Copied to
clipboard

G = C5⋊D12order 120 = 23·3·5

The semidirect product of C5 and D12 acting via D12/D6=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C52D12, C153D4, D62D5, Dic5⋊S3, D304C2, C10.6D6, C6.6D10, C30.6C22, C31(C5⋊D4), C2.6(S3×D5), (S3×C10)⋊2C2, (C3×Dic5)⋊3C2, SmallGroup(120,13)

Series: Derived Chief Lower central Upper central

C1C30 — C5⋊D12
C1C5C15C30C3×Dic5 — C5⋊D12
C15C30 — C5⋊D12
C1C2

Generators and relations for C5⋊D12
 G = < a,b,c | a5=b12=c2=1, bab-1=cac=a-1, cbc=b-1 >

6C2
30C2
3C22
5C4
15C22
2S3
10S3
6D5
6C10
15D4
5C12
5D6
3D10
3C2×C10
2C5×S3
2D15
5D12
3C5⋊D4

Character table of C5⋊D12

 class 12A2B2C345A5B610A10B10C10D10E10F12A12B15A15B30A30B
 size 1163021022222666610104444
ρ1111111111111111111111    trivial
ρ2111-11-1111111111-1-11111    linear of order 2
ρ311-111-111111-1-1-1-1-1-11111    linear of order 2
ρ411-1-11111111-1-1-1-1111111    linear of order 2
ρ52-2002022-2-2-200000022-2-2    orthogonal lifted from D4
ρ62200-1-222-122000011-1-1-1-1    orthogonal lifted from D6
ρ72200-1222-1220000-1-1-1-1-1-1    orthogonal lifted from S3
ρ82-200-10221-2-20000-33-1-111    orthogonal lifted from D12
ρ922-2020-1-5/2-1+5/22-1-5/2-1+5/21-5/21-5/21+5/21+5/200-1-5/2-1+5/2-1-5/2-1+5/2    orthogonal lifted from D10
ρ10222020-1+5/2-1-5/22-1+5/2-1-5/2-1-5/2-1-5/2-1+5/2-1+5/200-1+5/2-1-5/2-1+5/2-1-5/2    orthogonal lifted from D5
ρ11222020-1-5/2-1+5/22-1-5/2-1+5/2-1+5/2-1+5/2-1-5/2-1-5/200-1-5/2-1+5/2-1-5/2-1+5/2    orthogonal lifted from D5
ρ1222-2020-1+5/2-1-5/22-1+5/2-1-5/21+5/21+5/21-5/21-5/200-1+5/2-1-5/2-1+5/2-1-5/2    orthogonal lifted from D10
ρ132-200-10221-2-200003-3-1-111    orthogonal lifted from D12
ρ142-20020-1-5/2-1+5/2-21+5/21-5/2ζ5455455352ζ535200-1-5/2-1+5/21+5/21-5/2    complex lifted from C5⋊D4
ρ152-20020-1+5/2-1-5/2-21-5/21+5/25352ζ5352545ζ54500-1+5/2-1-5/21-5/21+5/2    complex lifted from C5⋊D4
ρ162-20020-1-5/2-1+5/2-21+5/21-5/2545ζ545ζ5352535200-1-5/2-1+5/21+5/21-5/2    complex lifted from C5⋊D4
ρ172-20020-1+5/2-1-5/2-21-5/21+5/2ζ53525352ζ54554500-1+5/2-1-5/21-5/21+5/2    complex lifted from C5⋊D4
ρ184-400-20-1-5-1+521+51-50000001+5/21-5/2-1-5/2-1+5/2    orthogonal faithful, Schur index 2
ρ194400-20-1-5-1+5-2-1-5-1+50000001+5/21-5/21+5/21-5/2    orthogonal lifted from S3×D5
ρ204-400-20-1+5-1-521-51+50000001-5/21+5/2-1+5/2-1-5/2    orthogonal faithful, Schur index 2
ρ214400-20-1+5-1-5-2-1+5-1-50000001-5/21+5/21-5/21+5/2    orthogonal lifted from S3×D5

Smallest permutation representation of C5⋊D12
On 60 points
Generators in S60
(1 26 54 44 20)(2 21 45 55 27)(3 28 56 46 22)(4 23 47 57 29)(5 30 58 48 24)(6 13 37 59 31)(7 32 60 38 14)(8 15 39 49 33)(9 34 50 40 16)(10 17 41 51 35)(11 36 52 42 18)(12 19 43 53 25)
(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 9)(2 8)(3 7)(4 6)(10 12)(13 29)(14 28)(15 27)(16 26)(17 25)(18 36)(19 35)(20 34)(21 33)(22 32)(23 31)(24 30)(37 57)(38 56)(39 55)(40 54)(41 53)(42 52)(43 51)(44 50)(45 49)(46 60)(47 59)(48 58)

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

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

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

C5⋊D12 is a maximal subgroup of
D12⋊D5  D60⋊C2  D6.D10  D5×D12  Dic3.D10  S3×C5⋊D4  D10⋊D6  C5⋊D36  C15⋊D12  D62D15  D30⋊S3  GL2(𝔽3)⋊D5  Dic5⋊S4
C5⋊D12 is a maximal quotient of
C5⋊D24  D12.D5  Dic6⋊D5  C5⋊Dic12  D6⋊Dic5  D304C4  C30.Q8  C5⋊D36  C15⋊D12  D62D15  D30⋊S3  Dic5⋊S4

Matrix representation of C5⋊D12 in GL4(𝔽61) generated by

1000
0100
00171
00600
,
0100
60100
003939
004722
,
16000
06000
001744
006044
G:=sub<GL(4,GF(61))| [1,0,0,0,0,1,0,0,0,0,17,60,0,0,1,0],[0,60,0,0,1,1,0,0,0,0,39,47,0,0,39,22],[1,0,0,0,60,60,0,0,0,0,17,60,0,0,44,44] >;

C5⋊D12 in GAP, Magma, Sage, TeX

C_5\rtimes D_{12}
% in TeX

G:=Group("C5:D12");
// GroupNames label

G:=SmallGroup(120,13);
// by ID

G=gap.SmallGroup(120,13);
# by ID

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

G:=Group<a,b,c|a^5=b^12=c^2=1,b*a*b^-1=c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C5⋊D12 in TeX
Character table of C5⋊D12 in TeX

׿
×
𝔽