C3xC4oD24 - GroupNames

G = C₃xC₄oD₂₄ order 288 = 2⁵·3²

Direct product of C₃ and C₄oD₂₄

direct product, metabelian, supersoluble, monomial

Aliases: C₃xC₄oD₂₄, D₂₄:₇C₆, C₂₄.₈₂D₆, Dic₁₂:₇C₆, C₁₂.₉₅D₁₂, C₆².₈₇D₄, (C₆xC₂₄):₉C₂, (C₂xC₂₄):₆C₆, (C₂xC₂₄):₉S₃, C₂₄:C₂:₇C₆, C₄oD₁₂:₁C₆, C₈.₁₇(S₃xC₆), C₆.₁₁(C₆xD₄), (C₃xD₂₄):₁₅C₂, C₂₄.₁₉(C₂xC₆), D₁₂.₇(C₂xC₆), C₁₂.₃₅(C₃xD₄), C₆.₉₉(C₂xD₁₂), (C₂xC₆).₁₉D₁₂, C₂.₁₃(C₆xD₁₂), C₄.₂₀(C₃xD₁₂), (C₃xC₁₂).₁₄₀D₄, (C₂xC₁₂).₄₄₃D₆, C₃²:₁₆(C₄oD₈), Dic₆.₆(C₂xC₆), C₂².₁(C₃xD₁₂), (C₃xDic₁₂):₁₅C₂, C₁₂.₃₀(C₂²xC₆), (C₃xC₂₄).₅₇C₂², (C₃xC₁₂).₁₆₂C₂³, C₁₂.₂₁₇(C₂²xS₃), (C₆xC₁₂).₃₁₄C₂², (C₃xD₁₂).₄₆C₂², (C₃xDic₆).₄₆C₂², (C₂xC₈):₄(C₃xS₃), C₃:₁(C₃xC₄oD₈), C₄.₂₈(S₃xC₂xC₆), (C₃xC₄oD₁₂):₅C₂, (C₂xC₄).₈₀(S₃xC₆), (C₂xC₆).₂₂(C₃xD₄), (C₃xC₂₄:C₂):₁₅C₂, (C₃xC₆).₁₈₁(C₂xD₄), (C₂xC₁₂).₁₁₅(C₂xC₆), SmallGroup(288,675)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₁₂ — C₃xC₄oD₂₄

Chief series

C₁ — C₃ — C₆ — C₁₂ — C₃xC₁₂ — C₃xD₁₂ — C₃xC₄oD₁₂ — C₃xC₄oD₂₄

Lower central

C₃ — C₆ — C₁₂ — C₃xC₄oD₂₄

Upper central

C₁ — C₁₂ — C₂xC₁₂ — C₂xC₂₄

Generators and relations for C₃xC₄oD₂₄
G = < a,b,c,d | a³=b⁴=d²=1, c¹²=b², ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b²c¹¹ >

Subgroups: 346 in 135 conjugacy classes, 58 normal (42 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, C₂xC₈, D₈, SD₁₆, Q₁₆, C₄oD₄, C₃xS₃, C₃xC₆, C₃xC₆, C₂₄, C₂₄, Dic₆, C₄xS₃, D₁₂, C₃:D₄, C₂xC₁₂, C₂xC₁₂, C₃xD₄, C₃xQ₈, C₄oD₈, C₃xDic₃, C₃xC₁₂, S₃xC₆, C₆², C₂₄:C₂, D₂₄, Dic₁₂, C₂xC₂₄, C₂xC₂₄, C₃xD₈, C₃xSD₁₆, C₃xQ₁₆, C₄oD₁₂, C₃xC₄oD₄, C₃xC₂₄, C₃xDic₆, S₃xC₁₂, C₃xD₁₂, C₃xC₃:D₄, C₆xC₁₂, C₄oD₂₄, C₃xC₄oD₈, C₃xC₂₄:C₂, C₃xD₂₄, C₃xDic₁₂, C₆xC₂₄, C₃xC₄oD₁₂, C₃xC₄oD₂₄
Quotients: C₁, C₂, C₃, C₂², S₃, C₆, D₄, C₂³, D₆, C₂xC₆, C₂xD₄, C₃xS₃, D₁₂, C₃xD₄, C₂²xS₃, C₂²xC₆, C₄oD₈, S₃xC₆, C₂xD₁₂, C₆xD₄, C₃xD₁₂, S₃xC₂xC₆, C₄oD₂₄, C₃xC₄oD₈, C₆xD₁₂, C₃xC₄oD₂₄

Smallest permutation representation of C₃xC₄oD₂₄
►On 48 points

Generators in S₄₈

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

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

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

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

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

90 conjugacy classes

class	1	2A	2B	2C	2D	3A	3B	3C	3D	3E	4A	4B	4C	4D	4E	6A	6B	6C	···	6M	6N	6O	6P	6Q	8A	8B	8C	8D	12A	12B	12C	12D	12E	···	12R	12S	12T	12U	12V	24A	···	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	8	8	8	8	12	12	12	12	12	···	12	12	12	12	12	24	···	24
size	1	1	2	12	12	1	1	2	2	2	1	1	2	12	12	1	1	2	···	2	12	12	12	12	2	2	2	2	1	1	1	1	2	···	2	12	12	12	12	2	···	2

90 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	2	2
type	+	+	+	+	+	+							+	+	+	+	+		+		+
image	C₁	C₂	C₂	C₂	C₂	C₂	C₃	C₆	C₆	C₆	C₆	C₆	S₃	D₄	D₄	D₆	D₆	C₃xS₃	D₁₂	C₃xD₄	D₁₂	C₃xD₄	C₄oD₈	S₃xC₆	S₃xC₆	C₃xD₁₂	C₃xD₁₂	C₄oD₂₄	C₃xC₄oD₈	C₃xC₄oD₂₄
kernel	C₃xC₄oD₂₄	C₃xC₂₄:C₂	C₃xD₂₄	C₃xDic₁₂	C₆xC₂₄	C₃xC₄oD₁₂	C₄oD₂₄	C₂₄:C₂	D₂₄	Dic₁₂	C₂xC₂₄	C₄oD₁₂	C₂xC₂₄	C₃xC₁₂	C₆²	C₂₄	C₂xC₁₂	C₂xC₈	C₁₂	C₁₂	C₂xC₆	C₂xC₆	C₃²	C₈	C₂xC₄	C₄	C₂²	C₃	C₃	C₁
# reps	1	2	1	1	1	2	2	4	2	2	2	4	1	1	1	2	1	2	2	2	2	2	4	4	2	4	4	8	8	16

Matrix representation of C₃xC₄oD₂₄ ►in GL₂(F₇₃) generated by

64	0
0	64

46	0
0	46

43	0
0	17

0	17
43	0

G:=sub<GL(2,GF(73))| [64,0,0,64],[46,0,0,46],[43,0,0,17],[0,43,17,0] >;

C₃xC₄oD₂₄ in GAP, Magma, Sage, TeX

C_3\times C_4\circ D_{24}

% in TeX

G:=Group("C3xC4oD24");

// GroupNames label

G:=SmallGroup(288,675);

// by ID

G=gap.SmallGroup(288,675);

# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,344,590,142,2524,102,9414]);

// Polycyclic

G:=Group<a,b,c,d|a^3=b^4=d^2=1,c^12=b^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=b^2*c^11>;

// generators/relations

G = C3xC4oD24 order 288 = 25·32

Direct product of C3 and C4oD24

G = C₃xC₄oD₂₄ order 288 = 2⁵·3²

Direct product of C₃ and C₄oD₂₄