S3xD4xC3^2 - GroupNames

G = S₃xD₄xC₃² order 432 = 2⁴·3³

Direct product of C₃², S₃ and D₄

direct product, metabelian, supersoluble, monomial

Aliases: S₃xD₄xC₃², C₁₂:C₆², D₆:₂C₆², C₆²:₂₈D₆, Dic₃:₁C₆², C₁₂:₅(S₃xC₆), (S₃xC₁₂):₅C₆, D₁₂:₃(C₃xC₆), (C₃xD₁₂):₈C₆, (C₃xC₁₂):₂₁D₆, (C₂xC₆):₂C₆², C₃³:₃₂(C₂xD₄), (S₃xC₆²):₈C₂, C₆²:₁₁(C₂xC₆), (D₄xC₃³):₄C₂, (D₄xC₃²):₉C₆, C₃²:₁₄(C₆xD₄), C₆.₅(C₂xC₆²), C₂.₆(S₃xC₆²), (C₃xC₆²):₇C₂², (C₃²xD₁₂):₁₄C₂, (C₃²xC₁₂):₇C₂², (C₃²xC₆).₇₉C₂³, (C₃²xDic₃):₂₂C₂², C₄:₁(S₃xC₃xC₆), C₃:₂(D₄xC₃xC₆), (S₃xC₂xC₆):₇C₆, (S₃xC₃xC₁₂):₉C₂, (C₂xC₆):₉(S₃xC₆), C₆.₇₇(S₃xC₂xC₆), C₂²:₃(S₃xC₃xC₆), (S₃xC₆):₈(C₂xC₆), (C₄xS₃):₁(C₃xC₆), (C₃xC₁₂):₇(C₂xC₆), (C₃xD₄):₂(C₃xC₆), (C₃xC₃:D₄):₅C₆, C₃:D₄:₁(C₃xC₆), (S₃xC₃xC₆):₂₂C₂², (C₂²xS₃):₃(C₃xC₆), (C₃xDic₃):₈(C₂xC₆), (C₃²xC₃:D₄):₉C₂, (C₃xC₆).₅₃(C₂²xC₆), (C₃xC₆).₁₉₈(C₂²xS₃), SmallGroup(432,704)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₆ — S₃xD₄xC₃²

Chief series

C₁ — C₃ — C₆ — C₃xC₆ — C₃²xC₆ — S₃xC₃xC₆ — S₃xC₆² — S₃xD₄xC₃²

Lower central

C₃ — C₆ — S₃xD₄xC₃²

Upper central

C₁ — C₃xC₆ — D₄xC₃²

Generators and relations for S₃xD₄xC₃²
G = < a,b,c,d,e,f | a³=b³=c³=d²=e⁴=f²=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, dcd=c^-1, ce=ec, cf=fc, de=ed, df=fd, fef=e^-1 >

Subgroups: 880 in 388 conjugacy classes, 150 normal (26 characteristic)
C₁, C₂, C₂, C₃, C₃, C₃, C₄, C₄, C₂², C₂², S₃, S₃, C₆, C₆, C₆, C₂xC₄, D₄, D₄, C₂³, C₃², C₃², C₃², Dic₃, C₁₂, C₁₂, C₁₂, D₆, D₆, D₆, C₂xC₆, C₂xC₆, C₂xD₄, C₃xS₃, C₃xS₃, C₃xC₆, C₃xC₆, C₃xC₆, C₄xS₃, D₁₂, C₃:D₄, C₂xC₁₂, C₃xD₄, C₃xD₄, C₃xD₄, C₂²xS₃, C₂²xC₆, C₃³, C₃xDic₃, C₃xC₁₂, C₃xC₁₂, C₃xC₁₂, S₃xC₆, S₃xC₆, C₆², C₆², S₃xD₄, C₆xD₄, S₃xC₃², S₃xC₃², C₃²xC₆, C₃²xC₆, S₃xC₁₂, C₃xD₁₂, C₃xC₃:D₄, C₆xC₁₂, D₄xC₃², D₄xC₃², D₄xC₃², S₃xC₂xC₆, C₂xC₆², C₃²xDic₃, C₃²xC₁₂, S₃xC₃xC₆, S₃xC₃xC₆, S₃xC₃xC₆, C₃xC₆², C₃xS₃xD₄, D₄xC₃xC₆, S₃xC₃xC₁₂, C₃²xD₁₂, C₃²xC₃:D₄, D₄xC₃³, S₃xC₆², S₃xD₄xC₃²
Quotients: C₁, C₂, C₃, C₂², S₃, C₆, D₄, C₂³, C₃², D₆, C₂xC₆, C₂xD₄, C₃xS₃, C₃xC₆, C₃xD₄, C₂²xS₃, C₂²xC₆, S₃xC₆, C₆², S₃xD₄, C₆xD₄, S₃xC₃², D₄xC₃², S₃xC₂xC₆, C₂xC₆², S₃xC₃xC₆, C₃xS₃xD₄, D₄xC₃xC₆, S₃xC₆², S₃xD₄xC₃²

Smallest permutation representation of S₃xD₄xC₃²
►On 72 points

Generators in S₇₂

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

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

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

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

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

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

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

135 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	3A	···	3H	3I	···	3Q	4A	4B	6A	···	6H	6I	···	6AG	6AH	···	6AW	6AX	···	6BO	6BP	···	6CE	12A	···	12H	12I	···	12Q	12R	···	12Y
order	1	2	2	2	2	2	2	2	3	···	3	3	···	3	4	4	6	···	6	6	···	6	6	···	6	6	···	6	6	···	6	12	···	12	12	···	12	12	···	12
size	1	1	2	2	3	3	6	6	1	···	1	2	···	2	2	6	1	···	1	2	···	2	3	···	3	4	···	4	6	···	6	2	···	2	4	···	4	6	···	6

135 irreducible representations

dim	1	1	1	1	1	1	1	1	1	1	1	1	2	2	2	2	2	2	2	2	4	4
type	+	+	+	+	+	+							+	+	+	+					+
image	C₁	C₂	C₂	C₂	C₂	C₂	C₃	C₆	C₆	C₆	C₆	C₆	S₃	D₄	D₆	D₆	C₃xS₃	C₃xD₄	S₃xC₆	S₃xC₆	S₃xD₄	C₃xS₃xD₄
kernel	S₃xD₄xC₃²	S₃xC₃xC₁₂	C₃²xD₁₂	C₃²xC₃:D₄	D₄xC₃³	S₃xC₆²	C₃xS₃xD₄	S₃xC₁₂	C₃xD₁₂	C₃xC₃:D₄	D₄xC₃²	S₃xC₂xC₆	D₄xC₃²	S₃xC₃²	C₃xC₁₂	C₆²	C₃xD₄	C₃xS₃	C₁₂	C₂xC₆	C₃²	C₃
# reps	1	1	1	2	1	2	8	8	8	16	8	16	1	2	1	2	8	16	8	16	1	8

Matrix representation of S₃xD₄xC₃² ►in GL₆(F₁₃)

3	0	0	0	0	0
0	3	0	0	0	0
0	0	9	0	0	0
0	0	0	9	0	0
0	0	0	0	1	0
0	0	0	0	0	1

9	0	0	0	0	0
0	9	0	0	0	0
0	0	9	0	0	0
0	0	0	9	0	0
0	0	0	0	1	0
0	0	0	0	0	1

9	0	0	0	0	0
0	3	0	0	0	0
0	0	9	4	0	0
0	0	0	3	0	0
0	0	0	0	1	0
0	0	0	0	0	1

0	1	0	0	0	0
1	0	0	0	0	0
0	0	12	0	0	0
0	0	8	1	0	0
0	0	0	0	12	0
0	0	0	0	0	12

1	0	0	0	0	0
0	1	0	0	0	0
0	0	12	0	0	0
0	0	0	12	0	0
0	0	0	0	1	11
0	0	0	0	1	12

1	0	0	0	0	0
0	1	0	0	0	0
0	0	12	0	0	0
0	0	0	12	0	0
0	0	0	0	1	11
0	0	0	0	0	12

G:=sub<GL(6,GF(13))| [3,0,0,0,0,0,0,3,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[9,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[9,0,0,0,0,0,0,3,0,0,0,0,0,0,9,0,0,0,0,0,4,3,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,12,8,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,1,0,0,0,0,11,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,11,12] >;

S₃xD₄xC₃² in GAP, Magma, Sage, TeX

S_3\times D_4\times C_3^2

% in TeX

G:=Group("S3xD4xC3^2");

// GroupNames label

G:=SmallGroup(432,704);

// by ID

G=gap.SmallGroup(432,704);

# by ID

G:=PCGroup([7,-2,-2,-2,-3,-3,-2,-3,807,14118]);

// Polycyclic

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

// generators/relations

G = S3xD4xC32 order 432 = 24·33

Direct product of C32, S3 and D4

G = S₃xD₄xC₃² order 432 = 2⁴·3³

Direct product of C₃², S₃ and D₄