C3:Dic30 - GroupNames

G = C₃⋊Dic₃₀ order 360 = 2³·3²·5

The semidirect product of C₃ and Dic₃₀ acting via Dic₃₀/Dic₁₅=C₂

Aliases: C₆.₇D₃₀, C₁₅⋊₂Dic₆, C₃⋊₁Dic₃₀, C₃₀.₂₇D₆, Dic₃.D₁₅, C₃²⋊₂Dic₁₀, Dic₁₅.₁S₃, C₁₀.₇S₃², (C₃×C₁₅)⋊₅Q₈, C₃⋊₁(C₁₅⋊Q₈), C₆.₇(S₃×D₅), (C₃×C₆).₇D₁₀, C₂.₇(S₃×D₁₅), C₅⋊₁(C₃²⋊₂Q₈), C₃⋊Dic₁₅.₄C₂, (C₅×Dic₃).₁S₃, (C₃×Dic₃).₁D₅, (C₃×C₃₀).₂₁C₂², (C₃×Dic₁₅).₃C₂, (Dic₃×C₁₅).₁C₂, SmallGroup(360,83)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃×C₃₀ — C₃⋊Dic₃₀

Chief series

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

Lower central

C₃×C₁₅ — C₃×C₃₀ — C₃⋊Dic₃₀

Upper central

C₁ — C₂

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

Subgroups: 316 in 54 conjugacy classes, 24 normal (all characteristic)
C₁, C₂, C₃, C₃, C₄, C₅, C₆, C₆, Q₈, C₃², C₁₀, Dic₃, Dic₃, C₁₂, C₁₅, C₁₅, C₃×C₆, Dic₅, C₂₀, Dic₆, C₃₀, C₃₀, C₃×Dic₃, C₃×Dic₃, C₃⋊Dic₃, Dic₁₀, C₃×C₁₅, C₅×Dic₃, C₃×Dic₅, Dic₁₅, Dic₁₅, C₆₀, C₃²⋊₂Q₈, C₃×C₃₀, C₁₅⋊Q₈, Dic₃₀, Dic₃×C₁₅, C₃×Dic₁₅, C₃⋊Dic₁₅, C₃⋊Dic₃₀
Quotients: C₁, C₂, C₂², S₃, Q₈, D₅, D₆, D₁₀, Dic₆, D₁₅, S₃², Dic₁₀, S₃×D₅, D₃₀, C₃²⋊₂Q₈, C₁₅⋊Q₈, Dic₃₀, S₃×D₁₅, C₃⋊Dic₃₀

Smallest permutation representation of C₃⋊Dic₃₀
►On 120 points

Generators in S₁₂₀

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

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

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

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

51 conjugacy classes

class	1	2	3A	3B	3C	4A	4B	4C	5A	5B	6A	6B	6C	10A	10B	12A	12B	12C	12D	15A	15B	15C	15D	15E	···	15J	20A	20B	20C	20D	30A	30B	30C	30D	30E	···	30J	60A	···	60H
order	1	2	3	3	3	4	4	4	5	5	6	6	6	10	10	12	12	12	12	15	15	15	15	15	···	15	20	20	20	20	30	30	30	30	30	···	30	60	···	60
size	1	1	2	2	4	6	30	90	2	2	2	2	4	2	2	6	6	30	30	2	2	2	2	4	···	4	6	6	6	6	2	2	2	2	4	···	4	6	···	6

51 irreducible representations

dim	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	S₃	S₃	Q₈	D₅	D₆	D₁₀	Dic₆	D₁₅	Dic₁₀	D₃₀	Dic₃₀	S₃²	S₃×D₅	C₃²⋊₂Q₈	C₁₅⋊Q₈	S₃×D₁₅	C₃⋊Dic₃₀
kernel	C₃⋊Dic₃₀	Dic₃×C₁₅	C₃×Dic₁₅	C₃⋊Dic₁₅	C₅×Dic₃	Dic₁₅	C₃×C₁₅	C₃×Dic₃	C₃₀	C₃×C₆	C₁₅	Dic₃	C₃²	C₆	C₃	C₁₀	C₆	C₅	C₃	C₂	C₁
# reps	1	1	1	1	1	1	1	2	2	2	4	4	4	4	8	1	2	1	2	4	4

Matrix representation of C₃⋊Dic₃₀ ►in GL₄(𝔽₆₁) generated by

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

0	1	0	0
1	0	0	0
0	0	36	8
0	0	53	44

60	0	0	0
0	60	0	0
0	0	40	23
0	0	2	21

G:=sub<GL(4,GF(61))| [0,1,0,0,60,60,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,36,53,0,0,8,44],[60,0,0,0,0,60,0,0,0,0,40,2,0,0,23,21] >;

C₃⋊Dic₃₀ in GAP, Magma, Sage, TeX

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

% in TeX

G:=Group("C3:Dic30");

// GroupNames label

G:=SmallGroup(360,83);

// by ID

G=gap.SmallGroup(360,83);

# by ID

G:=PCGroup([6,-2,-2,-2,-3,-3,-5,24,73,31,201,1444,10373]);

// Polycyclic

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

// generators/relations

G = C3⋊Dic30 order 360 = 23·32·5

The semidirect product of C3 and Dic30 acting via Dic30/Dic15=C2

G = C₃⋊Dic₃₀ order 360 = 2³·3²·5

The semidirect product of C₃ and Dic₃₀ acting via Dic₃₀/Dic₁₅=C₂