C2xA4xDic5 - GroupNames

G = C₂xA₄xDic₅ order 480 = 2⁵·3·5

Direct product of C₂, A₄ and Dic₅

direct product, metabelian, soluble, monomial, A-group

Aliases: C₂xA₄xDic₅, C₁₀:₂(C₄xA₄), (C₁₀xA₄):₄C₄, C₂⁴.(C₃xD₅), (C₂³xC₁₀).C₆, C₂²:(C₆xDic₅), (C₂³xDic₅):C₃, C₂².₉(D₅xA₄), (C₂²xC₁₀):₂C₁₂, (C₂xA₄).₁₆D₁₀, (C₂²xA₄).₂D₅, C₂³:₂(C₃xDic₅), C₂³.₁₃(C₆xD₅), C₁₀.₁₀(C₂²xA₄), (C₂²xDic₅):₃C₆, (C₁₀xA₄).₁₆C₂², C₅:₃(C₂xC₄xA₄), C₂.₂(C₂xD₅xA₄), (A₄xC₂xC₁₀).₂C₂, (C₅xA₄):₁₁(C₂xC₄), (C₂xC₁₀):₄(C₂xC₁₂), (C₂xC₁₀).₁₄(C₂xA₄), (C₂²xC₁₀).₄(C₂xC₆), SmallGroup(480,1044)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂xC₁₀ — C₂xA₄xDic₅

Chief series

C₁ — C₅ — C₂xC₁₀ — C₂²xC₁₀ — C₁₀xA₄ — A₄xDic₅ — C₂xA₄xDic₅

Lower central

C₂xC₁₀ — C₂xA₄xDic₅

Upper central

C₁ — C₂²

Generators and relations for C₂xA₄xDic₅
G = < a,b,c,d,e,f | a²=b²=c²=d³=e¹⁰=1, f²=e⁵, ab=ba, ac=ca, ad=da, ae=ea, af=fa, dbd^-1=bc=cb, be=eb, bf=fb, dcd^-1=b, ce=ec, cf=fc, de=ed, df=fd, fef^-1=e^-1 >

Subgroups: 580 in 132 conjugacy classes, 39 normal (21 characteristic)
C₁, C₂, C₂, C₂, C₃, C₄, C₂², C₂², C₅, C₆, C₂xC₄, C₂³, C₂³, C₂³, C₁₀, C₁₀, C₁₀, C₁₂, A₄, C₂xC₆, C₁₅, C₂²xC₄, C₂⁴, Dic₅, Dic₅, C₂xC₁₀, C₂xC₁₀, C₂xC₁₂, C₂xA₄, C₂xA₄, C₃₀, C₂³xC₄, C₂xDic₅, C₂xDic₅, C₂²xC₁₀, C₂²xC₁₀, C₂²xC₁₀, C₄xA₄, C₂²xA₄, C₃xDic₅, C₅xA₄, C₂xC₃₀, C₂²xDic₅, C₂²xDic₅, C₂³xC₁₀, C₂xC₄xA₄, C₆xDic₅, C₁₀xA₄, C₁₀xA₄, C₂³xDic₅, A₄xDic₅, A₄xC₂xC₁₀, C₂xA₄xDic₅
Quotients: C₁, C₂, C₃, C₄, C₂², C₆, C₂xC₄, D₅, C₁₂, A₄, C₂xC₆, Dic₅, D₁₀, C₂xC₁₂, C₂xA₄, C₃xD₅, C₂xDic₅, C₄xA₄, C₂²xA₄, C₃xDic₅, C₆xD₅, C₂xC₄xA₄, C₆xDic₅, D₅xA₄, A₄xDic₅, C₂xD₅xA₄, C₂xA₄xDic₅

Smallest permutation representation of C₂xA₄xDic₅
►On 120 points

Generators in S₁₂₀

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

(1 28)(2 29)(3 30)(4 21)(5 22)(6 23)(7 24)(8 25)(9 26)(10 27)(31 43)(32 44)(33 45)(34 46)(35 47)(36 48)(37 49)(38 50)(39 41)(40 42)(71 88)(72 89)(73 90)(74 81)(75 82)(76 83)(77 84)(78 85)(79 86)(80 87)(91 108)(92 109)(93 110)(94 101)(95 102)(96 103)(97 104)(98 105)(99 106)(100 107)

(1 28)(2 29)(3 30)(4 21)(5 22)(6 23)(7 24)(8 25)(9 26)(10 27)(11 115)(12 116)(13 117)(14 118)(15 119)(16 120)(17 111)(18 112)(19 113)(20 114)(51 70)(52 61)(53 62)(54 63)(55 64)(56 65)(57 66)(58 67)(59 68)(60 69)(71 88)(72 89)(73 90)(74 81)(75 82)(76 83)(77 84)(78 85)(79 86)(80 87)

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

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

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

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

64 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	3A	3B	4A	4B	4C	4D	4E	4F	4G	4H	5A	5B	6A	···	6F	10A	···	10F	10G	···	10N	12A	···	12H	15A	15B	15C	15D	30A	···	30L
order	1	2	2	2	2	2	2	2	3	3	4	4	4	4	4	4	4	4	5	5	6	···	6	10	···	10	10	···	10	12	···	12	15	15	15	15	30	···	30
size	1	1	1	1	3	3	3	3	4	4	5	5	5	5	15	15	15	15	2	2	4	···	4	2	···	2	6	···	6	20	···	20	8	8	8	8	8	···	8

64 irreducible representations

dim	1	1	1	1	1	1	1	1	2	2	2	2	2	2	3	3	3	3	6	6	6
type	+	+	+						+	-	+				+	+	+		+	-	+
image	C₁	C₂	C₂	C₃	C₄	C₆	C₆	C₁₂	D₅	Dic₅	D₁₀	C₃xD₅	C₃xDic₅	C₆xD₅	A₄	C₂xA₄	C₂xA₄	C₄xA₄	D₅xA₄	A₄xDic₅	C₂xD₅xA₄
kernel	C₂xA₄xDic₅	A₄xDic₅	A₄xC₂xC₁₀	C₂³xDic₅	C₁₀xA₄	C₂²xDic₅	C₂³xC₁₀	C₂²xC₁₀	C₂²xA₄	C₂xA₄	C₂xA₄	C₂⁴	C₂³	C₂³	C₂xDic₅	Dic₅	C₂xC₁₀	C₁₀	C₂²	C₂	C₂
# reps	1	2	1	2	4	4	2	8	2	4	2	4	8	4	1	2	1	4	2	4	2

Matrix representation of C₂xA₄xDic₅ ►in GL₇(F₆₁)

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

1	0	0	0	0	0	0
0	1	0	0	0	0	0
0	0	1	0	0	0	0
0	0	0	1	0	0	0
0	0	0	0	1	0	0
0	0	0	0	0	60	0
0	0	0	0	48	0	60

1	0	0	0	0	0	0
0	1	0	0	0	0	0
0	0	1	0	0	0	0
0	0	0	1	0	0	0
0	0	0	0	60	0	0
0	0	0	0	0	1	0
0	0	0	0	0	47	60

47	0	0	0	0	0	0
0	47	0	0	0	0	0
0	0	1	0	0	0	0
0	0	0	1	0	0	0
0	0	0	0	48	47	59
0	0	0	0	60	0	0
0	0	0	0	0	0	13

44	60	0	0	0	0	0
1	0	0	0	0	0	0
0	0	17	1	0	0	0
0	0	60	0	0	0	0
0	0	0	0	1	0	0
0	0	0	0	0	1	0
0	0	0	0	0	0	1

37	53	0	0	0	0	0
34	24	0	0	0	0	0
0	0	41	34	0	0	0
0	0	8	20	0	0	0
0	0	0	0	1	0	0
0	0	0	0	0	1	0
0	0	0	0	0	0	1

G:=sub<GL(7,GF(61))| [60,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,48,0,0,0,0,0,60,0,0,0,0,0,0,0,60],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,1,47,0,0,0,0,0,0,60],[47,0,0,0,0,0,0,0,47,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,48,60,0,0,0,0,0,47,0,0,0,0,0,0,59,0,13],[44,1,0,0,0,0,0,60,0,0,0,0,0,0,0,0,17,60,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[37,34,0,0,0,0,0,53,24,0,0,0,0,0,0,0,41,8,0,0,0,0,0,34,20,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1] >;

C₂xA₄xDic₅ in GAP, Magma, Sage, TeX

C_2\times A_4\times {\rm Dic}_5

% in TeX

G:=Group("C2xA4xDic5");

// GroupNames label

G:=SmallGroup(480,1044);

// by ID

G=gap.SmallGroup(480,1044);

# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,2,-5,84,648,271,18822]);

// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^3=e^10=1,f^2=e^5,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,d*b*d^-1=b*c=c*b,b*e=e*b,b*f=f*b,d*c*d^-1=b,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=e^-1>;

// generators/relations

G = C2xA4xDic5 order 480 = 25·3·5

Direct product of C2, A4 and Dic5

G = C₂xA₄xDic₅ order 480 = 2⁵·3·5

Direct product of C₂, A₄ and Dic₅