C3xD12.C4 - GroupNames

G = C₃xD₁₂.C₄ order 288 = 2⁵·3²

Direct product of C₃ and D₁₂.C₄

direct product, metabelian, supersoluble, monomial

Aliases: C₃xD₁₂.C₄, D₁₂.C₁₂, C₂₄.₆₆D₆, Dic₆.C₁₂, (S₃xC₈):₈C₆, C₃:D₄.C₁₂, C₈:S₃:₆C₆, C₈.₁₂(S₃xC₆), C₄.₅(S₃xC₁₂), (S₃xC₂₄):₁₇C₂, C₁₂.₅₃(C₄xS₃), C₂₄.₁₅(C₂xC₆), C₄oD₁₂.₃C₆, (C₃xD₁₂).₃C₄, D₆.₂(C₂xC₁₂), C₁₂.₁₃(C₂xC₁₂), (C₂xC₁₂).₃₂₂D₆, C₃²:₁₁(C₈oD₄), M₄(2):₅(C₃xS₃), (C₃xM₄(2)):₆C₆, C₆².₆₀(C₂xC₄), C₂².₁(S₃xC₁₂), (C₃xDic₆).₃C₄, (C₃xM₄(2)):₁₁S₃, C₁₂.₃₉(C₂²xC₆), (C₃xC₂₄).₄₈C₂², C₆.₁₆(C₂²xC₁₂), Dic₃.₄(C₂xC₁₂), (S₃xC₁₂).₆₀C₂², (C₃xC₁₂).₁₇₁C₂³, (C₆xC₁₂).₁₁₄C₂², C₁₂.₂₂₇(C₂²xS₃), (C₃²xM₄(2)):₈C₂, (C₂xC₃:C₈):₃C₆, (C₆xC₃:C₈):₂₄C₂, C₃:₂(C₃xC₈oD₄), C₄.₃₉(S₃xC₂xC₆), C₃:C₈.₁₂(C₂xC₆), C₂.₁₇(S₃xC₂xC₁₂), C₆.₁₁₅(S₃xC₂xC₄), (C₂xC₆).₆(C₂xC₁₂), (C₂xC₆).₂₄(C₄xS₃), (C₂xC₄).₄₆(S₃xC₆), (C₃xC₃:D₄).₃C₄, (C₃xC₈:S₃):₁₄C₂, (C₄xS₃).₁₇(C₂xC₆), (S₃xC₆).₁₄(C₂xC₄), (C₃xC₁₂).₆₈(C₂xC₄), (C₂xC₁₂).₂₅(C₂xC₆), (C₃xC₄oD₁₂).₉C₂, (C₃xC₃:C₈).₄₃C₂², (C₃xC₆).₈₇(C₂²xC₄), (C₃xDic₃).₂₀(C₂xC₄), SmallGroup(288,678)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₆ — C₃xD₁₂.C₄

Chief series

C₁ — C₃ — C₆ — C₁₂ — C₃xC₁₂ — S₃xC₁₂ — C₃xC₄oD₁₂ — C₃xD₁₂.C₄

Lower central

C₃ — C₆ — C₃xD₁₂.C₄

Upper central

C₁ — C₁₂ — C₃xM₄(2)

Generators and relations for C₃xD₁₂.C₄
G = < a,b,c,d | a³=b¹²=c²=1, d⁴=b⁶, ab=ba, ac=ca, ad=da, cbc=b^-1, dbd^-1=b⁷, dcd^-1=b⁶c >

Subgroups: 250 in 134 conjugacy classes, 74 normal (42 characteristic)
C₁, C₂, C₂, C₃, C₃, C₄, C₄, C₂², C₂², S₃, C₆, C₆, C₈, C₈, C₂xC₄, C₂xC₄, D₄, Q₈, C₃², Dic₃, C₁₂, C₁₂, D₆, C₂xC₆, C₂xC₆, C₂xC₈, M₄(2), M₄(2), C₄oD₄, C₃xS₃, C₃xC₆, C₃xC₆, C₃:C₈, C₂₄, C₂₄, Dic₆, C₄xS₃, D₁₂, C₃:D₄, C₂xC₁₂, C₂xC₁₂, C₃xD₄, C₃xQ₈, C₈oD₄, C₃xDic₃, C₃xC₁₂, S₃xC₆, C₆², S₃xC₈, C₈:S₃, C₂xC₃:C₈, C₂xC₂₄, C₃xM₄(2), C₃xM₄(2), C₄oD₁₂, C₃xC₄oD₄, C₃xC₃:C₈, C₃xC₂₄, C₃xDic₆, S₃xC₁₂, C₃xD₁₂, C₃xC₃:D₄, C₆xC₁₂, D₁₂.C₄, C₃xC₈oD₄, S₃xC₂₄, C₃xC₈:S₃, C₆xC₃:C₈, C₃²xM₄(2), C₃xC₄oD₁₂, C₃xD₁₂.C₄
Quotients: C₁, C₂, C₃, C₄, C₂², S₃, C₆, C₂xC₄, C₂³, C₁₂, D₆, C₂xC₆, C₂²xC₄, C₃xS₃, C₄xS₃, C₂xC₁₂, C₂²xS₃, C₂²xC₆, C₈oD₄, S₃xC₆, S₃xC₂xC₄, C₂²xC₁₂, S₃xC₁₂, S₃xC₂xC₆, D₁₂.C₄, C₃xC₈oD₄, S₃xC₂xC₁₂, C₃xD₁₂.C₄

Smallest permutation representation of C₃xD₁₂.C₄
►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 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 36)(2 35)(3 34)(4 33)(5 32)(6 31)(7 30)(8 29)(9 28)(10 27)(11 26)(12 25)(13 48)(14 47)(15 46)(16 45)(17 44)(18 43)(19 42)(20 41)(21 40)(22 39)(23 38)(24 37)

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

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

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

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

90 conjugacy classes

class	1	2A	2B	2C	2D	3A	3B	3C	3D	3E	4A	4B	4C	4D	4E	6A	6B	6C	···	6G	6H	6I	6J	6K	6L	6M	6N	8A	8B	8C	8D	8E	8F	8G	8H	8I	8J	12A	12B	12C	12D	12E	···	12L	12M	12N	12O	12P	12Q	12R	12S	24A	···	24H	24I	···	24P	24Q	···	24AB	24AC	24AD	24AE	24AF
order	1	2	2	2	2	3	3	3	3	3	4	4	4	4	4	6	6	6	···	6	6	6	6	6	6	6	6	8	8	8	8	8	8	8	8	8	8	12	12	12	12	12	···	12	12	12	12	12	12	12	12	24	···	24	24	···	24	24	···	24	24	24	24	24
size	1	1	2	6	6	1	1	2	2	2	1	1	2	6	6	1	1	2	···	2	4	4	4	6	6	6	6	2	2	2	2	3	3	3	3	6	6	1	1	1	1	2	···	2	4	4	4	6	6	6	6	2	···	2	3	···	3	4	···	4	6	6	6	6

90 irreducible representations

dim	1	1	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	4	4
type	+	+	+	+	+	+													+	+	+
image	C₁	C₂	C₂	C₂	C₂	C₂	C₃	C₄	C₄	C₄	C₆	C₆	C₆	C₆	C₆	C₁₂	C₁₂	C₁₂	S₃	D₆	D₆	C₃xS₃	C₄xS₃	C₄xS₃	C₈oD₄	S₃xC₆	S₃xC₆	S₃xC₁₂	S₃xC₁₂	C₃xC₈oD₄	D₁₂.C₄	C₃xD₁₂.C₄
kernel	C₃xD₁₂.C₄	S₃xC₂₄	C₃xC₈:S₃	C₆xC₃:C₈	C₃²xM₄(2)	C₃xC₄oD₁₂	D₁₂.C₄	C₃xDic₆	C₃xD₁₂	C₃xC₃:D₄	S₃xC₈	C₈:S₃	C₂xC₃:C₈	C₃xM₄(2)	C₄oD₁₂	Dic₆	D₁₂	C₃:D₄	C₃xM₄(2)	C₂₄	C₂xC₁₂	M₄(2)	C₁₂	C₂xC₆	C₃²	C₈	C₂xC₄	C₄	C₂²	C₃	C₃	C₁
# reps	1	2	2	1	1	1	2	2	2	4	4	4	2	2	2	4	4	8	1	2	1	2	2	2	4	4	2	4	4	8	2	4

Matrix representation of C₃xD₁₂.C₄ ►in GL₄(F₇₃) generated by

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

65	0	0	0
0	9	0	0
0	0	46	0
0	0	0	27

0	9	0	0
65	0	0	0
0	0	0	51
0	0	63	0

46	0	0	0
0	46	0	0
0	0	0	1
0	0	46	0

G:=sub<GL(4,GF(73))| [64,0,0,0,0,64,0,0,0,0,1,0,0,0,0,1],[65,0,0,0,0,9,0,0,0,0,46,0,0,0,0,27],[0,65,0,0,9,0,0,0,0,0,0,63,0,0,51,0],[46,0,0,0,0,46,0,0,0,0,0,46,0,0,1,0] >;

C₃xD₁₂.C₄ in GAP, Magma, Sage, TeX

C_3\times D_{12}.C_4

% in TeX

G:=Group("C3xD12.C4");

// GroupNames label

G:=SmallGroup(288,678);

// by ID

G=gap.SmallGroup(288,678);

# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,701,555,142,102,9414]);

// Polycyclic

G:=Group<a,b,c,d|a^3=b^12=c^2=1,d^4=b^6,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,d*b*d^-1=b^7,d*c*d^-1=b^6*c>;

// generators/relations

G = C3xD12.C4 order 288 = 25·32

Direct product of C3 and D12.C4

G = C₃xD₁₂.C₄ order 288 = 2⁵·3²

Direct product of C₃ and D₁₂.C₄