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G = C15⋊D4order 120 = 23·3·5

1st semidirect product of C15 and D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C151D4, D61D5, D101S3, C6.4D10, C10.4D6, Dic154C2, C30.4C22, (C6×D5)⋊1C2, C52(C3⋊D4), C32(C5⋊D4), C2.4(S3×D5), (S3×C10)⋊1C2, SmallGroup(120,11)

Series: Derived Chief Lower central Upper central

C1C30 — C15⋊D4
C1C5C15C30C6×D5 — C15⋊D4
C15C30 — C15⋊D4
C1C2

Generators and relations for C15⋊D4
 G = < a,b,c | a15=b4=c2=1, bab-1=a-1, cac=a4, cbc=b-1 >

6C2
10C2
3C22
5C22
15C4
2S3
10C6
2D5
6C10
15D4
5C2×C6
5Dic3
3Dic5
3C2×C10
2C5×S3
2C3×D5
5C3⋊D4
3C5⋊D4

Character table of C15⋊D4

 class 12A2B2C345A5B6A6B6C10A10B10C10D10E10F15A15B30A30B
 size 1161023022210102266664444
ρ1111111111111111111111    trivial
ρ211-111-11111111-1-1-1-11111    linear of order 2
ρ311-1-111111-1-111-1-1-1-11111    linear of order 2
ρ4111-11-1111-1-11111111111    linear of order 2
ρ5220-2-1022-111220000-1-1-1-1    orthogonal lifted from D6
ρ62202-1022-1-1-1220000-1-1-1-1    orthogonal lifted from S3
ρ72-2002022-200-2-2000022-2-2    orthogonal lifted from D4
ρ822-2020-1+5/2-1-5/2200-1-5/2-1+5/21+5/21+5/21-5/21-5/2-1-5/2-1+5/2-1-5/2-1+5/2    orthogonal lifted from D10
ρ9222020-1+5/2-1-5/2200-1-5/2-1+5/2-1-5/2-1-5/2-1+5/2-1+5/2-1-5/2-1+5/2-1-5/2-1+5/2    orthogonal lifted from D5
ρ1022-2020-1-5/2-1+5/2200-1+5/2-1-5/21-5/21-5/21+5/21+5/2-1+5/2-1-5/2-1+5/2-1-5/2    orthogonal lifted from D10
ρ11222020-1-5/2-1+5/2200-1+5/2-1-5/2-1+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
ρ122-200-10221-3--3-2-20000-1-111    complex lifted from C3⋊D4
ρ132-200-10221--3-3-2-20000-1-111    complex lifted from C3⋊D4
ρ142-20020-1+5/2-1-5/2-2001+5/21-5/2ζ53525352545ζ545-1-5/2-1+5/21+5/21-5/2    complex lifted from C5⋊D4
ρ152-20020-1-5/2-1+5/2-2001-5/21+5/2545ζ5455352ζ5352-1+5/2-1-5/21-5/21+5/2    complex lifted from C5⋊D4
ρ162-20020-1-5/2-1+5/2-2001-5/21+5/2ζ545545ζ53525352-1+5/2-1-5/21-5/21+5/2    complex lifted from C5⋊D4
ρ172-20020-1+5/2-1-5/2-2001+5/21-5/25352ζ5352ζ545545-1-5/2-1+5/21+5/21-5/2    complex lifted from C5⋊D4
ρ184400-20-1-5-1+5-200-1+5-1-500001-5/21+5/21-5/21+5/2    orthogonal lifted from S3×D5
ρ194400-20-1+5-1-5-200-1-5-1+500001+5/21-5/21+5/21-5/2    orthogonal lifted from S3×D5
ρ204-400-20-1-5-1+52001-51+500001-5/21+5/2-1+5/2-1-5/2    symplectic faithful, Schur index 2
ρ214-400-20-1+5-1-52001+51-500001+5/21-5/2-1-5/2-1+5/2    symplectic faithful, Schur index 2

Smallest permutation representation of C15⋊D4
On 60 points
Generators in S60
(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)
(1 32 17 51)(2 31 18 50)(3 45 19 49)(4 44 20 48)(5 43 21 47)(6 42 22 46)(7 41 23 60)(8 40 24 59)(9 39 25 58)(10 38 26 57)(11 37 27 56)(12 36 28 55)(13 35 29 54)(14 34 30 53)(15 33 16 52)
(2 5)(3 9)(4 13)(7 10)(8 14)(12 15)(16 28)(18 21)(19 25)(20 29)(23 26)(24 30)(31 47)(32 51)(33 55)(34 59)(35 48)(36 52)(37 56)(38 60)(39 49)(40 53)(41 57)(42 46)(43 50)(44 54)(45 58)

G:=sub<Sym(60)| (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), (1,32,17,51)(2,31,18,50)(3,45,19,49)(4,44,20,48)(5,43,21,47)(6,42,22,46)(7,41,23,60)(8,40,24,59)(9,39,25,58)(10,38,26,57)(11,37,27,56)(12,36,28,55)(13,35,29,54)(14,34,30,53)(15,33,16,52), (2,5)(3,9)(4,13)(7,10)(8,14)(12,15)(16,28)(18,21)(19,25)(20,29)(23,26)(24,30)(31,47)(32,51)(33,55)(34,59)(35,48)(36,52)(37,56)(38,60)(39,49)(40,53)(41,57)(42,46)(43,50)(44,54)(45,58)>;

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), (1,32,17,51)(2,31,18,50)(3,45,19,49)(4,44,20,48)(5,43,21,47)(6,42,22,46)(7,41,23,60)(8,40,24,59)(9,39,25,58)(10,38,26,57)(11,37,27,56)(12,36,28,55)(13,35,29,54)(14,34,30,53)(15,33,16,52), (2,5)(3,9)(4,13)(7,10)(8,14)(12,15)(16,28)(18,21)(19,25)(20,29)(23,26)(24,30)(31,47)(32,51)(33,55)(34,59)(35,48)(36,52)(37,56)(38,60)(39,49)(40,53)(41,57)(42,46)(43,50)(44,54)(45,58) );

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)], [(1,32,17,51),(2,31,18,50),(3,45,19,49),(4,44,20,48),(5,43,21,47),(6,42,22,46),(7,41,23,60),(8,40,24,59),(9,39,25,58),(10,38,26,57),(11,37,27,56),(12,36,28,55),(13,35,29,54),(14,34,30,53),(15,33,16,52)], [(2,5),(3,9),(4,13),(7,10),(8,14),(12,15),(16,28),(18,21),(19,25),(20,29),(23,26),(24,30),(31,47),(32,51),(33,55),(34,59),(35,48),(36,52),(37,56),(38,60),(39,49),(40,53),(41,57),(42,46),(43,50),(44,54),(45,58)])

C15⋊D4 is a maximal subgroup of
D205S3  D6.D10  D125D5  C20⋊D6  C30.C23  D5×C3⋊D4  S3×C5⋊D4  C45⋊D4  C30.12D6  D6⋊D15  D30⋊S3  D10.1S4  D10⋊S4
C15⋊D4 is a maximal quotient of
C15⋊D8  C30.D4  C20.D6  C15⋊Q16  D10⋊Dic3  D6⋊Dic5  Dic155C4  C45⋊D4  C30.12D6  D6⋊D15  D30⋊S3  D10⋊S4

Matrix representation of C15⋊D4 in GL4(𝔽61) generated by

601700
444400
00470
00213
,
144500
394700
005541
00116
,
1000
176000
0010
003660
G:=sub<GL(4,GF(61))| [60,44,0,0,17,44,0,0,0,0,47,2,0,0,0,13],[14,39,0,0,45,47,0,0,0,0,55,11,0,0,41,6],[1,17,0,0,0,60,0,0,0,0,1,36,0,0,0,60] >;

C15⋊D4 in GAP, Magma, Sage, TeX

C_{15}\rtimes D_4
% in TeX

G:=Group("C15:D4");
// GroupNames label

G:=SmallGroup(120,11);
// by ID

G=gap.SmallGroup(120,11);
# by ID

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

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

Export

Subgroup lattice of C15⋊D4 in TeX
Character table of C15⋊D4 in TeX

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