C12xD15 - GroupNames

G = C₁₂xD₁₅ order 360 = 2³·3²·5

Direct product of C₁₂ and D₁₅

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

Aliases: C₁₂xD₁₅, C₆₀:₂C₆, C₆₀:₅S₃, D₃₀.₂C₆, C₃₀.₅₂D₆, C₆.₂₀D₃₀, Dic₁₅:₅C₆, C₅:₃(S₃xC₁₂), C₂₀:₂(C₃xS₃), (C₃xC₆₀):₄C₂, C₃:₂(D₅xC₁₂), (C₃xC₁₂):₃D₅, C₁₂:₂(C₃xD₅), C₆.₉(C₆xD₅), C₁₅:₁₄(C₄xS₃), C₃²:₇(C₄xD₅), C₁₅:₇(C₂xC₁₂), C₁₀.₉(S₃xC₆), C₃₀.₉(C₂xC₆), C₂.₁(C₆xD₁₅), (C₆xD₁₅).₄C₂, (C₃xC₆).₂₈D₁₀, (C₃xDic₁₅):₁₁C₂, (C₃xC₃₀).₃₈C₂², (C₃xC₁₅):₂₇(C₂xC₄), SmallGroup(360,101)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₁₅ — C₁₂xD₁₅

Chief series

C₁ — C₅ — C₁₅ — C₃₀ — C₃xC₃₀ — C₆xD₁₅ — C₁₂xD₁₅

Lower central

C₁₅ — C₁₂xD₁₅

Upper central

C₁ — C₁₂

Generators and relations for C₁₂xD₁₅
G = < a,b,c | a¹²=b¹⁵=c²=1, ab=ba, ac=ca, cbc=b^-1 >

Subgroups: 276 in 70 conjugacy classes, 34 normal (30 characteristic)
C₁, C₂, C₂, C₃, C₃, C₄, C₄, C₂², C₅, S₃, C₆, C₆, C₂xC₄, C₃², D₅, C₁₀, Dic₃, C₁₂, C₁₂, D₆, C₂xC₆, C₁₅, C₁₅, C₃xS₃, C₃xC₆, Dic₅, C₂₀, D₁₀, C₄xS₃, C₂xC₁₂, C₃xD₅, D₁₅, C₃₀, C₃₀, C₃xDic₃, C₃xC₁₂, S₃xC₆, C₄xD₅, C₃xC₁₅, C₃xDic₅, Dic₁₅, C₆₀, C₆₀, C₆xD₅, D₃₀, S₃xC₁₂, C₃xD₁₅, C₃xC₃₀, D₅xC₁₂, C₄xD₁₅, C₃xDic₁₅, C₃xC₆₀, C₆xD₁₅, C₁₂xD₁₅
Quotients: C₁, C₂, C₃, C₄, C₂², S₃, C₆, C₂xC₄, D₅, C₁₂, D₆, C₂xC₆, C₃xS₃, D₁₀, C₄xS₃, C₂xC₁₂, C₃xD₅, D₁₅, S₃xC₆, C₄xD₅, C₆xD₅, D₃₀, S₃xC₁₂, C₃xD₁₅, D₅xC₁₂, C₄xD₁₅, C₆xD₁₅, C₁₂xD₁₅

Smallest permutation representation of C₁₂xD₁₅
►On 120 points

Generators in S₁₂₀

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

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

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

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

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

108 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	12A	12B	12C	12D	12E	···	12J	12K	12L	12M	12N	15A	···	15P	20A	20B	20C	20D	30A	···	30P	60A	···	60AF
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	12	12	12	12	12	···	12	12	12	12	12	15	···	15	20	20	20	20	30	···	30	60	···	60
size	1	1	15	15	1	1	2	2	2	1	1	15	15	2	2	1	1	2	2	2	15	15	15	15	2	2	1	1	1	1	2	···	2	15	15	15	15	2	···	2	2	2	2	2	2	···	2	2	···	2

108 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	2	2	2	2	2	2
type	+	+	+	+							+	+	+		+			+				+
image	C₁	C₂	C₂	C₂	C₃	C₄	C₆	C₆	C₆	C₁₂	S₃	D₅	D₆	C₃xS₃	D₁₀	C₄xS₃	C₃xD₅	D₁₅	S₃xC₆	C₄xD₅	C₆xD₅	D₃₀	S₃xC₁₂	C₃xD₁₅	D₅xC₁₂	C₄xD₁₅	C₆xD₁₅	C₁₂xD₁₅
kernel	C₁₂xD₁₅	C₃xDic₁₅	C₃xC₆₀	C₆xD₁₅	C₄xD₁₅	C₃xD₁₅	Dic₁₅	C₆₀	D₃₀	D₁₅	C₆₀	C₃xC₁₂	C₃₀	C₂₀	C₃xC₆	C₁₅	C₁₂	C₁₂	C₁₀	C₃²	C₆	C₆	C₅	C₄	C₃	C₃	C₂	C₁
# reps	1	1	1	1	2	4	2	2	2	8	1	2	1	2	2	2	4	4	2	4	4	4	4	8	8	8	8	16

Matrix representation of C₁₂xD₁₅ ►in GL₄(F₆₁) generated by

40	0	0	0
0	40	0	0
0	0	11	0
0	0	0	11

47	0	0	0
0	13	0	0
0	0	9	0
0	0	6	34

0	48	0	0
14	0	0	0
0	0	5	31
0	0	13	56

G:=sub<GL(4,GF(61))| [40,0,0,0,0,40,0,0,0,0,11,0,0,0,0,11],[47,0,0,0,0,13,0,0,0,0,9,6,0,0,0,34],[0,14,0,0,48,0,0,0,0,0,5,13,0,0,31,56] >;

C₁₂xD₁₅ in GAP, Magma, Sage, TeX

C_{12}\times D_{15}

% in TeX

G:=Group("C12xD15");

// GroupNames label

G:=SmallGroup(360,101);

// by ID

G=gap.SmallGroup(360,101);

# by ID

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

// Polycyclic

G:=Group<a,b,c|a^12=b^15=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;

// generators/relations

G = C12xD15 order 360 = 23·32·5

Direct product of C12 and D15

G = C₁₂xD₁₅ order 360 = 2³·3²·5

Direct product of C₁₂ and D₁₅