C3xC2^3.12D6 - GroupNames

G = C₃×C₂³.₁₂D₆ order 288 = 2⁵·3²

Direct product of C₃ and C₂³.₁₂D₆

direct product, metabelian, supersoluble, monomial

Aliases: C₃×C₂³.₁₂D₆, C₆².₂₀₁C₂³, (C₆×D₄).₅C₆, C₆.₄₈(C₆×D₄), (C₄×Dic₃)⋊₅C₆, (C₆×D₄).₂₈S₃, (C₃×C₁₂).₈₇D₄, C₁₂.₁₇(C₃×D₄), (C₆×Dic₆)⋊₁₅C₂, (C₂×Dic₆)⋊₁₀C₆, C₆.D₄⋊₉C₆, (C₂×C₁₂).₃₂₆D₆, C₂³.₁₂(S₃×C₆), (C₂²×C₆).₃₁D₆, (Dic₃×C₁₂)⋊₁₅C₂, C₁₂.₈₉(C₃⋊D₄), (C₆×C₁₂).₁₂₁C₂², C₃²⋊₁₅(C₄.₄D₄), (C₂×C₆²).₅₆C₂², C₆.₁₂₄(D₄⋊₂S₃), (C₆×Dic₃).₁₀₀C₂², (D₄×C₃×C₆).₇C₂, C₄.₇(C₃×C₃⋊D₄), (C₂×C₄).₅₀(S₃×C₆), (C₂×D₄).₆(C₃×S₃), C₆.₃₀(C₃×C₄○D₄), C₃⋊₃(C₃×C₄.₄D₄), C₂.₁₂(C₆×C₃⋊D₄), C₂².₅₈(S₃×C₂×C₆), (C₂×C₁₂).₃₂(C₂×C₆), (C₃×C₆).₂₅₈(C₂×D₄), C₆.₁₄₉(C₂×C₃⋊D₄), C₂.₁₆(C₃×D₄⋊₂S₃), (C₂×C₆).₅₆(C₂²×C₆), (C₂²×C₆).₃₀(C₂×C₆), (C₃×C₆).₁₃₈(C₄○D₄), (C₃×C₆.D₄)⋊₂₅C₂, (C₂×C₆).₃₃₄(C₂²×S₃), (C₂×Dic₃).₃₇(C₂×C₆), SmallGroup(288,707)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₆ — C₃×C₂³.₁₂D₆

Chief series

C₁ — C₃ — C₆ — C₂×C₆ — C₆² — C₆×Dic₃ — Dic₃×C₁₂ — C₃×C₂³.₁₂D₆

Lower central

C₃ — C₂×C₆ — C₃×C₂³.₁₂D₆

Upper central

C₁ — C₂×C₆ — C₆×D₄

Generators and relations for C₃×C₂³.₁₂D₆
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, ebe^-1=bc=cb, fbf^-1=bd=db, ce=ec, cf=fc, de=ed, df=fd, fef^-1=e⁵ >

Subgroups: 394 in 179 conjugacy classes, 66 normal (26 characteristic)
C₁, C₂, C₂, C₂, C₃, C₃, C₄, C₄, C₂², C₂², C₆, C₆, C₆, C₂×C₄, C₂×C₄, D₄, Q₈, C₂³, C₃², Dic₃, C₁₂, C₁₂, C₂×C₆, C₂×C₆, C₄², C₂²⋊C₄, C₂×D₄, C₂×Q₈, C₃×C₆, C₃×C₆, C₃×C₆, Dic₆, C₂×Dic₃, C₂×C₁₂, C₂×C₁₂, C₃×D₄, C₃×Q₈, C₂²×C₆, C₂²×C₆, C₄.₄D₄, C₃×Dic₃, C₃×C₁₂, C₆², C₆², C₄×Dic₃, C₆.D₄, C₄×C₁₂, C₃×C₂²⋊C₄, C₂×Dic₆, C₆×D₄, C₆×D₄, C₆×Q₈, C₃×Dic₆, C₆×Dic₃, C₆×C₁₂, D₄×C₃², C₂×C₆², C₂³.₁₂D₆, C₃×C₄.₄D₄, Dic₃×C₁₂, C₃×C₆.D₄, C₆×Dic₆, D₄×C₃×C₆, C₃×C₂³.₁₂D₆
Quotients: C₁, C₂, C₃, C₂², S₃, C₆, D₄, C₂³, D₆, C₂×C₆, C₂×D₄, C₄○D₄, C₃×S₃, C₃⋊D₄, C₃×D₄, C₂²×S₃, C₂²×C₆, C₄.₄D₄, S₃×C₆, D₄⋊₂S₃, C₂×C₃⋊D₄, C₆×D₄, C₃×C₄○D₄, C₃×C₃⋊D₄, S₃×C₂×C₆, C₂³.₁₂D₆, C₃×C₄.₄D₄, C₃×D₄⋊₂S₃, C₆×C₃⋊D₄, C₃×C₂³.₁₂D₆

Smallest permutation representation of C₃×C₂³.₁₂D₆
►On 48 points

Generators in S₄₈

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

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

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

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

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

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

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

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	···	6AE	12A	12B	12C	12D	12E	···	12J	12K	···	12R	12S	12T	12U	12V
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	12	12	12	12	12	···	12	12	···	12	12	12	12	12
size	1	1	1	1	4	4	1	1	2	2	2	2	2	6	6	6	6	12	12	1	···	1	2	···	2	4	···	4	2	2	2	2	4	···	4	6	···	6	12	12	12	12

72 irreducible representations

dim	1	1	1	1	1	1	1	1	1	1	2	2	2	2	2	2	2	2	2	2	2	2	4	4
type	+	+	+	+	+						+	+	+	+									-
image	C₁	C₂	C₂	C₂	C₂	C₃	C₆	C₆	C₆	C₆	S₃	D₄	D₆	D₆	C₄○D₄	C₃×S₃	C₃⋊D₄	C₃×D₄	S₃×C₆	S₃×C₆	C₃×C₄○D₄	C₃×C₃⋊D₄	D₄⋊₂S₃	C₃×D₄⋊₂S₃
kernel	C₃×C₂³.₁₂D₆	Dic₃×C₁₂	C₃×C₆.D₄	C₆×Dic₆	D₄×C₃×C₆	C₂³.₁₂D₆	C₄×Dic₃	C₆.D₄	C₂×Dic₆	C₆×D₄	C₆×D₄	C₃×C₁₂	C₂×C₁₂	C₂²×C₆	C₃×C₆	C₂×D₄	C₁₂	C₁₂	C₂×C₄	C₂³	C₆	C₄	C₆	C₂
# reps	1	1	4	1	1	2	2	8	2	2	1	2	1	2	4	2	4	4	2	4	8	8	2	4

Matrix representation of C₃×C₂³.₁₂D₆ ►in GL₆(𝔽₁₃)

1	0	0	0	0	0
0	1	0	0	0	0
0	0	3	0	0	0
0	0	0	3	0	0
0	0	0	0	1	0
0	0	0	0	0	1

1	0	0	0	0	0
3	12	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	1	12

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

12	0	0	0	0	0
0	12	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

1	0	0	0	0	0
0	1	0	0	0	0
0	0	10	0	0	0
0	0	10	4	0	0
0	0	0	0	12	2
0	0	0	0	12	1

8	12	0	0	0	0
0	5	0	0	0	0
0	0	4	8	0	0
0	0	3	9	0	0
0	0	0	0	5	3
0	0	0	0	5	8

G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,3,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,1,0,0,0,0,0,12],[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],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,10,10,0,0,0,0,0,4,0,0,0,0,0,0,12,12,0,0,0,0,2,1],[8,0,0,0,0,0,12,5,0,0,0,0,0,0,4,3,0,0,0,0,8,9,0,0,0,0,0,0,5,5,0,0,0,0,3,8] >;

C₃×C₂³.₁₂D₆ in GAP, Magma, Sage, TeX

C_3\times C_2^3._{12}D_6

% in TeX

G:=Group("C3xC2^3.12D6");

// GroupNames label

G:=SmallGroup(288,707);

// by ID

G=gap.SmallGroup(288,707);

# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,336,176,1598,303,9414]);

// 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,e*b*e^-1=b*c=c*b,f*b*f^-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=e^5>;

// generators/relations

G = C3×C23.12D6 order 288 = 25·32

Direct product of C3 and C23.12D6

G = C₃×C₂³.₁₂D₆ order 288 = 2⁵·3²

Direct product of C₃ and C₂³.₁₂D₆