Copied to
clipboard

G = C157D4order 120 = 23·3·5

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

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C157D4, D302C2, C2.5D30, C6.12D10, C10.12D6, C222D15, Dic151C2, C30.12C22, (C2×C6)⋊2D5, (C2×C10)⋊4S3, (C2×C30)⋊2C2, C33(C5⋊D4), C53(C3⋊D4), SmallGroup(120,30)

Series: Derived Chief Lower central Upper central

C1C30 — C157D4
C1C5C15C30D30 — C157D4
C15C30 — C157D4
C1C2C22

Generators and relations for C157D4
 G = < a,b,c | a15=b4=c2=1, bab-1=cac=a-1, cbc=b-1 >

2C2
30C2
15C22
15C4
2C6
10S3
2C10
6D5
15D4
5D6
5Dic3
3D10
3Dic5
2C30
2D15
5C3⋊D4
3C5⋊D4

Smallest permutation representation of C157D4
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 39 24 50)(2 38 25 49)(3 37 26 48)(4 36 27 47)(5 35 28 46)(6 34 29 60)(7 33 30 59)(8 32 16 58)(9 31 17 57)(10 45 18 56)(11 44 19 55)(12 43 20 54)(13 42 21 53)(14 41 22 52)(15 40 23 51)
(2 15)(3 14)(4 13)(5 12)(6 11)(7 10)(8 9)(16 17)(18 30)(19 29)(20 28)(21 27)(22 26)(23 25)(31 58)(32 57)(33 56)(34 55)(35 54)(36 53)(37 52)(38 51)(39 50)(40 49)(41 48)(42 47)(43 46)(44 60)(45 59)

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

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

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

C157D4 is a maximal subgroup of
Dic5.D6  Dic3.D10  D5×C3⋊D4  S3×C5⋊D4  D6011C2  D4×D15  D42D15  C457D4  C22⋊D45  D6⋊D15  D62D15  C62⋊D5  A4⋊D15  Q8.D30  C242D15
C157D4 is a maximal quotient of
C30.4Q8  D303C4  D4⋊D15  D4.D15  Q82D15  C157Q16  C30.38D4  C457D4  D6⋊D15  D62D15  C62⋊D5  C242D15

33 conjugacy classes

class 1 2A2B2C 3  4 5A5B6A6B6C10A···10F15A15B15C15D30A···30L
order1222345566610···101515151530···30
size11230230222222···222222···2

33 irreducible representations

dim11112222222222
type+++++++++++
imageC1C2C2C2S3D4D5D6D10C3⋊D4D15C5⋊D4D30C157D4
kernelC157D4Dic15D30C2×C30C2×C10C15C2×C6C10C6C5C22C3C2C1
# reps11111121224448

Matrix representation of C157D4 in GL2(𝔽31) generated by

23
35
,
515
2126
,
11
030
G:=sub<GL(2,GF(31))| [2,3,3,5],[5,21,15,26],[1,0,1,30] >;

C157D4 in GAP, Magma, Sage, TeX

C_{15}\rtimes_7D_4
% in TeX

G:=Group("C15:7D4");
// GroupNames label

G:=SmallGroup(120,30);
// by ID

G=gap.SmallGroup(120,30);
# by ID

G:=PCGroup([5,-2,-2,-2,-3,-5,61,323,2404]);
// Polycyclic

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

Export

Subgroup lattice of C157D4 in TeX

׿
×
𝔽