C3xDic30 - GroupNames

G = C₃×Dic₃₀ order 360 = 2³·3²·5

Direct product of C₃ and Dic₃₀

direct product, metacyclic, supersoluble, monomial

Aliases: C₃×Dic₃₀, C₆₀.₁C₆, C₆₀.₇S₃, C₁₅⋊₇Dic₆, C₆.₁₉D₃₀, C₃₀.₅₁D₆, C₁₂.₇D₁₅, C₃²⋊₆Dic₁₀, Dic₁₅.₁C₆, C₄.(C₃×D₁₅), (C₃×C₁₅)⋊₉Q₈, C₁₅⋊₂(C₃×Q₈), C₆.₈(C₆×D₅), C₅⋊₂(C₃×Dic₆), C₁₀.₈(S₃×C₆), C₂₀.₁(C₃×S₃), C₃₀.₈(C₂×C₆), (C₃×C₆₀).₂C₂, C₂.₃(C₆×D₁₅), (C₃×C₁₂).₂D₅, C₁₂.₁(C₃×D₅), C₃⋊₂(C₃×Dic₁₀), (C₃×C₆).₂₇D₁₀, (C₃×C₃₀).₃₇C₂², (C₃×Dic₁₅).₁C₂, SmallGroup(360,100)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃₀ — C₃×Dic₃₀

Chief series

C₁ — C₅ — C₁₅ — C₃₀ — C₃×C₃₀ — C₃×Dic₁₅ — C₃×Dic₃₀

Lower central

C₁₅ — C₃₀ — C₃×Dic₃₀

Upper central

C₁ — C₆ — C₁₂

Generators and relations for C₃×Dic₃₀
G = < a,b,c | a³=b⁶⁰=1, c²=b³⁰, ab=ba, ac=ca, cbc^-1=b^-1 >

Subgroups: 180 in 54 conjugacy classes, 30 normal (26 characteristic)
C₁, C₂, C₃, C₃, C₄, C₄, C₅, C₆, C₆, Q₈, C₃², C₁₀, Dic₃, C₁₂, C₁₂, C₁₅, C₁₅, C₃×C₆, Dic₅, C₂₀, Dic₆, C₃×Q₈, C₃₀, C₃₀, C₃×Dic₃, C₃×C₁₂, Dic₁₀, C₃×C₁₅, C₃×Dic₅, Dic₁₅, C₆₀, C₆₀, C₃×Dic₆, C₃×C₃₀, C₃×Dic₁₀, Dic₃₀, C₃×Dic₁₅, C₃×C₆₀, C₃×Dic₃₀
Quotients: C₁, C₂, C₃, C₂², S₃, C₆, Q₈, D₅, D₆, C₂×C₆, C₃×S₃, D₁₀, Dic₆, C₃×Q₈, C₃×D₅, D₁₅, S₃×C₆, Dic₁₀, C₆×D₅, D₃₀, C₃×Dic₆, C₃×D₁₅, C₃×Dic₁₀, Dic₃₀, C₆×D₁₅, C₃×Dic₃₀

Smallest permutation representation of C₃×Dic₃₀
►On 120 points

Generators in S₁₂₀

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

(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)

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

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

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

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

99 conjugacy classes

class	1	2	3A	3B	3C	3D	3E	4A	4B	4C	5A	5B	6A	6B	6C	6D	6E	10A	10B	12A	···	12H	12I	12J	12K	12L	15A	···	15P	20A	20B	20C	20D	30A	···	30P	60A	···	60AF
order	1	2	3	3	3	3	3	4	4	4	5	5	6	6	6	6	6	10	10	12	···	12	12	12	12	12	15	···	15	20	20	20	20	30	···	30	60	···	60
size	1	1	1	1	2	2	2	2	30	30	2	2	1	1	2	2	2	2	2	2	···	2	30	30	30	30	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	Q₈	D₅	D₆	C₃×S₃	D₁₀	Dic₆	C₃×Q₈	C₃×D₅	D₁₅	S₃×C₆	Dic₁₀	C₆×D₅	D₃₀	C₃×Dic₆	C₃×D₁₅	C₃×Dic₁₀	Dic₃₀	C₆×D₁₅	C₃×Dic₃₀
kernel	C₃×Dic₃₀	C₃×Dic₁₅	C₃×C₆₀	Dic₃₀	Dic₁₅	C₆₀	C₆₀	C₃×C₁₅	C₃×C₁₂	C₃₀	C₂₀	C₃×C₆	C₁₅	C₁₅	C₁₂	C₁₂	C₁₀	C₃²	C₆	C₆	C₅	C₄	C₃	C₃	C₂	C₁
# reps	1	2	1	2	4	2	1	1	2	1	2	2	2	2	4	4	2	4	4	4	4	8	8	8	8	16

Matrix representation of C₃×Dic₃₀ ►in GL₂(𝔽₆₁) generated by

47	0
0	47

18	0
34	17

11	23
0	50

G:=sub<GL(2,GF(61))| [47,0,0,47],[18,34,0,17],[11,0,23,50] >;

C₃×Dic₃₀ in GAP, Magma, Sage, TeX

C_3\times {\rm Dic}_{30}

% in TeX

G:=Group("C3xDic30");

// GroupNames label

G:=SmallGroup(360,100);

// by ID

G=gap.SmallGroup(360,100);

# by ID

G:=PCGroup([6,-2,-2,-3,-2,-3,-5,72,169,79,1444,10373]);

// Polycyclic

G:=Group<a,b,c|a^3=b^60=1,c^2=b^30,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;

// generators/relations

G = C3×Dic30 order 360 = 23·32·5

Direct product of C3 and Dic30

G = C₃×Dic₃₀ order 360 = 2³·3²·5

Direct product of C₃ and Dic₃₀