C3xQ8:2F5 - GroupNames

G = C₃xQ₈:₂F₅ order 480 = 2⁵·3·5

Direct product of C₃ and Q₈:₂F₅

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₃xQ₈:₂F₅, D₂₀:₂C₁₂, C₁₅:₁₄C₄wrC₂, (C₄xF₅):₂C₆, Q₈:₃(C₃xF₅), (C₃xQ₈):₅F₅, C₄.F₅:₂C₆, C₄.₄(C₆xF₅), (C₃xD₂₀):₅C₄, (C₁₂xF₅):₇C₂, (Q₈xC₁₅):₅C₄, (C₅xQ₈):₄C₁₂, C₆₀.₄₃(C₂xC₄), C₂₀.₄(C₂xC₁₂), (C₆xD₅).₃₈D₄, D₁₀.₃(C₃xD₄), C₁₂.₄₃(C₂xF₅), Q₈:₂D₅.₄C₆, (C₃xDic₅).₈₇D₄, Dic₅.₂₂(C₃xD₄), C₆.₃₄(C₂²:F₅), C₃₀.₃₄(C₂²:C₄), (D₅xC₁₂).₈₅C₂², C₅:₂(C₃xC₄wrC₂), (C₃xC₄.F₅):₈C₂, C₂.₉(C₃xC₂²:F₅), (C₄xD₅).₁₀(C₂xC₆), C₁₀.₈(C₃xC₂²:C₄), (C₃xQ₈:₂D₅).₅C₂, SmallGroup(480,290)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂₀ — C₃xQ₈:₂F₅

Chief series

C₁ — C₅ — C₁₀ — C₂₀ — C₄xD₅ — D₅xC₁₂ — C₃xC₄.F₅ — C₃xQ₈:₂F₅

Lower central

C₅ — C₁₀ — C₂₀ — C₃xQ₈:₂F₅

Upper central

C₁ — C₆ — C₁₂ — C₃xQ₈

Generators and relations for C₃xQ₈:₂F₅
G = < a,b,c,d,e | a³=b⁴=d⁵=e⁴=1, c²=b², ab=ba, ac=ca, ad=da, ae=ea, cbc^-1=b^-1, bd=db, be=eb, cd=dc, ece^-1=b^-1c, ede^-1=d³ >

Subgroups: 360 in 88 conjugacy classes, 32 normal (all characteristic)
C₁, C₂, C₂, C₃, C₄, C₄, C₂², C₅, C₆, C₆, C₈, C₂xC₄, D₄, Q₈, D₅, C₁₀, C₁₂, C₁₂, C₂xC₆, C₁₅, C₄², M₄(2), C₄oD₄, Dic₅, C₂₀, C₂₀, F₅, D₁₀, D₁₀, C₂₄, C₂xC₁₂, C₃xD₄, C₃xQ₈, C₃xD₅, C₃₀, C₄wrC₂, C₅:C₈, C₄xD₅, C₄xD₅, D₂₀, D₂₀, C₅xQ₈, C₂xF₅, C₄xC₁₂, C₃xM₄(2), C₃xC₄oD₄, C₃xDic₅, C₆₀, C₆₀, C₃xF₅, C₆xD₅, C₆xD₅, C₄.F₅, C₄xF₅, Q₈:₂D₅, C₃xC₄wrC₂, C₃xC₅:C₈, D₅xC₁₂, D₅xC₁₂, C₃xD₂₀, C₃xD₂₀, Q₈xC₁₅, C₆xF₅, Q₈:₂F₅, C₃xC₄.F₅, C₁₂xF₅, C₃xQ₈:₂D₅, C₃xQ₈:₂F₅
Quotients: C₁, C₂, C₃, C₄, C₂², C₆, C₂xC₄, D₄, C₁₂, C₂xC₆, C₂²:C₄, F₅, C₂xC₁₂, C₃xD₄, C₄wrC₂, C₂xF₅, C₃xC₂²:C₄, C₃xF₅, C₂²:F₅, C₃xC₄wrC₂, C₆xF₅, Q₈:₂F₅, C₃xC₂²:F₅, C₃xQ₈:₂F₅

Smallest permutation representation of C₃xQ₈:₂F₅
►On 120 points

Generators in S₁₂₀

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

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

(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)

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

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

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

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

57 conjugacy classes

class	1	2A	2B	2C	3A	3B	4A	4B	4C	4D	4E	4F	4G	4H	5	6A	6B	6C	6D	6E	6F	8A	8B	10	12A	12B	12C	12D	12E	12F	12G	12H	12I	···	12P	15A	15B	20A	20B	20C	24A	24B	24C	24D	30A	30B	60A	···	60F
order	1	2	2	2	3	3	4	4	4	4	4	4	4	4	5	6	6	6	6	6	6	8	8	10	12	12	12	12	12	12	12	12	12	···	12	15	15	20	20	20	24	24	24	24	30	30	60	···	60
size	1	1	10	20	1	1	2	4	5	5	10	10	10	10	4	1	1	10	10	20	20	20	20	4	2	2	4	4	5	5	5	5	10	···	10	4	4	8	8	8	20	20	20	20	4	4	8	···	8

57 irreducible representations

dim	1	1	1	1	1	1	1	1	1	1	1	1	2	2	2	2	2	2	4	4	4	4	4	4	8	8
type	+	+	+	+									+	+					+	+		+			+
image	C₁	C₂	C₂	C₂	C₃	C₄	C₄	C₆	C₆	C₆	C₁₂	C₁₂	D₄	D₄	C₃xD₄	C₃xD₄	C₄wrC₂	C₃xC₄wrC₂	F₅	C₂xF₅	C₃xF₅	C₂²:F₅	C₆xF₅	C₃xC₂²:F₅	Q₈:₂F₅	C₃xQ₈:₂F₅
kernel	C₃xQ₈:₂F₅	C₃xC₄.F₅	C₁₂xF₅	C₃xQ₈:₂D₅	Q₈:₂F₅	C₃xD₂₀	Q₈xC₁₅	C₄.F₅	C₄xF₅	Q₈:₂D₅	D₂₀	C₅xQ₈	C₃xDic₅	C₆xD₅	Dic₅	D₁₀	C₁₅	C₅	C₃xQ₈	C₁₂	Q₈	C₆	C₄	C₂	C₃	C₁
# reps	1	1	1	1	2	2	2	2	2	2	4	4	1	1	2	2	4	8	1	1	2	2	2	4	1	2

Matrix representation of C₃xQ₈:₂F₅ ►in GL₆(F₂₄₁)

1	0	0	0	0	0
0	1	0	0	0	0
0	0	15	0	0	0
0	0	0	15	0	0
0	0	0	0	15	0
0	0	0	0	0	15

64	0	0	0	0	0
2	177	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	1

79	5	0	0	0	0
53	162	0	0	0	0
0	0	240	0	0	0
0	0	0	240	0	0
0	0	0	0	240	0
0	0	0	0	0	240

1	0	0	0	0	0
0	1	0	0	0	0
0	0	0	1	0	0
0	0	0	0	1	0
0	0	0	0	0	1
0	0	240	240	240	240

1	0	0	0	0	0
176	64	0	0	0	0
0	0	1	0	0	0
0	0	0	0	0	1
0	0	0	1	0	0
0	0	240	240	240	240

G:=sub<GL(6,GF(241))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,15,0,0,0,0,0,0,15,0,0,0,0,0,0,15,0,0,0,0,0,0,15],[64,2,0,0,0,0,0,177,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[79,53,0,0,0,0,5,162,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,240],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,240,0,0,1,0,0,240,0,0,0,1,0,240,0,0,0,0,1,240],[1,176,0,0,0,0,0,64,0,0,0,0,0,0,1,0,0,240,0,0,0,0,1,240,0,0,0,0,0,240,0,0,0,1,0,240] >;

C₃xQ₈:₂F₅ in GAP, Magma, Sage, TeX

C_3\times Q_8\rtimes_2F_5

% in TeX

G:=Group("C3xQ8:2F5");

// GroupNames label

G:=SmallGroup(480,290);

// by ID

G=gap.SmallGroup(480,290);

# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-5,84,365,344,136,2524,1271,102,9414,1595]);

// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^4=d^5=e^4=1,c^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e^-1=b^-1*c,e*d*e^-1=d^3>;

// generators/relations

G = C3xQ8:2F5 order 480 = 25·3·5

Direct product of C3 and Q8:2F5

G = C₃xQ₈:₂F₅ order 480 = 2⁵·3·5

Direct product of C₃ and Q₈:₂F₅