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, C₃₆.₄₅D₆, Dic₆⋊₂C₁₈, C₃⋊C₈⋊₂C₁₈, D₄.(S₃×C₉), C₆.₈(D₄×C₉), C₄.₂(S₃×C₁₈), (D₄×C₉).₆S₃, C₃⋊₂(C₉×SD₁₆), (C₃×C₉)⋊₁₀SD₁₆, C₁₂.₅₀(S₃×C₆), C₁₂.₂(C₂×C₁₈), (C₉×Dic₆)⋊₈C₂, (C₃×D₄).₁C₁₈, (C₃×C₁₈).₃₅D₄, (C₃×Dic₆).₁C₆, (D₄×C₃²).₈C₆, C₁₈.₃₂(C₃⋊D₄), (C₃×C₃₆).₄₄C₂², C₃².₃(C₃×SD₁₆), (C₉×C₃⋊C₈)⋊₉C₂, (C₃×C₃⋊C₈).₅C₆, (D₄×C₃×C₉).₃C₂, (C₃×D₄.S₃).C₃, C₂.₅(C₉×C₃⋊D₄), (C₃×C₆).₅₆(C₃×D₄), C₆.₄₆(C₃×C₃⋊D₄), C₃.₄(C₃×D₄.S₃), (C₃×C₁₂).₂₈(C₂×C₆), (C₃×D₄).₁₅(C₃×S₃), SmallGroup(432,151)

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×Dic₆ — 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=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: 160 in 76 conjugacy classes, 33 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, C₃₆, C₃₆, C₂×C₁₈, C₃×Dic₃, C₃×C₁₂, C₆², D₄.S₃, C₃×SD₁₆, C₃×C₁₈, C₃×C₁₈, C₇₂, D₄×C₉, D₄×C₉, Q₈×C₉, C₃×C₃⋊C₈, C₃×Dic₆, D₄×C₃², C₉×Dic₃, C₃×C₃₆, C₆×C₁₈, C₉×SD₁₆, C₃×D₄.S₃, C₉×C₃⋊C₈, C₉×Dic₆, D₄×C₃×C₉, C₉×D₄.S₃
Quotients: C₁, C₂, C₃, C₂², S₃, C₆, D₄, C₉, D₆, C₂×C₆, SD₁₆, C₁₈, C₃×S₃, C₃⋊D₄, C₃×D₄, C₂×C₁₈, S₃×C₆, D₄.S₃, C₃×SD₁₆, S₃×C₉, D₄×C₉, C₃×C₃⋊D₄, S₃×C₁₈, C₉×SD₁₆, 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 47 41 28)(2 48 42 29)(3 49 43 30)(4 50 44 31)(5 51 45 32)(6 52 37 33)(7 53 38 34)(8 54 39 35)(9 46 40 36)(10 62 66 22)(11 63 67 23)(12 55 68 24)(13 56 69 25)(14 57 70 26)(15 58 71 27)(16 59 72 19)(17 60 64 20)(18 61 65 21)

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

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

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

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

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

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

108 conjugacy classes

class	1	2A	2B	3A	3B	3C	3D	3E	4A	4B	6A	6B	6C	6D	6E	6F	···	6M	8A	8B	9A	···	9F	9G	···	9L	12A	12B	12C	12D	12E	12F	12G	18A	···	18F	18G	···	18L	18M	···	18AD	24A	24B	24C	24D	36A	···	36F	36G	···	36L	36M	···	36R	72A	···	72L
order	1	2	2	3	3	3	3	3	4	4	6	6	6	6	6	6	···	6	8	8	9	···	9	9	···	9	12	12	12	12	12	12	12	18	···	18	18	···	18	18	···	18	24	24	24	24	36	···	36	36	···	36	36	···	36	72	···	72
size	1	1	4	1	1	2	2	2	2	12	1	1	2	2	2	4	···	4	6	6	1	···	1	2	···	2	2	2	4	4	4	12	12	1	···	1	2	···	2	4	···	4	6	6	6	6	2	···	2	4	···	4	12	···	12	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	SD₁₆	C₃×S₃	C₃⋊D₄	C₃×D₄	S₃×C₆	C₃×SD₁₆	S₃×C₉	D₄×C₉	C₃×C₃⋊D₄	S₃×C₁₈	C₉×SD₁₆	C₉×C₃⋊D₄	D₄.S₃	C₃×D₄.S₃	C₉×D₄.S₃
kernel	C₉×D₄.S₃	C₉×C₃⋊C₈	C₉×Dic₆	D₄×C₃×C₉	C₃×D₄.S₃	C₃×C₃⋊C₈	C₃×Dic₆	D₄×C₃²	D₄.S₃	C₃⋊C₈	Dic₆	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

16	0	0	0
0	16	0	0
0	0	64	0
0	0	0	64

72	0	0	0
0	72	0	0
0	0	1	72
0	0	2	72

72	0	0	0
0	1	0	0
0	0	72	0
0	0	71	1

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

0	1	0	0
1	0	0	0
0	0	12	67
0	0	12	61

G:=sub<GL(4,GF(73))| [16,0,0,0,0,16,0,0,0,0,64,0,0,0,0,64],[72,0,0,0,0,72,0,0,0,0,1,2,0,0,72,72],[72,0,0,0,0,1,0,0,0,0,72,71,0,0,0,1],[8,0,0,0,0,64,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,12,12,0,0,67,61] >;

C₉×D₄.S₃ in GAP, Magma, Sage, TeX

C_9\times D_4.S_3

% in TeX

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

// GroupNames label

G:=SmallGroup(432,151);

// by ID

G=gap.SmallGroup(432,151);

# by ID

G:=PCGroup([7,-2,-2,-3,-2,-3,-2,-3,504,197,142,2355,1186,192,14118]);

// Polycyclic

G:=Group<a,b,c,d,e|a^9=b^4=c^2=d^3=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 = 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