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G = C4×D15order 120 = 23·3·5

Direct product of C4 and D15

direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary

Aliases: C4×D15, C602C2, C202S3, C122D5, C4Dic15, C2.1D30, C10.9D6, C6.9D10, D30.2C2, Dic155C2, C30.9C22, C53(C4×S3), C32(C4×D5), C157(C2×C4), SmallGroup(120,27)

Series: Derived Chief Lower central Upper central

C1C15 — C4×D15
C1C5C15C30D30 — C4×D15
C15 — C4×D15
C1C4

Generators and relations for C4×D15
 G = < a,b,c | a4=b15=c2=1, ab=ba, ac=ca, cbc=b-1 >

15C2
15C2
15C4
15C22
5S3
5S3
3D5
3D5
15C2×C4
5Dic3
5D6
3D10
3Dic5
5C4×S3
3C4×D5

Smallest permutation representation of C4×D15
On 60 points
Generators in S60
(1 47 24 39)(2 48 25 40)(3 49 26 41)(4 50 27 42)(5 51 28 43)(6 52 29 44)(7 53 30 45)(8 54 16 31)(9 55 17 32)(10 56 18 33)(11 57 19 34)(12 58 20 35)(13 59 21 36)(14 60 22 37)(15 46 23 38)
(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 23)(2 22)(3 21)(4 20)(5 19)(6 18)(7 17)(8 16)(9 30)(10 29)(11 28)(12 27)(13 26)(14 25)(15 24)(31 54)(32 53)(33 52)(34 51)(35 50)(36 49)(37 48)(38 47)(39 46)(40 60)(41 59)(42 58)(43 57)(44 56)(45 55)

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

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

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

C4×D15 is a maximal subgroup of
D152C8  D30.5C4  C40⋊S3  D20⋊S3  D12⋊D5  D15⋊Q8  D6.D10  C4×S3×D5  C20⋊D6  D6011C2  D42D15  Q83D15  C6.D30  C20.6S4
C4×D15 is a maximal quotient of
C40⋊S3  C30.4Q8  D303C4  C6.D30

36 conjugacy classes

class 1 2A2B2C 3 4A4B4C4D5A5B 6 10A10B12A12B15A15B15C15D20A20B20C20D30A30B30C30D60A···60H
order1222344445561010121215151515202020203030303060···60
size111515211151522222222222222222222···2

36 irreducible representations

dim11111222222222
type++++++++++
imageC1C2C2C2C4S3D5D6D10C4×S3D15C4×D5D30C4×D15
kernelC4×D15Dic15C60D30D15C20C12C10C6C5C4C3C2C1
# reps11114121224448

Matrix representation of C4×D15 in GL2(𝔽29) generated by

170
017
,
2821
2122
,
2223
87
G:=sub<GL(2,GF(29))| [17,0,0,17],[28,21,21,22],[22,8,23,7] >;

C4×D15 in GAP, Magma, Sage, TeX

C_4\times D_{15}
% in TeX

G:=Group("C4xD15");
// GroupNames label

G:=SmallGroup(120,27);
// by ID

G=gap.SmallGroup(120,27);
# by ID

G:=PCGroup([5,-2,-2,-2,-3,-5,26,323,2404]);
// Polycyclic

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

Export

Subgroup lattice of C4×D15 in TeX

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