C3xS3xDic5 - GroupNames

G = C₃xS₃xDic₅ order 360 = 2³·3²·5

Direct product of C₃, S₃ and Dic₅

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

Aliases: C₃xS₃xDic₅, C₃₀.₂₉D₆, Dic₁₅:₃C₆, C₅:₄(S₃xC₁₂), D₆.(C₃xD₅), (S₃xC₁₀).C₆, C₆.₂(C₆xD₅), (C₅xS₃):₂C₁₂, (S₃xC₁₅):₄C₄, C₁₅:₁₇(C₄xS₃), C₁₅:₅(C₂xC₁₂), C₁₀.₂(S₃xC₆), C₃₀.₂(C₂xC₆), (S₃xC₆).₂D₅, C₃:₁(C₆xDic₅), C₆.₂₉(S₃xD₅), (S₃xC₃₀).₁C₂, (C₃xDic₅):₁C₆, (C₃xC₆).₁₄D₁₀, C₃²:₄(C₂xDic₅), (C₃xDic₁₅):₃C₂, (C₃xC₃₀).₂C₂², (C₃²xDic₅):₁C₂, C₂.₂(C₃xS₃xD₅), (C₃xC₁₅):₁₄(C₂xC₄), SmallGroup(360,59)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₁₅ — C₃xS₃xDic₅

Chief series

C₁ — C₅ — C₁₅ — C₃₀ — C₃xC₃₀ — C₃²xDic₅ — C₃xS₃xDic₅

Lower central

C₁₅ — C₃xS₃xDic₅

Upper central

C₁ — C₆

Generators and relations for C₃xS₃xDic₅
G = < a,b,c,d,e | a³=b³=c²=d¹⁰=1, e²=d⁵, ab=ba, ac=ca, ad=da, ae=ea, cbc=b^-1, bd=db, be=eb, cd=dc, ce=ec, ede^-1=d^-1 >

Subgroups: 204 in 70 conjugacy classes, 36 normal (28 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, Dic₅, C₂xC₁₀, C₄xS₃, C₂xC₁₂, C₅xS₃, C₃₀, C₃₀, C₃xDic₃, C₃xC₁₂, S₃xC₆, C₂xDic₅, C₃xC₁₅, C₃xDic₅, C₃xDic₅, Dic₁₅, S₃xC₁₀, C₂xC₃₀, S₃xC₁₂, S₃xC₁₅, C₃xC₃₀, S₃xDic₅, C₆xDic₅, C₃²xDic₅, C₃xDic₁₅, S₃xC₃₀, C₃xS₃xDic₅
Quotients: C₁, C₂, C₃, C₄, C₂², S₃, C₆, C₂xC₄, D₅, C₁₂, D₆, C₂xC₆, C₃xS₃, Dic₅, D₁₀, C₄xS₃, C₂xC₁₂, C₃xD₅, S₃xC₆, C₂xDic₅, C₃xDic₅, S₃xD₅, C₆xD₅, S₃xC₁₂, S₃xDic₅, C₆xDic₅, C₃xS₃xD₅, C₃xS₃xDic₅

Smallest permutation representation of C₃xS₃xDic₅
►On 120 points

Generators in S₁₂₀

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

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

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

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

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

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

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

72 conjugacy classes

class	1	2A	2B	2C	3A	3B	3C	3D	3E	4A	4B	4C	4D	5A	5B	6A	6B	6C	6D	6E	6F	6G	6H	6I	10A	10B	10C	10D	10E	10F	12A	12B	12C	12D	12E	···	12J	12K	12L	12M	12N	15A	15B	15C	15D	15E	···	15J	30A	30B	30C	30D	30E	···	30J	30K	···	30R
order	1	2	2	2	3	3	3	3	3	4	4	4	4	5	5	6	6	6	6	6	6	6	6	6	10	10	10	10	10	10	12	12	12	12	12	···	12	12	12	12	12	15	15	15	15	15	···	15	30	30	30	30	30	···	30	30	···	30
size	1	1	3	3	1	1	2	2	2	5	5	15	15	2	2	1	1	2	2	2	3	3	3	3	2	2	6	6	6	6	5	5	5	5	10	···	10	15	15	15	15	2	2	2	2	4	···	4	2	2	2	2	4	···	4	6	···	6

72 irreducible representations

dim	1	1	1	1	1	1	1	1	1	1	2	2	2	2	2	2	2	2	2	2	2	2	4	4	4	4
type	+	+	+	+							+	+	+		-	+							+	-
image	C₁	C₂	C₂	C₂	C₃	C₄	C₆	C₆	C₆	C₁₂	S₃	D₅	D₆	C₃xS₃	Dic₅	D₁₀	C₄xS₃	C₃xD₅	S₃xC₆	C₃xDic₅	C₆xD₅	S₃xC₁₂	S₃xD₅	S₃xDic₅	C₃xS₃xD₅	C₃xS₃xDic₅
kernel	C₃xS₃xDic₅	C₃²xDic₅	C₃xDic₁₅	S₃xC₃₀	S₃xDic₅	S₃xC₁₅	C₃xDic₅	Dic₁₅	S₃xC₁₀	C₅xS₃	C₃xDic₅	S₃xC₆	C₃₀	Dic₅	C₃xS₃	C₃xC₆	C₁₅	D₆	C₁₀	S₃	C₆	C₅	C₆	C₃	C₂	C₁
# reps	1	1	1	1	2	4	2	2	2	8	1	2	1	2	4	2	2	4	2	8	4	4	2	2	4	4

Matrix representation of C₃xS₃xDic₅ ►in GL₄(F₆₁) generated by

1	0	0	0
0	1	0	0
0	0	47	0
0	0	0	47

1	0	0	0
0	1	0	0
0	0	47	0
0	0	13	13

1	0	0	0
0	1	0	0
0	0	1	49
0	0	0	60

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

57	57	0	0
50	4	0	0
0	0	11	0
0	0	0	11

G:=sub<GL(4,GF(61))| [1,0,0,0,0,1,0,0,0,0,47,0,0,0,0,47],[1,0,0,0,0,1,0,0,0,0,47,13,0,0,0,13],[1,0,0,0,0,1,0,0,0,0,1,0,0,0,49,60],[0,1,0,0,60,44,0,0,0,0,60,0,0,0,0,60],[57,50,0,0,57,4,0,0,0,0,11,0,0,0,0,11] >;

C₃xS₃xDic₅ in GAP, Magma, Sage, TeX

C_3\times S_3\times {\rm Dic}_5

% in TeX

G:=Group("C3xS3xDic5");

// GroupNames label

G:=SmallGroup(360,59);

// by ID

G=gap.SmallGroup(360,59);

# by ID

G:=PCGroup([6,-2,-2,-3,-2,-3,-5,72,730,10373]);

// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^3=c^2=d^10=1,e^2=d^5,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^-1>;

// generators/relations

G = C3xS3xDic5 order 360 = 23·32·5

Direct product of C3, S3 and Dic5

G = C₃xS₃xDic₅ order 360 = 2³·3²·5

Direct product of C₃, S₃ and Dic₅