C₃×Dic₃, C₃×C₁₂, S₃×C₆, C₆², C₆², C₄×Dic₃, D₆⋊C₄, C₆.D₄, C₄×C₁₂, C₃×C₂²⋊C₄, C₃×C₂²⋊C₄, C₂×Dic₆, C₂×C₃⋊D₄, C₆×D₄, C₆×Q₈, C₃×Dic₆, C₆×Dic₃, C₃×C₃⋊D₄, C₆×C₁₂, S₃×C₂×C₆, C₂×C₆², C₂³.₁₁D₆, C₃×C₄.₄D₄, Dic₃×C₁₂, C₃×D₆⋊C₄, C₃×C₆.D₄, C₃²×C₂²⋊C₄, C₆×Dic₆, C₆×C₃⋊D₄, C₃×C₂³.₁₁D₆
Quotients: C₁, C₂, C₃, C₂², S₃, C₆, D₄, C₂³, D₆, C₂×C₆, C₂×D₄, C₄○D₄, C₃×S₃, C₃×D₄, C₂²×S₃, C₂²×C₆, C₄.₄D₄, S₃×C₆, C₄○D₁₂, S₃×D₄, D₄⋊₂S₃, C₆×D₄, C₃×C₄○D₄, S₃×C₂×C₆, C₂³.₁₁D₆, C₃×C₄.₄D₄, C₃×C₄○D₁₂, C₃×S₃×D₄, C₃×D₄⋊₂S₃, C₃×C₂³.₁₁D₆

Smallest permutation representation of C₃×C₂³.₁₁D₆
►On 48 points

Generators in S₄₈

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

(2 46)(4 48)(6 38)(8 40)(10 42)(12 44)(13 32)(14 20)(15 34)(16 22)(17 36)(18 24)(19 26)(21 28)(23 30)(25 31)(27 33)(29 35)

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

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

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

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

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

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

72 conjugacy classes

class	1	2A	2B	2C	2D	2E	3A	3B	3C	3D	3E	4A	4B	4C	4D	4E	4F	4G	4H	6A	···	6F	6G	···	6O	6P	···	6W	6X	6Y	12A	12B	12C	12D	12E	···	12R	12S	···	12Z	12AA	12AB
order	1	2	2	2	2	2	3	3	3	3	3	4	4	4	4	4	4	4	4	6	···	6	6	···	6	6	···	6	6	6	12	12	12	12	12	···	12	12	···	12	12	12
size	1	1	1	1	4	12	1	1	2	2	2	2	2	4	6	6	6	6	12	1	···	1	2	···	2	4	···	4	12	12	2	2	2	2	4	···	4	6	···	6	12	12

72 irreducible representations

dim	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	4	4	4	4
type	+	+	+	+	+	+	+								+	+	+	+									+	-
image	C₁	C₂	C₂	C₂	C₂	C₂	C₂	C₃	C₆	C₆	C₆	C₆	C₆	C₆	S₃	D₄	D₆	D₆	C₄○D₄	C₃×S₃	C₃×D₄	S₃×C₆	S₃×C₆	C₄○D₁₂	C₃×C₄○D₄	C₃×C₄○D₁₂	S₃×D₄	D₄⋊₂S₃	C₃×S₃×D₄	C₃×D₄⋊₂S₃
kernel	C₃×C₂³.₁₁D₆	Dic₃×C₁₂	C₃×D₆⋊C₄	C₃×C₆.D₄	C₃²×C₂²⋊C₄	C₆×Dic₆	C₆×C₃⋊D₄	C₂³.₁₁D₆	C₄×Dic₃	D₆⋊C₄	C₆.D₄	C₃×C₂²⋊C₄	C₂×Dic₆	C₂×C₃⋊D₄	C₃×C₂²⋊C₄	C₃×Dic₃	C₂×C₁₂	C₂²×C₆	C₃×C₆	C₂²⋊C₄	Dic₃	C₂×C₄	C₂³	C₆	C₆	C₂	C₆	C₆	C₂	C₂
# reps	1	1	2	1	1	1	1	2	2	4	2	2	2	2	1	2	2	1	4	2	4	4	2	4	8	8	1	1	2	2

Matrix representation of C₃×C₂³.₁₁D₆ ►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

1	0	0	0	0	0
0	1	0	0	0	0
0	0	1	0	0	0
0	0	5	12	0	0
0	0	0	0	1	0
0	0	0	0	10	12

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

9	0	0	0	0	0
0	3	0	0	0	0
0	0	5	0	0	0
0	0	0	5	0	0
0	0	0	0	1	5
0	0	0	0	0	12

0	3	0	0	0	0
9	0	0	0	0	0
0	0	8	2	0	0
0	0	0	5	0	0
0	0	0	0	8	0
0	0	0	0	2	5

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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,5,0,0,0,0,0,12,0,0,0,0,0,0,1,10,0,0,0,0,0,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[9,0,0,0,0,0,0,3,0,0,0,0,0,0,5,0,0,0,0,0,0,5,0,0,0,0,0,0,1,0,0,0,0,0,5,12],[0,9,0,0,0,0,3,0,0,0,0,0,0,0,8,0,0,0,0,0,2,5,0,0,0,0,0,0,8,2,0,0,0,0,0,5] >;

C₃×C₂³.₁₁D₆ in GAP, Magma, Sage, TeX

C_3\times C_2^3._{11}D_6

% in TeX

G:=Group("C3xC2^3.11D6");

// GroupNames label

G:=SmallGroup(288,656);

// by ID

G=gap.SmallGroup(288,656);

# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,701,176,1598,555,9414]);

// Polycyclic

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

// generators/relations

G = C3×C23.11D6 order 288 = 25·32

Direct product of C3 and C23.11D6

G = C₃×C₂³.₁₁D₆ order 288 = 2⁵·3²

Direct product of C₃ and C₂³.₁₁D₆