C3xD4.D9 - GroupNames

G = C₃×D₄.D₉ order 432 = 2⁴·3³

Direct product of C₃ and D₄.D₉

direct product, metabelian, supersoluble, monomial

Aliases: C₃×D₄.D₉, Dic₁₈⋊₈C₆, C₁₂.₄₄D₁₈, C₉⋊C₈⋊₈C₆, D₄.(C₃×D₉), C₄.₁(C₆×D₉), C₁₂.₁(S₃×C₆), (D₄×C₉).₃C₆, (C₃×D₄).₆D₉, C₉⋊₆(C₃×SD₁₆), (C₃×C₉)⋊₁₁SD₁₆, C₃₆.₂₁(C₂×C₆), (C₃×C₁₂).₈₁D₆, (C₃×C₁₈).₄₀D₄, C₁₈.₂₃(C₃×D₄), (C₃×Dic₁₈)⋊₆C₂, C₆.₃₁(C₉⋊D₄), (D₄×C₃²).₇S₃, (C₃×C₃₆).₁₆C₂², C₃².₄(D₄.S₃), (C₃×C₉⋊C₈)⋊₄C₂, (D₄×C₃×C₉).₁C₂, C₂.₄(C₃×C₉⋊D₄), (C₃×D₄).₁(C₃×S₃), C₆.₁₈(C₃×C₃⋊D₄), C₃.₁(C₃×D₄.S₃), (C₃×C₆).₉₂(C₃⋊D₄), SmallGroup(432,148)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃₆ — C₃×D₄.D₉

Chief series

C₁ — C₃ — C₉ — C₁₈ — C₃₆ — C₃×C₃₆ — C₃×Dic₁₈ — C₃×D₄.D₉

Lower central

C₉ — C₁₈ — C₃₆ — C₃×D₄.D₉

Upper central

C₁ — C₆ — C₁₂ — C₃×D₄

Generators and relations for C₃×D₄.D₉
G = < a,b,c,d,e | a³=b⁴=c²=d⁹=1, e²=b², ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe^-1=b^-1, bd=db, cd=dc, ece^-1=bc, ede^-1=d^-1 >

Subgroups: 230 in 76 conjugacy classes, 30 normal (all characteristic)
C₁, C₂, C₂, C₃, C₃, C₄, C₄, C₂², C₆, C₆, C₈, D₄, Q₈, C₉, C₉, C₃², Dic₃, C₁₂, C₁₂, C₂×C₆, SD₁₆, C₁₈, C₁₈, C₃×C₆, C₃×C₆, C₃⋊C₈, C₂₄, Dic₆, C₃×D₄, C₃×D₄, C₃×Q₈, C₃×C₉, Dic₉, C₃₆, C₃₆, C₂×C₁₈, C₃×Dic₃, C₃×C₁₂, C₆², D₄.S₃, C₃×SD₁₆, C₃×C₁₈, C₃×C₁₈, C₉⋊C₈, Dic₁₈, D₄×C₉, D₄×C₉, C₃×C₃⋊C₈, C₃×Dic₆, D₄×C₃², C₃×Dic₉, C₃×C₃₆, C₆×C₁₈, D₄.D₉, C₃×D₄.S₃, C₃×C₉⋊C₈, C₃×Dic₁₈, D₄×C₃×C₉, C₃×D₄.D₉
Quotients: C₁, C₂, C₃, C₂², S₃, C₆, D₄, D₆, C₂×C₆, SD₁₆, D₉, C₃×S₃, C₃⋊D₄, C₃×D₄, D₁₈, S₃×C₆, D₄.S₃, C₃×SD₁₆, C₃×D₉, C₉⋊D₄, C₃×C₃⋊D₄, C₆×D₉, D₄.D₉, C₃×D₄.S₃, C₃×C₉⋊D₄, C₃×D₄.D₉

Smallest permutation representation of C₃×D₄.D₉
►On 72 points

Generators in S₇₂

(1 7 4)(2 8 5)(3 9 6)(10 16 13)(11 17 14)(12 18 15)(19 25 22)(20 26 23)(21 27 24)(28 34 31)(29 35 32)(30 36 33)(37 40 43)(38 41 44)(39 42 45)(46 49 52)(47 50 53)(48 51 54)(55 58 61)(56 59 62)(57 60 63)(64 67 70)(65 68 71)(66 69 72)

(1 28 10 19)(2 29 11 20)(3 30 12 21)(4 31 13 22)(5 32 14 23)(6 33 15 24)(7 34 16 25)(8 35 17 26)(9 36 18 27)(37 55 46 64)(38 56 47 65)(39 57 48 66)(40 58 49 67)(41 59 50 68)(42 60 51 69)(43 61 52 70)(44 62 53 71)(45 63 54 72)

(1 19)(2 20)(3 21)(4 22)(5 23)(6 24)(7 25)(8 26)(9 27)(10 28)(11 29)(12 30)(13 31)(14 32)(15 33)(16 34)(17 35)(18 36)(37 46)(38 47)(39 48)(40 49)(41 50)(42 51)(43 52)(44 53)(45 54)

(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 49 50 51 52 53 54)(55 56 57 58 59 60 61 62 63)(64 65 66 67 68 69 70 71 72)

(1 52 10 43)(2 51 11 42)(3 50 12 41)(4 49 13 40)(5 48 14 39)(6 47 15 38)(7 46 16 37)(8 54 17 45)(9 53 18 44)(19 70 28 61)(20 69 29 60)(21 68 30 59)(22 67 31 58)(23 66 32 57)(24 65 33 56)(25 64 34 55)(26 72 35 63)(27 71 36 62)

G:=sub<Sym(72)| (1,7,4)(2,8,5)(3,9,6)(10,16,13)(11,17,14)(12,18,15)(19,25,22)(20,26,23)(21,27,24)(28,34,31)(29,35,32)(30,36,33)(37,40,43)(38,41,44)(39,42,45)(46,49,52)(47,50,53)(48,51,54)(55,58,61)(56,59,62)(57,60,63)(64,67,70)(65,68,71)(66,69,72), (1,28,10,19)(2,29,11,20)(3,30,12,21)(4,31,13,22)(5,32,14,23)(6,33,15,24)(7,34,16,25)(8,35,17,26)(9,36,18,27)(37,55,46,64)(38,56,47,65)(39,57,48,66)(40,58,49,67)(41,59,50,68)(42,60,51,69)(43,61,52,70)(44,62,53,71)(45,63,54,72), (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,25)(8,26)(9,27)(10,28)(11,29)(12,30)(13,31)(14,32)(15,33)(16,34)(17,35)(18,36)(37,46)(38,47)(39,48)(40,49)(41,50)(42,51)(43,52)(44,53)(45,54), (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)(46,47,48,49,50,51,52,53,54)(55,56,57,58,59,60,61,62,63)(64,65,66,67,68,69,70,71,72), (1,52,10,43)(2,51,11,42)(3,50,12,41)(4,49,13,40)(5,48,14,39)(6,47,15,38)(7,46,16,37)(8,54,17,45)(9,53,18,44)(19,70,28,61)(20,69,29,60)(21,68,30,59)(22,67,31,58)(23,66,32,57)(24,65,33,56)(25,64,34,55)(26,72,35,63)(27,71,36,62)>;

G:=Group( (1,7,4)(2,8,5)(3,9,6)(10,16,13)(11,17,14)(12,18,15)(19,25,22)(20,26,23)(21,27,24)(28,34,31)(29,35,32)(30,36,33)(37,40,43)(38,41,44)(39,42,45)(46,49,52)(47,50,53)(48,51,54)(55,58,61)(56,59,62)(57,60,63)(64,67,70)(65,68,71)(66,69,72), (1,28,10,19)(2,29,11,20)(3,30,12,21)(4,31,13,22)(5,32,14,23)(6,33,15,24)(7,34,16,25)(8,35,17,26)(9,36,18,27)(37,55,46,64)(38,56,47,65)(39,57,48,66)(40,58,49,67)(41,59,50,68)(42,60,51,69)(43,61,52,70)(44,62,53,71)(45,63,54,72), (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,25)(8,26)(9,27)(10,28)(11,29)(12,30)(13,31)(14,32)(15,33)(16,34)(17,35)(18,36)(37,46)(38,47)(39,48)(40,49)(41,50)(42,51)(43,52)(44,53)(45,54), (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)(46,47,48,49,50,51,52,53,54)(55,56,57,58,59,60,61,62,63)(64,65,66,67,68,69,70,71,72), (1,52,10,43)(2,51,11,42)(3,50,12,41)(4,49,13,40)(5,48,14,39)(6,47,15,38)(7,46,16,37)(8,54,17,45)(9,53,18,44)(19,70,28,61)(20,69,29,60)(21,68,30,59)(22,67,31,58)(23,66,32,57)(24,65,33,56)(25,64,34,55)(26,72,35,63)(27,71,36,62) );

G=PermutationGroup([[(1,7,4),(2,8,5),(3,9,6),(10,16,13),(11,17,14),(12,18,15),(19,25,22),(20,26,23),(21,27,24),(28,34,31),(29,35,32),(30,36,33),(37,40,43),(38,41,44),(39,42,45),(46,49,52),(47,50,53),(48,51,54),(55,58,61),(56,59,62),(57,60,63),(64,67,70),(65,68,71),(66,69,72)], [(1,28,10,19),(2,29,11,20),(3,30,12,21),(4,31,13,22),(5,32,14,23),(6,33,15,24),(7,34,16,25),(8,35,17,26),(9,36,18,27),(37,55,46,64),(38,56,47,65),(39,57,48,66),(40,58,49,67),(41,59,50,68),(42,60,51,69),(43,61,52,70),(44,62,53,71),(45,63,54,72)], [(1,19),(2,20),(3,21),(4,22),(5,23),(6,24),(7,25),(8,26),(9,27),(10,28),(11,29),(12,30),(13,31),(14,32),(15,33),(16,34),(17,35),(18,36),(37,46),(38,47),(39,48),(40,49),(41,50),(42,51),(43,52),(44,53),(45,54)], [(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),(46,47,48,49,50,51,52,53,54),(55,56,57,58,59,60,61,62,63),(64,65,66,67,68,69,70,71,72)], [(1,52,10,43),(2,51,11,42),(3,50,12,41),(4,49,13,40),(5,48,14,39),(6,47,15,38),(7,46,16,37),(8,54,17,45),(9,53,18,44),(19,70,28,61),(20,69,29,60),(21,68,30,59),(22,67,31,58),(23,66,32,57),(24,65,33,56),(25,64,34,55),(26,72,35,63),(27,71,36,62)]])

81 conjugacy classes

class	1	2A	2B	3A	3B	3C	3D	3E	4A	4B	6A	6B	6C	6D	6E	6F	···	6M	8A	8B	9A	···	9I	12A	12B	12C	12D	12E	12F	12G	18A	···	18I	18J	···	18AA	24A	24B	24C	24D	36A	···	36I
order	1	2	2	3	3	3	3	3	4	4	6	6	6	6	6	6	···	6	8	8	9	···	9	12	12	12	12	12	12	12	18	···	18	18	···	18	24	24	24	24	36	···	36
size	1	1	4	1	1	2	2	2	2	36	1	1	2	2	2	4	···	4	18	18	2	···	2	2	2	4	4	4	36	36	2	···	2	4	···	4	18	18	18	18	4	···	4

81 irreducible representations

dim	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	4	4
type	+	+	+	+					+	+	+		+				+								-	-
image	C₁	C₂	C₂	C₂	C₃	C₆	C₆	C₆	S₃	D₄	D₆	SD₁₆	D₉	C₃×S₃	C₃×D₄	C₃⋊D₄	D₁₈	S₃×C₆	C₃×SD₁₆	C₃×D₉	C₉⋊D₄	C₃×C₃⋊D₄	C₆×D₉	C₃×C₉⋊D₄	D₄.S₃	D₄.D₉	C₃×D₄.S₃	C₃×D₄.D₉
kernel	C₃×D₄.D₉	C₃×C₉⋊C₈	C₃×Dic₁₈	D₄×C₃×C₉	D₄.D₉	C₉⋊C₈	Dic₁₈	D₄×C₉	D₄×C₃²	C₃×C₁₈	C₃×C₁₂	C₃×C₉	C₃×D₄	C₃×D₄	C₁₈	C₃×C₆	C₁₂	C₁₂	C₉	D₄	C₆	C₆	C₄	C₂	C₃²	C₃	C₃	C₁
# reps	1	1	1	1	2	2	2	2	1	1	1	2	3	2	2	2	3	2	4	6	6	4	6	12	1	3	2	6

Matrix representation of C₃×D₄.D₉ ►in GL₄(𝔽₇₃) generated by

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

72	0	0	0
0	72	0	0
0	0	6	3
0	0	12	67

72	0	0	0
0	1	0	0
0	0	6	3
0	0	37	67

2	0	0	0
0	37	0	0
0	0	1	0
0	0	0	1

0	37	0	0
2	0	0	0
0	0	30	18
0	0	27	43

G:=sub<GL(4,GF(73))| [64,0,0,0,0,64,0,0,0,0,1,0,0,0,0,1],[72,0,0,0,0,72,0,0,0,0,6,12,0,0,3,67],[72,0,0,0,0,1,0,0,0,0,6,37,0,0,3,67],[2,0,0,0,0,37,0,0,0,0,1,0,0,0,0,1],[0,2,0,0,37,0,0,0,0,0,30,27,0,0,18,43] >;

C₃×D₄.D₉ in GAP, Magma, Sage, TeX

C_3\times D_4.D_9

% in TeX

G:=Group("C3xD4.D9");

// GroupNames label

G:=SmallGroup(432,148);

// by ID

G=gap.SmallGroup(432,148);

# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,168,197,1011,514,80,10085,292,14118]);

// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^4=c^2=d^9=1,e^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=e*b*e^-1=b^-1,b*d=d*b,c*d=d*c,e*c*e^-1=b*c,e*d*e^-1=d^-1>;

// generators/relations

G = C3×D4.D9 order 432 = 24·33

Direct product of C3 and D4.D9

G = C₃×D₄.D₉ order 432 = 2⁴·3³

Direct product of C₃ and D₄.D₉