C9xD4:S3 - GroupNames

G = C₉×D₄⋊S₃ order 432 = 2⁴·3³

Direct product of C₉ and D₄⋊S₃

direct product, metabelian, supersoluble, monomial

Aliases: C₉×D₄⋊S₃, D₁₂⋊₂C₁₈, C₃₆.₄₄D₆, D₄⋊(S₃×C₉), (C₃×C₉)⋊₇D₈, C₃⋊C₈⋊₁C₁₈, C₃⋊₂(C₉×D₈), (D₄×C₉)⋊₄S₃, C₆.₇(D₄×C₉), (C₉×D₁₂)⋊₈C₂, (C₃×D₄)⋊₁C₁₈, C₄.₁(S₃×C₁₈), C₁₂.₄₉(S₃×C₆), C₁₂.₁(C₂×C₁₈), (C₃×D₁₂).₁C₆, (C₃×C₁₈).₃₄D₄, C₃².₃(C₃×D₈), (D₄×C₃²).₇C₆, C₁₈.₃₁(C₃⋊D₄), (C₃×C₃₆).₄₃C₂², (C₉×C₃⋊C₈)⋊₈C₂, (D₄×C₃×C₉)⋊₇C₂, (C₃×C₃⋊C₈).₂C₆, (C₃×D₄⋊S₃).C₃, C₃.₄(C₃×D₄⋊S₃), C₂.₄(C₉×C₃⋊D₄), (C₃×C₆).₅₅(C₃×D₄), C₆.₄₅(C₃×C₃⋊D₄), (C₃×C₁₂).₂₇(C₂×C₆), (C₃×D₄).₁₄(C₃×S₃), SmallGroup(432,150)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₁₂ — C₉×D₄⋊S₃

Chief series

C₁ — C₃ — C₆ — C₃×C₆ — C₃×C₁₂ — C₃×C₃₆ — C₉×D₁₂ — C₉×D₄⋊S₃

Lower central

C₃ — C₆ — C₁₂ — C₉×D₄⋊S₃

Upper central

C₁ — C₁₈ — C₃₆ — D₄×C₉

Generators and relations for C₉×D₄⋊S₃
G = < a,b,c,d,e | a⁹=b⁴=c²=d³=e²=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe=b^-1, bd=db, cd=dc, ece=bc, ede=d^-1 >

Subgroups: 208 in 82 conjugacy classes, 33 normal (all characteristic)
C₁, C₂, C₂, C₃, C₃, C₄, C₂², S₃, C₆, C₆, C₈, D₄, D₄, C₉, C₉, C₃², C₁₂, C₁₂, D₆, C₂×C₆, D₈, C₁₈, C₁₈, C₃×S₃, C₃×C₆, C₃×C₆, C₃⋊C₈, C₂₄, D₁₂, C₃×D₄, C₃×D₄, C₃×C₉, C₃₆, C₃₆, C₂×C₁₈, C₃×C₁₂, S₃×C₆, C₆², D₄⋊S₃, C₃×D₈, S₃×C₉, C₃×C₁₈, C₃×C₁₈, C₇₂, D₄×C₉, D₄×C₉, C₃×C₃⋊C₈, C₃×D₁₂, D₄×C₃², C₃×C₃₆, S₃×C₁₈, C₆×C₁₈, C₉×D₈, C₃×D₄⋊S₃, C₉×C₃⋊C₈, C₉×D₁₂, D₄×C₃×C₉, C₉×D₄⋊S₃
Quotients: C₁, C₂, C₃, C₂², S₃, C₆, D₄, C₉, D₆, C₂×C₆, D₈, C₁₈, C₃×S₃, C₃⋊D₄, C₃×D₄, C₂×C₁₈, S₃×C₆, D₄⋊S₃, C₃×D₈, S₃×C₉, D₄×C₉, C₃×C₃⋊D₄, S₃×C₁₈, C₉×D₈, C₃×D₄⋊S₃, C₉×C₃⋊D₄, C₉×D₄⋊S₃

Smallest permutation representation of C₉×D₄⋊S₃
►On 72 points

Generators in S₇₂

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

(1 38)(2 39)(3 40)(4 41)(5 42)(6 43)(7 44)(8 45)(9 37)(10 67)(11 68)(12 69)(13 70)(14 71)(15 72)(16 64)(17 65)(18 66)(28 50)(29 51)(30 52)(31 53)(32 54)(33 46)(34 47)(35 48)(36 49)

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

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

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

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

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

108 conjugacy classes

class	1	2A	2B	2C	3A	3B	3C	3D	3E	4	6A	6B	6C	6D	6E	6F	···	6M	6N	6O	8A	8B	9A	···	9F	9G	···	9L	12A	12B	12C	12D	12E	18A	···	18F	18G	···	18L	18M	···	18AD	18AE	···	18AJ	24A	24B	24C	24D	36A	···	36F	36G	···	36L	72A	···	72L
order	1	2	2	2	3	3	3	3	3	4	6	6	6	6	6	6	···	6	6	6	8	8	9	···	9	9	···	9	12	12	12	12	12	18	···	18	18	···	18	18	···	18	18	···	18	24	24	24	24	36	···	36	36	···	36	72	···	72
size	1	1	4	12	1	1	2	2	2	2	1	1	2	2	2	4	···	4	12	12	6	6	1	···	1	2	···	2	2	2	4	4	4	1	···	1	2	···	2	4	···	4	12	···	12	6	6	6	6	2	···	2	4	···	4	6	···	6

108 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	4	4	4
type	+	+	+	+									+	+	+	+												+
image	C₁	C₂	C₂	C₂	C₃	C₆	C₆	C₆	C₉	C₁₈	C₁₈	C₁₈	S₃	D₄	D₆	D₈	C₃×S₃	C₃⋊D₄	C₃×D₄	S₃×C₆	C₃×D₈	S₃×C₉	D₄×C₉	C₃×C₃⋊D₄	S₃×C₁₈	C₉×D₈	C₉×C₃⋊D₄	D₄⋊S₃	C₃×D₄⋊S₃	C₉×D₄⋊S₃
kernel	C₉×D₄⋊S₃	C₉×C₃⋊C₈	C₉×D₁₂	D₄×C₃×C₉	C₃×D₄⋊S₃	C₃×C₃⋊C₈	C₃×D₁₂	D₄×C₃²	D₄⋊S₃	C₃⋊C₈	D₁₂	C₃×D₄	D₄×C₉	C₃×C₁₈	C₃₆	C₃×C₉	C₃×D₄	C₁₈	C₃×C₆	C₁₂	C₃²	D₄	C₆	C₆	C₄	C₃	C₂	C₉	C₃	C₁
# reps	1	1	1	1	2	2	2	2	6	6	6	6	1	1	1	2	2	2	2	2	4	6	6	4	6	12	12	1	2	6

Matrix representation of C₉×D₄⋊S₃ ►in GL₄(𝔽₇₃) generated by

4	0	0	0
0	4	0	0
0	0	8	0
0	0	0	8

1	0	0	0
0	1	0	0
0	0	0	1
0	0	72	0

1	0	0	0
0	1	0	0
0	0	0	1
0	0	1	0

8	0	0	0
8	64	0	0
0	0	1	0
0	0	0	1

1	7	0	0
0	72	0	0
0	0	16	57
0	0	57	57

G:=sub<GL(4,GF(73))| [4,0,0,0,0,4,0,0,0,0,8,0,0,0,0,8],[1,0,0,0,0,1,0,0,0,0,0,72,0,0,1,0],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0],[8,8,0,0,0,64,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,7,72,0,0,0,0,16,57,0,0,57,57] >;

C₉×D₄⋊S₃ in GAP, Magma, Sage, TeX

C_9\times D_4\rtimes S_3

% in TeX

G:=Group("C9xD4:S3");

// GroupNames label

G:=SmallGroup(432,150);

// by ID

G=gap.SmallGroup(432,150);

# by ID

G:=PCGroup([7,-2,-2,-3,-2,-3,-2,-3,197,142,2355,1186,192,14118]);

// Polycyclic

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

// generators/relations

G = C9×D4⋊S3 order 432 = 24·33

Direct product of C9 and D4⋊S3

G = C₉×D₄⋊S₃ order 432 = 2⁴·3³

Direct product of C₉ and D₄⋊S₃