Copied to
clipboard

G = C2×C15⋊D4order 240 = 24·3·5

Direct product of C2 and C15⋊D4

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

Aliases: C2×C15⋊D4, C301D4, D65D10, D105D6, C30.21C23, Dic159C22, C154(C2×D4), C62(C5⋊D4), C102(C3⋊D4), (C2×C10).16D6, (C2×C6).16D10, (C22×D5)⋊2S3, (C22×S3)⋊1D5, (C6×D5)⋊5C22, (S3×C10)⋊5C22, C22.14(S3×D5), C6.21(C22×D5), (C2×Dic15)⋊10C2, C10.21(C22×S3), (C2×C30).15C22, (D5×C2×C6)⋊1C2, C53(C2×C3⋊D4), C33(C2×C5⋊D4), (S3×C2×C10)⋊1C2, C2.21(C2×S3×D5), SmallGroup(240,145)

Series: Derived Chief Lower central Upper central

C1C30 — C2×C15⋊D4
C1C5C15C30C6×D5C15⋊D4 — C2×C15⋊D4
C15C30 — C2×C15⋊D4
C1C22

Generators and relations for C2×C15⋊D4
 G = < a,b,c,d | a2=b15=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=b-1, dbd=b4, dcd=c-1 >

Subgroups: 432 in 108 conjugacy classes, 40 normal (18 characteristic)
C1, C2, C2 [×2], C2 [×4], C3, C4 [×2], C22, C22 [×8], C5, S3 [×2], C6, C6 [×2], C6 [×2], C2×C4, D4 [×4], C23 [×2], D5 [×2], C10, C10 [×2], C10 [×2], Dic3 [×2], D6 [×2], D6 [×2], C2×C6, C2×C6 [×4], C15, C2×D4, Dic5 [×2], D10 [×2], D10 [×2], C2×C10, C2×C10 [×4], C2×Dic3, C3⋊D4 [×4], C22×S3, C22×C6, C5×S3 [×2], C3×D5 [×2], C30, C30 [×2], C2×Dic5, C5⋊D4 [×4], C22×D5, C22×C10, C2×C3⋊D4, Dic15 [×2], C6×D5 [×2], C6×D5 [×2], S3×C10 [×2], S3×C10 [×2], C2×C30, C2×C5⋊D4, C15⋊D4 [×4], C2×Dic15, D5×C2×C6, S3×C2×C10, C2×C15⋊D4
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], C2×D4, D10 [×3], C3⋊D4 [×2], C22×S3, C5⋊D4 [×2], C22×D5, C2×C3⋊D4, S3×D5, C2×C5⋊D4, C15⋊D4 [×2], C2×S3×D5, C2×C15⋊D4

Smallest permutation representation of C2×C15⋊D4
On 120 points
Generators in S120
(1 44)(2 45)(3 31)(4 32)(5 33)(6 34)(7 35)(8 36)(9 37)(10 38)(11 39)(12 40)(13 41)(14 42)(15 43)(16 56)(17 57)(18 58)(19 59)(20 60)(21 46)(22 47)(23 48)(24 49)(25 50)(26 51)(27 52)(28 53)(29 54)(30 55)(61 91)(62 92)(63 93)(64 94)(65 95)(66 96)(67 97)(68 98)(69 99)(70 100)(71 101)(72 102)(73 103)(74 104)(75 105)(76 119)(77 120)(78 106)(79 107)(80 108)(81 109)(82 110)(83 111)(84 112)(85 113)(86 114)(87 115)(88 116)(89 117)(90 118)
(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 68 28 85)(2 67 29 84)(3 66 30 83)(4 65 16 82)(5 64 17 81)(6 63 18 80)(7 62 19 79)(8 61 20 78)(9 75 21 77)(10 74 22 76)(11 73 23 90)(12 72 24 89)(13 71 25 88)(14 70 26 87)(15 69 27 86)(31 96 55 111)(32 95 56 110)(33 94 57 109)(34 93 58 108)(35 92 59 107)(36 91 60 106)(37 105 46 120)(38 104 47 119)(39 103 48 118)(40 102 49 117)(41 101 50 116)(42 100 51 115)(43 99 52 114)(44 98 53 113)(45 97 54 112)
(1 44)(2 33)(3 37)(4 41)(5 45)(6 34)(7 38)(8 42)(9 31)(10 35)(11 39)(12 43)(13 32)(14 36)(15 40)(16 50)(17 54)(18 58)(19 47)(20 51)(21 55)(22 59)(23 48)(24 52)(25 56)(26 60)(27 49)(28 53)(29 57)(30 46)(61 115)(62 119)(63 108)(64 112)(65 116)(66 120)(67 109)(68 113)(69 117)(70 106)(71 110)(72 114)(73 118)(74 107)(75 111)(76 92)(77 96)(78 100)(79 104)(80 93)(81 97)(82 101)(83 105)(84 94)(85 98)(86 102)(87 91)(88 95)(89 99)(90 103)

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

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

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

C2×C15⋊D4 is a maximal subgroup of
D10.D12  Dic15⋊D4  D10.17D12  D6⋊(C4×D5)  C1517(C4×D4)  Dic159D4  D6⋊C4⋊D5  D10⋊D12  C60⋊D4  D10⋊C4⋊S3  Dic152D4  D6⋊D20  C604D4  D6.9D20  D3012D4  Dic15.10D4  C6010D4  Dic15.31D4  D64D20  D304D4  (C6×D5)⋊D4  (C2×C30)⋊D4  (S3×C10)⋊D4  Dic155D4  C15⋊C22≀C2  Dic1518D4  D2013D6  C2×D5×C3⋊D4  C2×S3×C5⋊D4
C2×C15⋊D4 is a maximal quotient of
D20.34D6  C60.36D4  D2021D6  D20.37D6  D12.37D10  C60.67D4  C60.88D4  C60.46D4  C60.89D4  Dic158Q8  C60⋊D4  C604D4  C6010D4  (C2×C30).D4  C30.(C2×D4)  (C2×C30)⋊D4  C15⋊C22≀C2  Dic1518D4

42 conjugacy classes

class 1 2A2B2C2D2E2F2G 3 4A4B5A5B6A6B6C6D6E6F6G10A···10F10G···10N15A15B30A···30F
order1222222234455666666610···1010···10151530···30
size11116610102303022222101010102···26···6444···4

42 irreducible representations

dim11111222222222444
type+++++++++++++-+
imageC1C2C2C2C2S3D4D5D6D6D10D10C3⋊D4C5⋊D4S3×D5C15⋊D4C2×S3×D5
kernelC2×C15⋊D4C15⋊D4C2×Dic15D5×C2×C6S3×C2×C10C22×D5C30C22×S3D10C2×C10D6C2×C6C10C6C22C2C2
# reps14111122214248242

Matrix representation of C2×C15⋊D4 in GL5(𝔽61)

600000
060000
006000
00010
00001
,
10000
047000
0271300
0004444
0001760
,
10000
0151500
0504600
00010
0001760
,
10000
060000
02100
00010
0001760

G:=sub<GL(5,GF(61))| [60,0,0,0,0,0,60,0,0,0,0,0,60,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,47,27,0,0,0,0,13,0,0,0,0,0,44,17,0,0,0,44,60],[1,0,0,0,0,0,15,50,0,0,0,15,46,0,0,0,0,0,1,17,0,0,0,0,60],[1,0,0,0,0,0,60,2,0,0,0,0,1,0,0,0,0,0,1,17,0,0,0,0,60] >;

C2×C15⋊D4 in GAP, Magma, Sage, TeX

C_2\times C_{15}\rtimes D_4
% in TeX

G:=Group("C2xC15:D4");
// GroupNames label

G:=SmallGroup(240,145);
// by ID

G=gap.SmallGroup(240,145);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-5,121,490,6917]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^15=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,d*b*d=b^4,d*c*d=c^-1>;
// generators/relations

׿
×
𝔽