S3xC2xC30 - GroupNames

G = S₃xC₂xC₃₀ order 360 = 2³·3²·5

Direct product of C₂xC₃₀ and S₃

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

Aliases: S₃xC₂xC₃₀, C₆²:₄C₁₀, C₆:(C₂xC₃₀), C₃₀:₄(C₂xC₆), (C₂xC₆):₅C₃₀, C₃:(C₂²xC₃₀), (C₂xC₃₀):₁₁C₆, (C₆xC₃₀):₁₀C₂, C₁₅:₄(C₂²xC₆), (C₃xC₃₀):₉C₂², (C₃xC₁₅):₁₀C₂³, C₃²:₂(C₂²xC₁₀), (C₃xC₆):₂(C₂xC₁₀), SmallGroup(360,158)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃ — S₃xC₂xC₃₀

Chief series

C₁ — C₃ — C₁₅ — C₃xC₁₅ — S₃xC₁₅ — S₃xC₃₀ — S₃xC₂xC₃₀

Lower central

C₃ — S₃xC₂xC₃₀

Upper central

C₁ — C₂xC₃₀

Generators and relations for S₃xC₂xC₃₀
G = < a,b,c,d | a²=b³⁰=c³=d²=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c^-1 >

Subgroups: 236 in 138 conjugacy classes, 84 normal (20 characteristic)
C₁, C₂, C₂, C₃, C₃, C₂², C₂², C₅, S₃, C₆, C₆, C₂³, C₃², C₁₀, C₁₀, D₆, C₂xC₆, C₂xC₆, C₁₅, C₁₅, C₃xS₃, C₃xC₆, C₂xC₁₀, C₂xC₁₀, C₂²xS₃, C₂²xC₆, C₅xS₃, C₃₀, C₃₀, S₃xC₆, C₆², C₂²xC₁₀, C₃xC₁₅, S₃xC₁₀, C₂xC₃₀, C₂xC₃₀, S₃xC₂xC₆, S₃xC₁₅, C₃xC₃₀, S₃xC₂xC₁₀, C₂²xC₃₀, S₃xC₃₀, C₆xC₃₀, S₃xC₂xC₃₀
Quotients: C₁, C₂, C₃, C₂², C₅, S₃, C₆, C₂³, C₁₀, D₆, C₂xC₆, C₁₅, C₃xS₃, C₂xC₁₀, C₂²xS₃, C₂²xC₆, C₅xS₃, C₃₀, S₃xC₆, C₂²xC₁₀, S₃xC₁₀, C₂xC₃₀, S₃xC₂xC₆, S₃xC₁₅, S₃xC₂xC₁₀, C₂²xC₃₀, S₃xC₃₀, S₃xC₂xC₃₀

Smallest permutation representation of S₃xC₂xC₃₀
►On 120 points

Generators in S₁₂₀

(1 52)(2 53)(3 54)(4 55)(5 56)(6 57)(7 58)(8 59)(9 60)(10 31)(11 32)(12 33)(13 34)(14 35)(15 36)(16 37)(17 38)(18 39)(19 40)(20 41)(21 42)(22 43)(23 44)(24 45)(25 46)(26 47)(27 48)(28 49)(29 50)(30 51)(61 98)(62 99)(63 100)(64 101)(65 102)(66 103)(67 104)(68 105)(69 106)(70 107)(71 108)(72 109)(73 110)(74 111)(75 112)(76 113)(77 114)(78 115)(79 116)(80 117)(81 118)(82 119)(83 120)(84 91)(85 92)(86 93)(87 94)(88 95)(89 96)(90 97)

(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 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120)

(1 21 11)(2 22 12)(3 23 13)(4 24 14)(5 25 15)(6 26 16)(7 27 17)(8 28 18)(9 29 19)(10 30 20)(31 51 41)(32 52 42)(33 53 43)(34 54 44)(35 55 45)(36 56 46)(37 57 47)(38 58 48)(39 59 49)(40 60 50)(61 71 81)(62 72 82)(63 73 83)(64 74 84)(65 75 85)(66 76 86)(67 77 87)(68 78 88)(69 79 89)(70 80 90)(91 101 111)(92 102 112)(93 103 113)(94 104 114)(95 105 115)(96 106 116)(97 107 117)(98 108 118)(99 109 119)(100 110 120)

(1 70)(2 71)(3 72)(4 73)(5 74)(6 75)(7 76)(8 77)(9 78)(10 79)(11 80)(12 81)(13 82)(14 83)(15 84)(16 85)(17 86)(18 87)(19 88)(20 89)(21 90)(22 61)(23 62)(24 63)(25 64)(26 65)(27 66)(28 67)(29 68)(30 69)(31 116)(32 117)(33 118)(34 119)(35 120)(36 91)(37 92)(38 93)(39 94)(40 95)(41 96)(42 97)(43 98)(44 99)(45 100)(46 101)(47 102)(48 103)(49 104)(50 105)(51 106)(52 107)(53 108)(54 109)(55 110)(56 111)(57 112)(58 113)(59 114)(60 115)

G:=sub<Sym(120)| (1,52)(2,53)(3,54)(4,55)(5,56)(6,57)(7,58)(8,59)(9,60)(10,31)(11,32)(12,33)(13,34)(14,35)(15,36)(16,37)(17,38)(18,39)(19,40)(20,41)(21,42)(22,43)(23,44)(24,45)(25,46)(26,47)(27,48)(28,49)(29,50)(30,51)(61,98)(62,99)(63,100)(64,101)(65,102)(66,103)(67,104)(68,105)(69,106)(70,107)(71,108)(72,109)(73,110)(74,111)(75,112)(76,113)(77,114)(78,115)(79,116)(80,117)(81,118)(82,119)(83,120)(84,91)(85,92)(86,93)(87,94)(88,95)(89,96)(90,97), (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)(91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120), (1,21,11)(2,22,12)(3,23,13)(4,24,14)(5,25,15)(6,26,16)(7,27,17)(8,28,18)(9,29,19)(10,30,20)(31,51,41)(32,52,42)(33,53,43)(34,54,44)(35,55,45)(36,56,46)(37,57,47)(38,58,48)(39,59,49)(40,60,50)(61,71,81)(62,72,82)(63,73,83)(64,74,84)(65,75,85)(66,76,86)(67,77,87)(68,78,88)(69,79,89)(70,80,90)(91,101,111)(92,102,112)(93,103,113)(94,104,114)(95,105,115)(96,106,116)(97,107,117)(98,108,118)(99,109,119)(100,110,120), (1,70)(2,71)(3,72)(4,73)(5,74)(6,75)(7,76)(8,77)(9,78)(10,79)(11,80)(12,81)(13,82)(14,83)(15,84)(16,85)(17,86)(18,87)(19,88)(20,89)(21,90)(22,61)(23,62)(24,63)(25,64)(26,65)(27,66)(28,67)(29,68)(30,69)(31,116)(32,117)(33,118)(34,119)(35,120)(36,91)(37,92)(38,93)(39,94)(40,95)(41,96)(42,97)(43,98)(44,99)(45,100)(46,101)(47,102)(48,103)(49,104)(50,105)(51,106)(52,107)(53,108)(54,109)(55,110)(56,111)(57,112)(58,113)(59,114)(60,115)>;

G:=Group( (1,52)(2,53)(3,54)(4,55)(5,56)(6,57)(7,58)(8,59)(9,60)(10,31)(11,32)(12,33)(13,34)(14,35)(15,36)(16,37)(17,38)(18,39)(19,40)(20,41)(21,42)(22,43)(23,44)(24,45)(25,46)(26,47)(27,48)(28,49)(29,50)(30,51)(61,98)(62,99)(63,100)(64,101)(65,102)(66,103)(67,104)(68,105)(69,106)(70,107)(71,108)(72,109)(73,110)(74,111)(75,112)(76,113)(77,114)(78,115)(79,116)(80,117)(81,118)(82,119)(83,120)(84,91)(85,92)(86,93)(87,94)(88,95)(89,96)(90,97), (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)(91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120), (1,21,11)(2,22,12)(3,23,13)(4,24,14)(5,25,15)(6,26,16)(7,27,17)(8,28,18)(9,29,19)(10,30,20)(31,51,41)(32,52,42)(33,53,43)(34,54,44)(35,55,45)(36,56,46)(37,57,47)(38,58,48)(39,59,49)(40,60,50)(61,71,81)(62,72,82)(63,73,83)(64,74,84)(65,75,85)(66,76,86)(67,77,87)(68,78,88)(69,79,89)(70,80,90)(91,101,111)(92,102,112)(93,103,113)(94,104,114)(95,105,115)(96,106,116)(97,107,117)(98,108,118)(99,109,119)(100,110,120), (1,70)(2,71)(3,72)(4,73)(5,74)(6,75)(7,76)(8,77)(9,78)(10,79)(11,80)(12,81)(13,82)(14,83)(15,84)(16,85)(17,86)(18,87)(19,88)(20,89)(21,90)(22,61)(23,62)(24,63)(25,64)(26,65)(27,66)(28,67)(29,68)(30,69)(31,116)(32,117)(33,118)(34,119)(35,120)(36,91)(37,92)(38,93)(39,94)(40,95)(41,96)(42,97)(43,98)(44,99)(45,100)(46,101)(47,102)(48,103)(49,104)(50,105)(51,106)(52,107)(53,108)(54,109)(55,110)(56,111)(57,112)(58,113)(59,114)(60,115) );

G=PermutationGroup([[(1,52),(2,53),(3,54),(4,55),(5,56),(6,57),(7,58),(8,59),(9,60),(10,31),(11,32),(12,33),(13,34),(14,35),(15,36),(16,37),(17,38),(18,39),(19,40),(20,41),(21,42),(22,43),(23,44),(24,45),(25,46),(26,47),(27,48),(28,49),(29,50),(30,51),(61,98),(62,99),(63,100),(64,101),(65,102),(66,103),(67,104),(68,105),(69,106),(70,107),(71,108),(72,109),(73,110),(74,111),(75,112),(76,113),(77,114),(78,115),(79,116),(80,117),(81,118),(82,119),(83,120),(84,91),(85,92),(86,93),(87,94),(88,95),(89,96),(90,97)], [(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),(91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120)], [(1,21,11),(2,22,12),(3,23,13),(4,24,14),(5,25,15),(6,26,16),(7,27,17),(8,28,18),(9,29,19),(10,30,20),(31,51,41),(32,52,42),(33,53,43),(34,54,44),(35,55,45),(36,56,46),(37,57,47),(38,58,48),(39,59,49),(40,60,50),(61,71,81),(62,72,82),(63,73,83),(64,74,84),(65,75,85),(66,76,86),(67,77,87),(68,78,88),(69,79,89),(70,80,90),(91,101,111),(92,102,112),(93,103,113),(94,104,114),(95,105,115),(96,106,116),(97,107,117),(98,108,118),(99,109,119),(100,110,120)], [(1,70),(2,71),(3,72),(4,73),(5,74),(6,75),(7,76),(8,77),(9,78),(10,79),(11,80),(12,81),(13,82),(14,83),(15,84),(16,85),(17,86),(18,87),(19,88),(20,89),(21,90),(22,61),(23,62),(24,63),(25,64),(26,65),(27,66),(28,67),(29,68),(30,69),(31,116),(32,117),(33,118),(34,119),(35,120),(36,91),(37,92),(38,93),(39,94),(40,95),(41,96),(42,97),(43,98),(44,99),(45,100),(46,101),(47,102),(48,103),(49,104),(50,105),(51,106),(52,107),(53,108),(54,109),(55,110),(56,111),(57,112),(58,113),(59,114),(60,115)]])

180 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	3A	3B	3C	3D	3E	5A	5B	5C	5D	6A	···	6F	6G	···	6O	6P	···	6W	10A	···	10L	10M	···	10AB	15A	···	15H	15I	···	15T	30A	···	30X	30Y	···	30BH	30BI	···	30CN
order	1	2	2	2	2	2	2	2	3	3	3	3	3	5	5	5	5	6	···	6	6	···	6	6	···	6	10	···	10	10	···	10	15	···	15	15	···	15	30	···	30	30	···	30	30	···	30
size	1	1	1	1	3	3	3	3	1	1	2	2	2	1	1	1	1	1	···	1	2	···	2	3	···	3	1	···	1	3	···	3	1	···	1	2	···	2	1	···	1	2	···	2	3	···	3

180 irreducible representations

dim	1	1	1	1	1	1	1	1	1	1	1	1	2	2	2	2	2	2	2	2
type	+	+	+										+	+
image	C₁	C₂	C₂	C₃	C₅	C₆	C₆	C₁₀	C₁₀	C₁₅	C₃₀	C₃₀	S₃	D₆	C₃xS₃	C₅xS₃	S₃xC₆	S₃xC₁₀	S₃xC₁₅	S₃xC₃₀
kernel	S₃xC₂xC₃₀	S₃xC₃₀	C₆xC₃₀	S₃xC₂xC₁₀	S₃xC₂xC₆	S₃xC₁₀	C₂xC₃₀	S₃xC₆	C₆²	C₂²xS₃	D₆	C₂xC₆	C₂xC₃₀	C₃₀	C₂xC₁₀	C₂xC₆	C₁₀	C₆	C₂²	C₂
# reps	1	6	1	2	4	12	2	24	4	8	48	8	1	3	2	4	6	12	8	24

Matrix representation of S₃xC₂xC₃₀ ►in GL₄(F₃₁) generated by

1	0	0	0
0	30	0	0
0	0	1	0
0	0	0	1

22	0	0	0
0	5	0	0
0	0	28	0
0	0	0	28

1	0	0	0
0	1	0	0
0	0	5	0
0	0	0	25

30	0	0	0
0	30	0	0
0	0	0	1
0	0	1	0

G:=sub<GL(4,GF(31))| [1,0,0,0,0,30,0,0,0,0,1,0,0,0,0,1],[22,0,0,0,0,5,0,0,0,0,28,0,0,0,0,28],[1,0,0,0,0,1,0,0,0,0,5,0,0,0,0,25],[30,0,0,0,0,30,0,0,0,0,0,1,0,0,1,0] >;

S₃xC₂xC₃₀ in GAP, Magma, Sage, TeX

S_3\times C_2\times C_{30}

% in TeX

G:=Group("S3xC2xC30");

// GroupNames label

G:=SmallGroup(360,158);

// by ID

G=gap.SmallGroup(360,158);

# by ID

G:=PCGroup([6,-2,-2,-2,-3,-5,-3,8645]);

// Polycyclic

G:=Group<a,b,c,d|a^2=b^30=c^3=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;

// generators/relations

G = S3xC2xC30 order 360 = 23·32·5

Direct product of C2xC30 and S3

G = S₃xC₂xC₃₀ order 360 = 2³·3²·5

Direct product of C₂xC₃₀ and S₃