C3xD60 - GroupNames

G = C₃xD₆₀ order 360 = 2³·3²·5

Direct product of C₃ and D₆₀

direct product, metacyclic, supersoluble, monomial

Aliases: C₃xD₆₀, C₆₀:₁C₆, C₆₀:₃S₃, D₃₀:₁C₆, C₁₅:₅D₁₂, C₁₂:₃D₁₅, C₃²:₄D₂₀, C₆.₂₁D₃₀, C₃₀.₅₃D₆, C₄:(C₃xD₁₅), C₅:₁(C₃xD₁₂), C₂₀:₁(C₃xS₃), (C₃xC₆₀):₂C₂, C₁₅:₄(C₃xD₄), C₁₂:₁(C₃xD₅), C₃:₁(C₃xD₂₀), (C₃xC₁₂):₂D₅, (C₃xC₁₅):₁₈D₄, (C₆xD₁₅):₁C₂, C₂.₄(C₆xD₁₅), C₆.₁₀(C₆xD₅), C₁₀.₁₀(S₃xC₆), C₃₀.₁₀(C₂xC₆), (C₃xC₆).₂₉D₁₀, (C₃xC₃₀).₃₉C₂², SmallGroup(360,102)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃₀ — C₃xD₆₀

Chief series

C₁ — C₅ — C₁₅ — C₃₀ — C₃xC₃₀ — C₆xD₁₅ — C₃xD₆₀

Lower central

C₁₅ — C₃₀ — C₃xD₆₀

Upper central

C₁ — C₆ — C₁₂

Generators and relations for C₃xD₆₀
G = < a,b,c | a³=b⁶⁰=c²=1, ab=ba, ac=ca, cbc=b^-1 >

Subgroups: 372 in 70 conjugacy classes, 30 normal (26 characteristic)
C₁, C₂, C₂, C₃, C₃, C₄, C₂², C₅, S₃, C₆, C₆, D₄, C₃², D₅, C₁₀, C₁₂, C₁₂, D₆, C₂xC₆, C₁₅, C₁₅, C₃xS₃, C₃xC₆, C₂₀, D₁₀, D₁₂, C₃xD₄, C₃xD₅, D₁₅, C₃₀, C₃₀, C₃xC₁₂, S₃xC₆, D₂₀, C₃xC₁₅, C₆₀, C₆₀, C₆xD₅, D₃₀, C₃xD₁₂, C₃xD₁₅, C₃xC₃₀, C₃xD₂₀, D₆₀, C₃xC₆₀, C₆xD₁₅, C₃xD₆₀
Quotients: C₁, C₂, C₃, C₂², S₃, C₆, D₄, D₅, D₆, C₂xC₆, C₃xS₃, D₁₀, D₁₂, C₃xD₄, C₃xD₅, D₁₅, S₃xC₆, D₂₀, C₆xD₅, D₃₀, C₃xD₁₂, C₃xD₁₅, C₃xD₂₀, D₆₀, C₆xD₁₅, C₃xD₆₀

Smallest permutation representation of C₃xD₆₀
►On 120 points

Generators in S₁₂₀

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

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

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

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

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

99 conjugacy classes

class	1	2A	2B	2C	3A	3B	3C	3D	3E	4	5A	5B	6A	6B	6C	6D	6E	6F	6G	6H	6I	10A	10B	12A	···	12H	15A	···	15P	20A	20B	20C	20D	30A	···	30P	60A	···	60AF
order	1	2	2	2	3	3	3	3	3	4	5	5	6	6	6	6	6	6	6	6	6	10	10	12	···	12	15	···	15	20	20	20	20	30	···	30	60	···	60
size	1	1	30	30	1	1	2	2	2	2	2	2	1	1	2	2	2	30	30	30	30	2	2	2	···	2	2	···	2	2	2	2	2	2	···	2	2	···	2

99 irreducible representations

dim	1	1	1	1	1	1	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2
type	+	+	+				+	+	+	+		+	+			+		+		+				+
image	C₁	C₂	C₂	C₃	C₆	C₆	S₃	D₄	D₅	D₆	C₃xS₃	D₁₀	D₁₂	C₃xD₄	C₃xD₅	D₁₅	S₃xC₆	D₂₀	C₆xD₅	D₃₀	C₃xD₁₂	C₃xD₁₅	C₃xD₂₀	D₆₀	C₆xD₁₅	C₃xD₆₀
kernel	C₃xD₆₀	C₃xC₆₀	C₆xD₁₅	D₆₀	C₆₀	D₃₀	C₆₀	C₃xC₁₅	C₃xC₁₂	C₃₀	C₂₀	C₃xC₆	C₁₅	C₁₅	C₁₂	C₁₂	C₁₀	C₃²	C₆	C₆	C₅	C₄	C₃	C₃	C₂	C₁
# reps	1	1	2	2	2	4	1	1	2	1	2	2	2	2	4	4	2	4	4	4	4	8	8	8	8	16

Matrix representation of C₃xD₆₀ ►in GL₂(F₆₁) generated by

47	0
0	47

44	0
0	43

0	43
44	0

G:=sub<GL(2,GF(61))| [47,0,0,47],[44,0,0,43],[0,44,43,0] >;

C₃xD₆₀ in GAP, Magma, Sage, TeX

C_3\times D_{60}

% in TeX

G:=Group("C3xD60");

// GroupNames label

G:=SmallGroup(360,102);

// by ID

G=gap.SmallGroup(360,102);

# by ID

G:=PCGroup([6,-2,-2,-3,-2,-3,-5,169,79,1444,10373]);

// Polycyclic

G:=Group<a,b,c|a^3=b^60=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;

// generators/relations

G = C3xD60 order 360 = 23·32·5

Direct product of C3 and D60

G = C₃xD₆₀ order 360 = 2³·3²·5

Direct product of C₃ and D₆₀