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, D₁₂:₄C₁₂, Dic₆:₄C₁₂, C₆².₃₄D₄, C₃²:₁₀C₄wrC₂, (C₃xD₁₂):₆C₄, C₄.₃(S₃xC₁₂), C₁₂.₅₁(C₄xS₃), C₁₂.₆(C₂xC₁₂), (C₃xDic₆):₆C₄, C₄oD₁₂.₂C₆, (C₄xDic₃):₁C₆, C₁₂.₆₃(C₃xD₄), (C₂xC₆).₄₅D₁₂, C₆.₅₁(D₆:C₄), (Dic₃xC₁₂):₆C₂, (C₃xC₁₂).₁₆₅D₄, (C₂xC₁₂).₃₁₆D₆, (C₃xM₄(2)):₈C₆, M₄(2):₄(C₃xS₃), (C₃xM₄(2)):₈S₃, C₂².₃(C₃xD₁₂), (C₆xC₁₂).₄₆C₂², C₁₂.₁₄₆(C₃:D₄), (C₃²xM₄(2)):₁₂C₂, C₃:₂(C₃xC₄wrC₂), (C₂xC₆).₂(C₃xD₄), (C₂xC₄).₃₇(S₃xC₆), C₂.₁₁(C₃xD₆:C₄), C₄.₂₉(C₃xC₃:D₄), (C₂xC₁₂).₁₆(C₂xC₆), (C₃xC₁₂).₄₂(C₂xC₄), (C₃xC₄oD₁₂).₄C₂, C₆.₁₀(C₃xC₂²:C₄), (C₃xC₆).₅₀(C₂²:C₄), SmallGroup(288,259)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₁₂ — C₃xD₁₂:C₄

Chief series

C₁ — C₃ — C₆ — C₁₂ — C₂xC₁₂ — C₆xC₁₂ — C₃xC₄oD₁₂ — C₃xD₁₂:C₄

Lower central

C₃ — C₆ — C₁₂ — C₃xD₁₂:C₄

Upper central

C₁ — C₁₂ — C₂xC₁₂ — C₃xM₄(2)

Generators and relations for C₃xD₁₂:C₄
G = < a,b,c,d | a³=b¹²=c²=d⁴=1, ab=ba, ac=ca, ad=da, cbc=b^-1, dbd^-1=b⁵, dcd^-1=b⁷c >

Subgroups: 250 in 98 conjugacy classes, 38 normal (all characteristic)
C₁, C₂, C₂, C₃, C₃, C₄, C₄, C₂², C₂², S₃, C₆, C₆, C₈, C₂xC₄, C₂xC₄, D₄, Q₈, C₃², Dic₃, C₁₂, C₁₂, D₆, C₂xC₆, C₂xC₆, C₄², M₄(2), C₄oD₄, C₃xS₃, C₃xC₆, C₃xC₆, C₂₄, Dic₆, C₄xS₃, D₁₂, C₂xDic₃, C₃:D₄, C₂xC₁₂, C₂xC₁₂, C₃xD₄, C₃xQ₈, C₄wrC₂, C₃xDic₃, C₃xC₁₂, S₃xC₆, C₆², C₄xDic₃, C₄xC₁₂, C₃xM₄(2), C₃xM₄(2), C₄oD₁₂, C₃xC₄oD₄, C₃xC₂₄, C₃xDic₆, S₃xC₁₂, C₃xD₁₂, C₆xDic₃, C₃xC₃:D₄, C₆xC₁₂, D₁₂:C₄, C₃xC₄wrC₂, Dic₃xC₁₂, C₃²xM₄(2), C₃xC₄oD₁₂, C₃xD₁₂:C₄
Quotients: C₁, C₂, C₃, C₄, C₂², S₃, C₆, C₂xC₄, D₄, C₁₂, D₆, C₂xC₆, C₂²:C₄, C₃xS₃, C₄xS₃, D₁₂, C₃:D₄, C₂xC₁₂, C₃xD₄, C₄wrC₂, S₃xC₆, D₆:C₄, C₃xC₂²:C₄, S₃xC₁₂, C₃xD₁₂, C₃xC₃:D₄, D₁₂:C₄, C₃xC₄wrC₂, C₃xD₆:C₄, 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 27)(2 26)(3 25)(4 36)(5 35)(6 34)(7 33)(8 32)(9 31)(10 30)(11 29)(12 28)(13 39)(14 38)(15 37)(16 48)(17 47)(18 46)(19 45)(20 44)(21 43)(22 42)(23 41)(24 40)

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

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

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

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

72 conjugacy classes

class	1	2A	2B	2C	3A	3B	3C	3D	3E	4A	4B	4C	4D	4E	4F	4G	4H	6A	6B	6C	···	6G	6H	6I	6J	6K	6L	8A	8B	12A	12B	12C	12D	12E	···	12L	12M	12N	12O	12P	···	12W	12X	12Y	24A	···	24P
order	1	2	2	2	3	3	3	3	3	4	4	4	4	4	4	4	4	6	6	6	···	6	6	6	6	6	6	8	8	12	12	12	12	12	···	12	12	12	12	12	···	12	12	12	24	···	24
size	1	1	2	12	1	1	2	2	2	1	1	2	6	6	6	6	12	1	1	2	···	2	4	4	4	12	12	4	4	1	1	1	1	2	···	2	4	4	4	6	···	6	12	12	4	···	4

72 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
type	+	+	+	+									+	+	+	+					+
image	C₁	C₂	C₂	C₂	C₃	C₄	C₄	C₆	C₆	C₆	C₁₂	C₁₂	S₃	D₄	D₄	D₆	C₃xS₃	C₄xS₃	C₃:D₄	C₃xD₄	D₁₂	C₃xD₄	C₄wrC₂	S₃xC₆	S₃xC₁₂	C₃xC₃:D₄	C₃xD₁₂	C₃xC₄wrC₂	D₁₂:C₄	C₃xD₁₂:C₄
kernel	C₃xD₁₂:C₄	Dic₃xC₁₂	C₃²xM₄(2)	C₃xC₄oD₁₂	D₁₂:C₄	C₃xDic₆	C₃xD₁₂	C₄xDic₃	C₃xM₄(2)	C₄oD₁₂	Dic₆	D₁₂	C₃xM₄(2)	C₃xC₁₂	C₆²	C₂xC₁₂	M₄(2)	C₁₂	C₁₂	C₁₂	C₂xC₆	C₂xC₆	C₃²	C₂xC₄	C₄	C₄	C₂²	C₃	C₃	C₁
# reps	1	1	1	1	2	2	2	2	2	2	4	4	1	1	1	1	2	2	2	2	2	2	4	2	4	4	4	8	2	4

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

8	0	0	0
0	8	0	0
0	0	1	0
0	0	0	1

9	0	0	0
0	65	0	0
0	0	46	44
0	0	0	27

0	65	0	0
9	0	0	0
0	0	46	44
0	0	10	27

0	46	0	0
27	0	0	0
0	0	27	41
0	0	0	1

G:=sub<GL(4,GF(73))| [8,0,0,0,0,8,0,0,0,0,1,0,0,0,0,1],[9,0,0,0,0,65,0,0,0,0,46,0,0,0,44,27],[0,9,0,0,65,0,0,0,0,0,46,10,0,0,44,27],[0,27,0,0,46,0,0,0,0,0,27,0,0,0,41,1] >;

C₃xD₁₂:C₄ in GAP, Magma, Sage, TeX

C_3\times D_{12}\rtimes C_4

% in TeX

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

// GroupNames label

G:=SmallGroup(288,259);

// by ID

G=gap.SmallGroup(288,259);

# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-3,365,92,136,2524,1271,102,9414]);

// Polycyclic

G:=Group<a,b,c,d|a^3=b^12=c^2=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,d*b*d^-1=b^5,d*c*d^-1=b^7*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