S3xDic15 - GroupNames

G = S₃xDic₁₅ order 360 = 2³·3²·5

Direct product of S₃ and Dic₁₅

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

Aliases: S₃xDic₁₅, D₆.D₁₅, C₆.₂D₃₀, C₃₀.₂₂D₆, C₁₀.₂S₃², (S₃xC₆).D₅, (S₃xC₁₀).S₃, (C₃xS₃):Dic₅, C₆.₂(S₃xD₅), C₅:₄(S₃xDic₃), (S₃xC₁₅):₃C₄, C₁₅:₁₆(C₄xS₃), C₃:₃(S₃xDic₅), C₂.₂(S₃xD₁₅), (C₃xC₆).₂D₁₀, (S₃xC₃₀).₂C₂, C₃:Dic₁₅:₆C₂, (C₅xS₃):₂Dic₃, C₁₅:₅(C₂xDic₃), C₃:₁(C₂xDic₁₅), C₃²:₂(C₂xDic₅), (C₃xDic₁₅):₅C₂, (C₃xC₃₀).₁₆C₂², (C₃xC₁₅):₂₀(C₂xC₄), SmallGroup(360,78)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃xC₁₅ — S₃xDic₁₅

Chief series

C₁ — C₅ — C₁₅ — C₃xC₁₅ — C₃xC₃₀ — C₃xDic₁₅ — S₃xDic₁₅

Lower central

C₃xC₁₅ — S₃xDic₁₅

Upper central

C₁ — C₂

Generators and relations for S₃xDic₁₅
G = < a,b,c,d | a³=b²=c³⁰=1, d²=c¹⁵, bab=a^-1, ac=ca, ad=da, bc=cb, bd=db, dcd^-1=c^-1 >

Subgroups: 348 in 70 conjugacy classes, 32 normal (24 characteristic)
C₁, C₂, C₂, C₃, C₃, C₄, C₂², C₅, S₃, C₆, C₆, C₂xC₄, C₃², C₁₀, C₁₀, Dic₃, C₁₂, D₆, C₂xC₆, C₁₅, C₁₅, C₃xS₃, C₃xC₆, Dic₅, C₂xC₁₀, C₄xS₃, C₂xDic₃, C₅xS₃, C₃₀, C₃₀, C₃xDic₃, C₃:Dic₃, S₃xC₆, C₂xDic₅, C₃xC₁₅, C₃xDic₅, Dic₁₅, Dic₁₅, S₃xC₁₀, C₂xC₃₀, S₃xDic₃, S₃xC₁₅, C₃xC₃₀, S₃xDic₅, C₂xDic₁₅, C₃xDic₁₅, C₃:Dic₁₅, S₃xC₃₀, S₃xDic₁₅
Quotients: C₁, C₂, C₄, C₂², S₃, C₂xC₄, D₅, Dic₃, D₆, Dic₅, D₁₀, C₄xS₃, C₂xDic₃, D₁₅, S₃², C₂xDic₅, Dic₁₅, S₃xD₅, D₃₀, S₃xDic₃, S₃xDic₅, C₂xDic₁₅, S₃xD₁₅, S₃xDic₁₅

Smallest permutation representation of S₃xDic₁₅
►On 120 points

Generators in S₁₂₀

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

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

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

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

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

54 conjugacy classes

class	1	2A	2B	2C	3A	3B	3C	4A	4B	4C	4D	5A	5B	6A	6B	6C	6D	6E	10A	10B	10C	10D	10E	10F	12A	12B	15A	15B	15C	15D	15E	···	15J	30A	30B	30C	30D	30E	···	30J	30K	···	30R
order	1	2	2	2	3	3	3	4	4	4	4	5	5	6	6	6	6	6	10	10	10	10	10	10	12	12	15	15	15	15	15	···	15	30	30	30	30	30	···	30	30	···	30
size	1	1	3	3	2	2	4	15	15	45	45	2	2	2	2	4	6	6	2	2	6	6	6	6	30	30	2	2	2	2	4	···	4	2	2	2	2	4	···	4	6	···	6

54 irreducible representations

dim	1	1	1	1	1	2	2	2	2	2	2	2	2	2	2	2	4	4	4	4	4	4
type	+	+	+	+		+	+	+	-	+	-	+		+	-	+	+	+	-	-	+	-
image	C₁	C₂	C₂	C₂	C₄	S₃	S₃	D₅	Dic₃	D₆	Dic₅	D₁₀	C₄xS₃	D₁₅	Dic₁₅	D₃₀	S₃²	S₃xD₅	S₃xDic₃	S₃xDic₅	S₃xD₁₅	S₃xDic₁₅
kernel	S₃xDic₁₅	C₃xDic₁₅	C₃:Dic₁₅	S₃xC₃₀	S₃xC₁₅	Dic₁₅	S₃xC₁₀	S₃xC₆	C₅xS₃	C₃₀	C₃xS₃	C₃xC₆	C₁₅	D₆	S₃	C₆	C₁₀	C₆	C₅	C₃	C₂	C₁
# reps	1	1	1	1	4	1	1	2	2	2	4	2	2	4	8	4	1	2	1	2	4	4

Matrix representation of S₃xDic₁₅ ►in GL₆(F₆₁)

1	0	0	0	0	0
0	1	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	0	1
0	0	0	0	60	60

60	0	0	0	0	0
0	60	0	0	0	0
0	0	60	0	0	0
0	0	0	60	0	0
0	0	0	0	1	0
0	0	0	0	60	60

44	60	0	0	0	0
1	0	0	0	0	0
0	0	60	60	0	0
0	0	1	0	0	0
0	0	0	0	1	0
0	0	0	0	0	1

22	47	0	0	0	0
39	39	0	0	0	0
0	0	60	0	0	0
0	0	1	1	0	0
0	0	0	0	1	0
0	0	0	0	0	1

G:=sub<GL(6,GF(61))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,60,0,0,0,0,1,60],[60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,60,0,0,0,0,0,60],[44,1,0,0,0,0,60,0,0,0,0,0,0,0,60,1,0,0,0,0,60,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[22,39,0,0,0,0,47,39,0,0,0,0,0,0,60,1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

S₃xDic₁₅ in GAP, Magma, Sage, TeX

S_3\times {\rm Dic}_{15}

% in TeX

G:=Group("S3xDic15");

// GroupNames label

G:=SmallGroup(360,78);

// by ID

G=gap.SmallGroup(360,78);

# by ID

G:=PCGroup([6,-2,-2,-2,-3,-3,-5,24,201,1444,10373]);

// Polycyclic

G:=Group<a,b,c,d|a^3=b^2=c^30=1,d^2=c^15,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;

// generators/relations

G = S3xDic15 order 360 = 23·32·5

Direct product of S3 and Dic15

G = S₃xDic₁₅ order 360 = 2³·3²·5

Direct product of S₃ and Dic₁₅