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G = C3×Q8⋊D5order 240 = 24·3·5

Direct product of C3 and Q8⋊D5

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

Aliases: C3×Q8⋊D5, D20.2C6, C30.42D4, C1514SD16, C12.38D10, C60.38C22, C52C83C6, (C3×Q8)⋊4D5, Q82(C3×D5), (C5×Q8)⋊3C6, C4.3(C6×D5), C53(C3×SD16), C20.3(C2×C6), (Q8×C15)⋊4C2, C10.9(C3×D4), (C3×D20).4C2, C6.25(C5⋊D4), (C3×C52C8)⋊10C2, C2.6(C3×C5⋊D4), SmallGroup(240,46)

Series: Derived Chief Lower central Upper central

C1C20 — C3×Q8⋊D5
C1C5C10C20C60C3×D20 — C3×Q8⋊D5
C5C10C20 — C3×Q8⋊D5
C1C6C12C3×Q8

Generators and relations for C3×Q8⋊D5
 G = < a,b,c,d,e | a3=b4=d5=e2=1, c2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=ebe=b-1, bd=db, cd=dc, ece=b-1c, ede=d-1 >

20C2
2C4
10C22
20C6
4D5
5D4
5C8
2C12
10C2×C6
2C20
2D10
4C3×D5
5SD16
5C3×D4
5C24
2C60
2C6×D5
5C3×SD16

Smallest permutation representation of C3×Q8⋊D5
On 120 points
Generators in S120
(1 41 21)(2 42 22)(3 43 23)(4 44 24)(5 45 25)(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 16 6 11)(2 17 7 12)(3 18 8 13)(4 19 9 14)(5 20 10 15)(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 66 6 61)(2 67 7 62)(3 68 8 63)(4 69 9 64)(5 70 10 65)(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)
(1 5)(2 4)(6 10)(7 9)(11 20)(12 19)(13 18)(14 17)(15 16)(21 25)(22 24)(26 30)(27 29)(31 40)(32 39)(33 38)(34 37)(35 36)(41 45)(42 44)(46 50)(47 49)(51 60)(52 59)(53 58)(54 57)(55 56)(61 75)(62 74)(63 73)(64 72)(65 71)(66 80)(67 79)(68 78)(69 77)(70 76)(81 95)(82 94)(83 93)(84 92)(85 91)(86 100)(87 99)(88 98)(89 97)(90 96)(101 115)(102 114)(103 113)(104 112)(105 111)(106 120)(107 119)(108 118)(109 117)(110 116)

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

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

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

C3×Q8⋊D5 is a maximal subgroup of
D12⋊D10  D15⋊SD16  D60⋊C22  D20.27D6  D20.28D6  D20.16D6  D20.17D6  C3×D5×SD16

51 conjugacy classes

class 1 2A2B3A3B4A4B5A5B6A6B6C6D8A8B10A10B12A12B12C12D15A15B15C15D20A···20F24A24B24C24D30A30B30C30D60A···60L
order1223344556666881010121212121515151520···20242424243030303060···60
size1120112422112020101022224422224···41010101022224···4

51 irreducible representations

dim11111111222222222244
type++++++++
imageC1C2C2C2C3C6C6C6D4D5SD16D10C3×D4C3×D5C5⋊D4C3×SD16C6×D5C3×C5⋊D4Q8⋊D5C3×Q8⋊D5
kernelC3×Q8⋊D5C3×C52C8C3×D20Q8×C15Q8⋊D5C52C8D20C5×Q8C30C3×Q8C15C12C10Q8C6C5C4C2C3C1
# reps11112222122224444824

Matrix representation of C3×Q8⋊D5 in GL4(𝔽241) generated by

15000
01500
002250
000225
,
1000
0100
0001
002400
,
240000
024000
00222222
0022219
,
024000
118900
0010
0001
,
5224000
5218900
0010
000240
G:=sub<GL(4,GF(241))| [15,0,0,0,0,15,0,0,0,0,225,0,0,0,0,225],[1,0,0,0,0,1,0,0,0,0,0,240,0,0,1,0],[240,0,0,0,0,240,0,0,0,0,222,222,0,0,222,19],[0,1,0,0,240,189,0,0,0,0,1,0,0,0,0,1],[52,52,0,0,240,189,0,0,0,0,1,0,0,0,0,240] >;

C3×Q8⋊D5 in GAP, Magma, Sage, TeX

C_3\times Q_8\rtimes D_5
% in TeX

G:=Group("C3xQ8:D5");
// GroupNames label

G:=SmallGroup(240,46);
// by ID

G=gap.SmallGroup(240,46);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-2,-5,169,151,867,441,69,6917]);
// Polycyclic

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

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Subgroup lattice of C3×Q8⋊D5 in TeX

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