Copied to
clipboard

G = C15⋊Q8order 120 = 23·3·5

The semidirect product of C15 and Q8 acting via Q8/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C15⋊Q8, C51Dic6, C6.7D10, C10.7D6, C31Dic10, Dic3.D5, C30.7C22, Dic5.1S3, Dic15.2C2, C2.7(S3×D5), (C5×Dic3).1C2, (C3×Dic5).1C2, SmallGroup(120,14)

Series: Derived Chief Lower central Upper central

C1C30 — C15⋊Q8
C1C5C15C30C3×Dic5 — C15⋊Q8
C15C30 — C15⋊Q8
C1C2

Generators and relations for C15⋊Q8
 G = < a,b,c | a15=b4=1, c2=b2, bab-1=a11, cac-1=a4, cbc-1=b-1 >

3C4
5C4
15C4
15Q8
5C12
5Dic3
3C20
3Dic5
5Dic6
3Dic10

Character table of C15⋊Q8

 class 1234A4B4C5A5B610A10B12A12B15A15B20A20B20C20D30A30B
 size 1126103022222101044666644
ρ1111111111111111111111    trivial
ρ2111-11-1111111111-1-1-1-111    linear of order 2
ρ3111-1-1111111-1-111-1-1-1-111    linear of order 2
ρ41111-1-111111-1-111111111    linear of order 2
ρ522-10-2022-12211-1-10000-1-1    orthogonal lifted from D6
ρ6222200-1+5/2-1-5/22-1+5/2-1-5/200-1-5/2-1+5/2-1-5/2-1+5/2-1+5/2-1-5/2-1-5/2-1+5/2    orthogonal lifted from D5
ρ722-102022-122-1-1-1-10000-1-1    orthogonal lifted from S3
ρ8222-200-1-5/2-1+5/22-1-5/2-1+5/200-1+5/2-1-5/21-5/21+5/21+5/21-5/2-1+5/2-1-5/2    orthogonal lifted from D10
ρ9222-200-1+5/2-1-5/22-1+5/2-1-5/200-1-5/2-1+5/21+5/21-5/21-5/21+5/2-1-5/2-1+5/2    orthogonal lifted from D10
ρ10222200-1-5/2-1+5/22-1-5/2-1+5/200-1+5/2-1-5/2-1+5/2-1-5/2-1-5/2-1+5/2-1+5/2-1-5/2    orthogonal lifted from D5
ρ112-2200022-2-2-200220000-2-2    symplectic lifted from Q8, Schur index 2
ρ122-2-1000221-2-2-33-1-1000011    symplectic lifted from Dic6, Schur index 2
ρ132-2-1000221-2-23-3-1-1000011    symplectic lifted from Dic6, Schur index 2
ρ142-22000-1+5/2-1-5/2-21-5/21+5/200-1-5/2-1+5/2ζ4ζ534ζ5243ζ5443ζ5ζ43ζ5443ζ54ζ534ζ521+5/21-5/2    symplectic lifted from Dic10, Schur index 2
ρ152-22000-1+5/2-1-5/2-21-5/21+5/200-1-5/2-1+5/24ζ534ζ52ζ43ζ5443ζ543ζ5443ζ5ζ4ζ534ζ521+5/21-5/2    symplectic lifted from Dic10, Schur index 2
ρ162-22000-1-5/2-1+5/2-21+5/21-5/200-1+5/2-1-5/243ζ5443ζ54ζ534ζ52ζ4ζ534ζ52ζ43ζ5443ζ51-5/21+5/2    symplectic lifted from Dic10, Schur index 2
ρ172-22000-1-5/2-1+5/2-21+5/21-5/200-1+5/2-1-5/2ζ43ζ5443ζ5ζ4ζ534ζ524ζ534ζ5243ζ5443ζ51-5/21+5/2    symplectic lifted from Dic10, Schur index 2
ρ1844-2000-1-5-1+5-2-1-5-1+5001-5/21+5/200001-5/21+5/2    orthogonal lifted from S3×D5
ρ1944-2000-1+5-1-5-2-1+5-1-5001+5/21-5/200001+5/21-5/2    orthogonal lifted from S3×D5
ρ204-4-2000-1-5-1+521+51-5001-5/21+5/20000-1+5/2-1-5/2    symplectic faithful, Schur index 2
ρ214-4-2000-1+5-1-521-51+5001+5/21-5/20000-1-5/2-1+5/2    symplectic faithful, Schur index 2

Smallest permutation representation of C15⋊Q8
Regular action on 120 points
Generators in S120
(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 58 17 38)(2 54 18 34)(3 50 19 45)(4 46 20 41)(5 57 21 37)(6 53 22 33)(7 49 23 44)(8 60 24 40)(9 56 25 36)(10 52 26 32)(11 48 27 43)(12 59 28 39)(13 55 29 35)(14 51 30 31)(15 47 16 42)(61 104 85 113)(62 100 86 109)(63 96 87 120)(64 92 88 116)(65 103 89 112)(66 99 90 108)(67 95 76 119)(68 91 77 115)(69 102 78 111)(70 98 79 107)(71 94 80 118)(72 105 81 114)(73 101 82 110)(74 97 83 106)(75 93 84 117)
(1 83 17 74)(2 87 18 63)(3 76 19 67)(4 80 20 71)(5 84 21 75)(6 88 22 64)(7 77 23 68)(8 81 24 72)(9 85 25 61)(10 89 26 65)(11 78 27 69)(12 82 28 73)(13 86 29 62)(14 90 30 66)(15 79 16 70)(31 108 51 99)(32 112 52 103)(33 116 53 92)(34 120 54 96)(35 109 55 100)(36 113 56 104)(37 117 57 93)(38 106 58 97)(39 110 59 101)(40 114 60 105)(41 118 46 94)(42 107 47 98)(43 111 48 102)(44 115 49 91)(45 119 50 95)

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

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

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

C15⋊Q8 is a maximal subgroup of
D5×Dic6  S3×Dic10  D15⋊Q8  D6.D10  Dic5.D6  C30.C23  Dic3.D10  C45⋊Q8  C15⋊Dic6  C3⋊Dic30  C323Dic10  CSU2(𝔽3)⋊D5  A4⋊Dic10
C15⋊Q8 is a maximal quotient of
C30.Q8  Dic155C4  C6.Dic10  C45⋊Q8  C15⋊Dic6  C3⋊Dic30  C323Dic10  A4⋊Dic10

Matrix representation of C15⋊Q8 in GL4(𝔽61) generated by

601700
444400
00601
00600
,
295400
73200
001150
00050
,
205800
324100
002315
004638
G:=sub<GL(4,GF(61))| [60,44,0,0,17,44,0,0,0,0,60,60,0,0,1,0],[29,7,0,0,54,32,0,0,0,0,11,0,0,0,50,50],[20,32,0,0,58,41,0,0,0,0,23,46,0,0,15,38] >;

C15⋊Q8 in GAP, Magma, Sage, TeX

C_{15}\rtimes Q_8
% in TeX

G:=Group("C15:Q8");
// GroupNames label

G:=SmallGroup(120,14);
// by ID

G=gap.SmallGroup(120,14);
# by ID

G:=PCGroup([5,-2,-2,-2,-3,-5,20,61,26,168,2404]);
// Polycyclic

G:=Group<a,b,c|a^15=b^4=1,c^2=b^2,b*a*b^-1=a^11,c*a*c^-1=a^4,c*b*c^-1=b^-1>;
// generators/relations

Export

Subgroup lattice of C15⋊Q8 in TeX
Character table of C15⋊Q8 in TeX

׿
×
𝔽