S3xC2^3xC6 - GroupNames

G = S₃×C₂³×C₆ order 288 = 2⁵·3²

Direct product of C₂³×C₆ and S₃

direct product, metabelian, supersoluble, monomial, A-group

Aliases: S₃×C₂³×C₆, C₃²⋊₂C₂⁵, C₆²⋊₁₀C₂³, C₃⋊(C₂⁴×C₆), C₆⋊(C₂³×C₆), (C₃×C₆)⋊₂C₂⁴, (C₂³×C₆)⋊₁₃C₆, (C₂²×C₆²)⋊₇C₂, (C₂×C₆²)⋊₁₇C₂², (C₂×C₆)⋊₆(C₂²×C₆), (C₂²×C₆)⋊₁₀(C₂×C₆), SmallGroup(288,1043)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃ — S₃×C₂³×C₆

Chief series

C₁ — C₃ — C₃² — C₃×S₃ — S₃×C₆ — S₃×C₂×C₆ — S₃×C₂²×C₆ — S₃×C₂³×C₆

Lower central

C₃ — S₃×C₂³×C₆

Upper central

C₁ — C₂³×C₆

Generators and relations for S₃×C₂³×C₆
G = < a,b,c,d,e,f | a²=b²=c²=d⁶=e³=f²=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef=e^-1 >

Subgroups: 2858 in 1563 conjugacy classes, 882 normal (10 characteristic)
C₁, C₂, C₂, C₃, C₃, C₂², C₂², S₃, C₆, C₆, C₂³, C₂³, C₃², D₆, C₂×C₆, C₂×C₆, C₂⁴, C₂⁴, C₃×S₃, C₃×C₆, C₂²×S₃, C₂²×C₆, C₂²×C₆, C₂⁵, S₃×C₆, C₆², S₃×C₂³, C₂³×C₆, C₂³×C₆, S₃×C₂×C₆, C₂×C₆², S₃×C₂⁴, C₂⁴×C₆, S₃×C₂²×C₆, C₂²×C₆², S₃×C₂³×C₆
Quotients: C₁, C₂, C₃, C₂², S₃, C₆, C₂³, D₆, C₂×C₆, C₂⁴, C₃×S₃, C₂²×S₃, C₂²×C₆, C₂⁵, S₃×C₆, S₃×C₂³, C₂³×C₆, S₃×C₂×C₆, S₃×C₂⁴, C₂⁴×C₆, S₃×C₂²×C₆, S₃×C₂³×C₆

Smallest permutation representation of S₃×C₂³×C₆
►On 96 points

Generators in S₉₆

(1 47)(2 48)(3 43)(4 44)(5 45)(6 46)(7 37)(8 38)(9 39)(10 40)(11 41)(12 42)(13 31)(14 32)(15 33)(16 34)(17 35)(18 36)(19 25)(20 26)(21 27)(22 28)(23 29)(24 30)(49 91)(50 92)(51 93)(52 94)(53 95)(54 96)(55 85)(56 86)(57 87)(58 88)(59 89)(60 90)(61 79)(62 80)(63 81)(64 82)(65 83)(66 84)(67 73)(68 74)(69 75)(70 76)(71 77)(72 78)

(1 23)(2 24)(3 19)(4 20)(5 21)(6 22)(7 13)(8 14)(9 15)(10 16)(11 17)(12 18)(25 43)(26 44)(27 45)(28 46)(29 47)(30 48)(31 37)(32 38)(33 39)(34 40)(35 41)(36 42)(49 67)(50 68)(51 69)(52 70)(53 71)(54 72)(55 61)(56 62)(57 63)(58 64)(59 65)(60 66)(73 91)(74 92)(75 93)(76 94)(77 95)(78 96)(79 85)(80 86)(81 87)(82 88)(83 89)(84 90)

(1 8)(2 9)(3 10)(4 11)(5 12)(6 7)(13 22)(14 23)(15 24)(16 19)(17 20)(18 21)(25 34)(26 35)(27 36)(28 31)(29 32)(30 33)(37 46)(38 47)(39 48)(40 43)(41 44)(42 45)(49 58)(50 59)(51 60)(52 55)(53 56)(54 57)(61 70)(62 71)(63 72)(64 67)(65 68)(66 69)(73 82)(74 83)(75 84)(76 79)(77 80)(78 81)(85 94)(86 95)(87 96)(88 91)(89 92)(90 93)

(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)(73 74 75 76 77 78)(79 80 81 82 83 84)(85 86 87 88 89 90)(91 92 93 94 95 96)

(1 3 5)(2 4 6)(7 9 11)(8 10 12)(13 15 17)(14 16 18)(19 21 23)(20 22 24)(25 27 29)(26 28 30)(31 33 35)(32 34 36)(37 39 41)(38 40 42)(43 45 47)(44 46 48)(49 53 51)(50 54 52)(55 59 57)(56 60 58)(61 65 63)(62 66 64)(67 71 69)(68 72 70)(73 77 75)(74 78 76)(79 83 81)(80 84 82)(85 89 87)(86 90 88)(91 95 93)(92 96 94)

(1 53)(2 54)(3 49)(4 50)(5 51)(6 52)(7 55)(8 56)(9 57)(10 58)(11 59)(12 60)(13 61)(14 62)(15 63)(16 64)(17 65)(18 66)(19 67)(20 68)(21 69)(22 70)(23 71)(24 72)(25 73)(26 74)(27 75)(28 76)(29 77)(30 78)(31 79)(32 80)(33 81)(34 82)(35 83)(36 84)(37 85)(38 86)(39 87)(40 88)(41 89)(42 90)(43 91)(44 92)(45 93)(46 94)(47 95)(48 96)

G:=sub<Sym(96)| (1,47)(2,48)(3,43)(4,44)(5,45)(6,46)(7,37)(8,38)(9,39)(10,40)(11,41)(12,42)(13,31)(14,32)(15,33)(16,34)(17,35)(18,36)(19,25)(20,26)(21,27)(22,28)(23,29)(24,30)(49,91)(50,92)(51,93)(52,94)(53,95)(54,96)(55,85)(56,86)(57,87)(58,88)(59,89)(60,90)(61,79)(62,80)(63,81)(64,82)(65,83)(66,84)(67,73)(68,74)(69,75)(70,76)(71,77)(72,78), (1,23)(2,24)(3,19)(4,20)(5,21)(6,22)(7,13)(8,14)(9,15)(10,16)(11,17)(12,18)(25,43)(26,44)(27,45)(28,46)(29,47)(30,48)(31,37)(32,38)(33,39)(34,40)(35,41)(36,42)(49,67)(50,68)(51,69)(52,70)(53,71)(54,72)(55,61)(56,62)(57,63)(58,64)(59,65)(60,66)(73,91)(74,92)(75,93)(76,94)(77,95)(78,96)(79,85)(80,86)(81,87)(82,88)(83,89)(84,90), (1,8)(2,9)(3,10)(4,11)(5,12)(6,7)(13,22)(14,23)(15,24)(16,19)(17,20)(18,21)(25,34)(26,35)(27,36)(28,31)(29,32)(30,33)(37,46)(38,47)(39,48)(40,43)(41,44)(42,45)(49,58)(50,59)(51,60)(52,55)(53,56)(54,57)(61,70)(62,71)(63,72)(64,67)(65,68)(66,69)(73,82)(74,83)(75,84)(76,79)(77,80)(78,81)(85,94)(86,95)(87,96)(88,91)(89,92)(90,93), (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)(73,74,75,76,77,78)(79,80,81,82,83,84)(85,86,87,88,89,90)(91,92,93,94,95,96), (1,3,5)(2,4,6)(7,9,11)(8,10,12)(13,15,17)(14,16,18)(19,21,23)(20,22,24)(25,27,29)(26,28,30)(31,33,35)(32,34,36)(37,39,41)(38,40,42)(43,45,47)(44,46,48)(49,53,51)(50,54,52)(55,59,57)(56,60,58)(61,65,63)(62,66,64)(67,71,69)(68,72,70)(73,77,75)(74,78,76)(79,83,81)(80,84,82)(85,89,87)(86,90,88)(91,95,93)(92,96,94), (1,53)(2,54)(3,49)(4,50)(5,51)(6,52)(7,55)(8,56)(9,57)(10,58)(11,59)(12,60)(13,61)(14,62)(15,63)(16,64)(17,65)(18,66)(19,67)(20,68)(21,69)(22,70)(23,71)(24,72)(25,73)(26,74)(27,75)(28,76)(29,77)(30,78)(31,79)(32,80)(33,81)(34,82)(35,83)(36,84)(37,85)(38,86)(39,87)(40,88)(41,89)(42,90)(43,91)(44,92)(45,93)(46,94)(47,95)(48,96)>;

G:=Group( (1,47)(2,48)(3,43)(4,44)(5,45)(6,46)(7,37)(8,38)(9,39)(10,40)(11,41)(12,42)(13,31)(14,32)(15,33)(16,34)(17,35)(18,36)(19,25)(20,26)(21,27)(22,28)(23,29)(24,30)(49,91)(50,92)(51,93)(52,94)(53,95)(54,96)(55,85)(56,86)(57,87)(58,88)(59,89)(60,90)(61,79)(62,80)(63,81)(64,82)(65,83)(66,84)(67,73)(68,74)(69,75)(70,76)(71,77)(72,78), (1,23)(2,24)(3,19)(4,20)(5,21)(6,22)(7,13)(8,14)(9,15)(10,16)(11,17)(12,18)(25,43)(26,44)(27,45)(28,46)(29,47)(30,48)(31,37)(32,38)(33,39)(34,40)(35,41)(36,42)(49,67)(50,68)(51,69)(52,70)(53,71)(54,72)(55,61)(56,62)(57,63)(58,64)(59,65)(60,66)(73,91)(74,92)(75,93)(76,94)(77,95)(78,96)(79,85)(80,86)(81,87)(82,88)(83,89)(84,90), (1,8)(2,9)(3,10)(4,11)(5,12)(6,7)(13,22)(14,23)(15,24)(16,19)(17,20)(18,21)(25,34)(26,35)(27,36)(28,31)(29,32)(30,33)(37,46)(38,47)(39,48)(40,43)(41,44)(42,45)(49,58)(50,59)(51,60)(52,55)(53,56)(54,57)(61,70)(62,71)(63,72)(64,67)(65,68)(66,69)(73,82)(74,83)(75,84)(76,79)(77,80)(78,81)(85,94)(86,95)(87,96)(88,91)(89,92)(90,93), (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)(73,74,75,76,77,78)(79,80,81,82,83,84)(85,86,87,88,89,90)(91,92,93,94,95,96), (1,3,5)(2,4,6)(7,9,11)(8,10,12)(13,15,17)(14,16,18)(19,21,23)(20,22,24)(25,27,29)(26,28,30)(31,33,35)(32,34,36)(37,39,41)(38,40,42)(43,45,47)(44,46,48)(49,53,51)(50,54,52)(55,59,57)(56,60,58)(61,65,63)(62,66,64)(67,71,69)(68,72,70)(73,77,75)(74,78,76)(79,83,81)(80,84,82)(85,89,87)(86,90,88)(91,95,93)(92,96,94), (1,53)(2,54)(3,49)(4,50)(5,51)(6,52)(7,55)(8,56)(9,57)(10,58)(11,59)(12,60)(13,61)(14,62)(15,63)(16,64)(17,65)(18,66)(19,67)(20,68)(21,69)(22,70)(23,71)(24,72)(25,73)(26,74)(27,75)(28,76)(29,77)(30,78)(31,79)(32,80)(33,81)(34,82)(35,83)(36,84)(37,85)(38,86)(39,87)(40,88)(41,89)(42,90)(43,91)(44,92)(45,93)(46,94)(47,95)(48,96) );

G=PermutationGroup([[(1,47),(2,48),(3,43),(4,44),(5,45),(6,46),(7,37),(8,38),(9,39),(10,40),(11,41),(12,42),(13,31),(14,32),(15,33),(16,34),(17,35),(18,36),(19,25),(20,26),(21,27),(22,28),(23,29),(24,30),(49,91),(50,92),(51,93),(52,94),(53,95),(54,96),(55,85),(56,86),(57,87),(58,88),(59,89),(60,90),(61,79),(62,80),(63,81),(64,82),(65,83),(66,84),(67,73),(68,74),(69,75),(70,76),(71,77),(72,78)], [(1,23),(2,24),(3,19),(4,20),(5,21),(6,22),(7,13),(8,14),(9,15),(10,16),(11,17),(12,18),(25,43),(26,44),(27,45),(28,46),(29,47),(30,48),(31,37),(32,38),(33,39),(34,40),(35,41),(36,42),(49,67),(50,68),(51,69),(52,70),(53,71),(54,72),(55,61),(56,62),(57,63),(58,64),(59,65),(60,66),(73,91),(74,92),(75,93),(76,94),(77,95),(78,96),(79,85),(80,86),(81,87),(82,88),(83,89),(84,90)], [(1,8),(2,9),(3,10),(4,11),(5,12),(6,7),(13,22),(14,23),(15,24),(16,19),(17,20),(18,21),(25,34),(26,35),(27,36),(28,31),(29,32),(30,33),(37,46),(38,47),(39,48),(40,43),(41,44),(42,45),(49,58),(50,59),(51,60),(52,55),(53,56),(54,57),(61,70),(62,71),(63,72),(64,67),(65,68),(66,69),(73,82),(74,83),(75,84),(76,79),(77,80),(78,81),(85,94),(86,95),(87,96),(88,91),(89,92),(90,93)], [(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),(73,74,75,76,77,78),(79,80,81,82,83,84),(85,86,87,88,89,90),(91,92,93,94,95,96)], [(1,3,5),(2,4,6),(7,9,11),(8,10,12),(13,15,17),(14,16,18),(19,21,23),(20,22,24),(25,27,29),(26,28,30),(31,33,35),(32,34,36),(37,39,41),(38,40,42),(43,45,47),(44,46,48),(49,53,51),(50,54,52),(55,59,57),(56,60,58),(61,65,63),(62,66,64),(67,71,69),(68,72,70),(73,77,75),(74,78,76),(79,83,81),(80,84,82),(85,89,87),(86,90,88),(91,95,93),(92,96,94)], [(1,53),(2,54),(3,49),(4,50),(5,51),(6,52),(7,55),(8,56),(9,57),(10,58),(11,59),(12,60),(13,61),(14,62),(15,63),(16,64),(17,65),(18,66),(19,67),(20,68),(21,69),(22,70),(23,71),(24,72),(25,73),(26,74),(27,75),(28,76),(29,77),(30,78),(31,79),(32,80),(33,81),(34,82),(35,83),(36,84),(37,85),(38,86),(39,87),(40,88),(41,89),(42,90),(43,91),(44,92),(45,93),(46,94),(47,95),(48,96)]])

144 conjugacy classes

class	1	2A	···	2O	2P	···	2AE	3A	3B	3C	3D	3E	6A	···	6AD	6AE	···	6BW	6BX	···	6DC
order	1	2	···	2	2	···	2	3	3	3	3	3	6	···	6	6	···	6	6	···	6
size	1	1	···	1	3	···	3	1	1	2	2	2	1	···	1	2	···	2	3	···	3

144 irreducible representations

dim	1	1	1	1	1	1	2	2	2	2
type	+	+	+				+	+
image	C₁	C₂	C₂	C₃	C₆	C₆	S₃	D₆	C₃×S₃	S₃×C₆
kernel	S₃×C₂³×C₆	S₃×C₂²×C₆	C₂²×C₆²	S₃×C₂⁴	S₃×C₂³	C₂³×C₆	C₂³×C₆	C₂²×C₆	C₂⁴	C₂³
# reps	1	30	1	2	60	2	1	15	2	30

Matrix representation of S₃×C₂³×C₆ ►in GL₅(𝔽₇)

1	0	0	0	0
0	6	0	0	0
0	0	6	0	0
0	0	0	1	0
0	0	0	0	1

1	0	0	0	0
0	6	0	0	0
0	0	1	0	0
0	0	0	6	0
0	0	0	0	6

6	0	0	0	0
0	1	0	0	0
0	0	1	0	0
0	0	0	6	0
0	0	0	0	6

4	0	0	0	0
0	2	0	0	0
0	0	4	0	0
0	0	0	5	0
0	0	0	0	5

1	0	0	0	0
0	1	0	0	0
0	0	1	0	0
0	0	0	4	0
0	0	0	0	2

1	0	0	0	0
0	1	0	0	0
0	0	6	0	0
0	0	0	0	1
0	0	0	1	0

G:=sub<GL(5,GF(7))| [1,0,0,0,0,0,6,0,0,0,0,0,6,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,6,0,0,0,0,0,1,0,0,0,0,0,6,0,0,0,0,0,6],[6,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,6,0,0,0,0,0,6],[4,0,0,0,0,0,2,0,0,0,0,0,4,0,0,0,0,0,5,0,0,0,0,0,5],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,4,0,0,0,0,0,2],[1,0,0,0,0,0,1,0,0,0,0,0,6,0,0,0,0,0,0,1,0,0,0,1,0] >;

S₃×C₂³×C₆ in GAP, Magma, Sage, TeX

S_3\times C_2^3\times C_6

% in TeX

G:=Group("S3xC2^3xC6");

// GroupNames label

G:=SmallGroup(288,1043);

// by ID

G=gap.SmallGroup(288,1043);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,9414]);

// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^6=e^3=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,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f=e^-1>;

// generators/relations

G = S3×C23×C6 order 288 = 25·32

Direct product of C23×C6 and S3

G = S₃×C₂³×C₆ order 288 = 2⁵·3²

Direct product of C₂³×C₆ and S₃