C3^2:3D20 - GroupNames

G = C₃²⋊₃D₂₀ order 360 = 2³·3²·5

2^nd semidirect product of C₃² and D₂₀ acting via D₂₀/C₁₀=C₂²

Aliases: D₃₀⋊₄S₃, C₃²⋊₃D₂₀, C₃₀.₂₀D₆, C₁₀.₁₁S₃², (C₃×C₁₅)⋊₁₄D₄, (C₆×D₁₅)⋊₈C₂, C₃⋊Dic₃⋊₃D₅, C₆.₂₆(S₃×D₅), C₅⋊₁(D₆⋊S₃), C₁₅⋊₆(C₃⋊D₄), C₃⋊₃(C₃⋊D₂₀), (C₃×C₆).₁₁D₁₀, C₂.₄(D₁₅⋊S₃), (C₃×C₃₀).₂₅C₂², (C₅×C₃⋊Dic₃)⋊₅C₂, SmallGroup(360,87)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃×C₃₀ — C₃²⋊₃D₂₀

Chief series

C₁ — C₅ — C₁₅ — C₃×C₁₅ — C₃×C₃₀ — C₆×D₁₅ — C₃²⋊₃D₂₀

Lower central

C₃×C₁₅ — C₃×C₃₀ — C₃²⋊₃D₂₀

Upper central

C₁ — C₂

Generators and relations for C₃²⋊₃D₂₀
G = < a,b,c,d | a³=b³=c²⁰=d²=1, ab=ba, cac^-1=a^-1, ad=da, cbc^-1=dbd=b^-1, dcd=c^-1 >

Subgroups: 452 in 70 conjugacy classes, 23 normal (11 characteristic)
C₁, C₂, C₂, C₃, C₃, C₄, C₂², C₅, S₃, C₆, C₆, D₄, C₃², D₅, C₁₀, Dic₃, D₆, C₂×C₆, C₁₅, C₁₅, C₃×S₃, C₃×C₆, C₂₀, D₁₀, C₃⋊D₄, C₃×D₅, D₁₅, C₃₀, C₃₀, C₃⋊Dic₃, S₃×C₆, D₂₀, C₃×C₁₅, C₅×Dic₃, C₆×D₅, D₃₀, D₆⋊S₃, C₃×D₁₅, C₃×C₃₀, C₃⋊D₂₀, C₅×C₃⋊Dic₃, C₆×D₁₅, C₃²⋊₃D₂₀
Quotients: C₁, C₂, C₂², S₃, D₄, D₅, D₆, D₁₀, C₃⋊D₄, S₃², D₂₀, S₃×D₅, D₆⋊S₃, C₃⋊D₂₀, D₁₅⋊S₃, C₃²⋊₃D₂₀

Smallest permutation representation of C₃²⋊₃D₂₀
►On 120 points

Generators in S₁₂₀

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

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

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

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

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

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

39 conjugacy classes

class	1	2A	2B	2C	3A	3B	3C	4	5A	5B	6A	6B	6C	6D	6E	6F	6G	10A	10B	15A	···	15H	20A	20B	20C	20D	30A	···	30H
order	1	2	2	2	3	3	3	4	5	5	6	6	6	6	6	6	6	10	10	15	···	15	20	20	20	20	30	···	30
size	1	1	30	30	2	2	4	18	2	2	2	2	4	30	30	30	30	2	2	4	···	4	18	18	18	18	4	···	4

39 irreducible representations

dim	1	1	1	2	2	2	2	2	2	2	4	4	4	4	4	4
type	+	+	+	+	+	+	+	+		+	+	+	-	+
image	C₁	C₂	C₂	S₃	D₄	D₅	D₆	D₁₀	C₃⋊D₄	D₂₀	S₃²	S₃×D₅	D₆⋊S₃	C₃⋊D₂₀	D₁₅⋊S₃	C₃²⋊₃D₂₀
kernel	C₃²⋊₃D₂₀	C₅×C₃⋊Dic₃	C₆×D₁₅	D₃₀	C₃×C₁₅	C₃⋊Dic₃	C₃₀	C₃×C₆	C₁₅	C₃²	C₁₀	C₆	C₅	C₃	C₂	C₁
# reps	1	1	2	2	1	2	2	2	4	4	1	4	1	4	4	4

Matrix representation of C₃²⋊₃D₂₀ ►in GL₆(𝔽₆₁)

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

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

27	36	0	0	0	0
25	4	0	0	0	0
0	0	0	60	0	0
0	0	60	0	0	0
0	0	0	0	1	0
0	0	0	0	60	60

52	45	0	0	0	0
5	9	0	0	0	0
0	0	0	1	0	0
0	0	1	0	0	0
0	0	0	0	1	0
0	0	0	0	0	1

G:=sub<GL(6,GF(61))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,1,0,0,0,0,60,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,60,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[27,25,0,0,0,0,36,4,0,0,0,0,0,0,0,60,0,0,0,0,60,0,0,0,0,0,0,0,1,60,0,0,0,0,0,60],[52,5,0,0,0,0,45,9,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

C₃²⋊₃D₂₀ in GAP, Magma, Sage, TeX

C_3^2\rtimes_3D_{20}

% in TeX

G:=Group("C3^2:3D20");

// GroupNames label

G:=SmallGroup(360,87);

// by ID

G=gap.SmallGroup(360,87);

# by ID

G:=PCGroup([6,-2,-2,-2,-3,-3,-5,73,31,387,201,730,10373]);

// Polycyclic

G:=Group<a,b,c,d|a^3=b^3=c^20=d^2=1,a*b=b*a,c*a*c^-1=a^-1,a*d=d*a,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;

// generators/relations

G = C32⋊3D20 order 360 = 23·32·5

2nd semidirect product of C32 and D20 acting via D20/C10=C22

G = C₃²⋊₃D₂₀ order 360 = 2³·3²·5

2^nd semidirect product of C₃² and D₂₀ acting via D₂₀/C₁₀=C₂²