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G = C3xC15:D4order 360 = 23·32·5

Direct product of C3 and C15:D4

direct product, metabelian, supersoluble, monomial

Aliases: C3xC15:D4, C30.31D6, Dic15:4C6, (C3xC15):4D4, (C6xD5):4S3, (S3xC6):1D5, (C6xD5):1C6, D6:1(C3xD5), C15:1(C3xD4), C6.4(C6xD5), (S3xC10):1C6, (S3xC30):1C2, D10:1(C3xS3), C10.4(S3xC6), C30.4(C2xC6), C6.31(S3xD5), (C3xC6).16D10, C32:7(C5:D4), C15:10(C3:D4), (C3xDic15):4C2, (C3xC30).4C22, (D5xC3xC6):1C2, C5:2(C3xC3:D4), C2.4(C3xS3xD5), C3:2(C3xC5:D4), SmallGroup(360,61)

Series: Derived Chief Lower central Upper central

C1C30 — C3xC15:D4
C1C5C15C30C3xC30D5xC3xC6 — C3xC15:D4
C15C30 — C3xC15:D4
C1C6

Generators and relations for C3xC15:D4
 G = < a,b,c,d | a3=b15=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=b-1, dbd=b4, dcd=c-1 >

Subgroups: 276 in 74 conjugacy classes, 28 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C22, C5, S3, C6, C6, D4, C32, D5, C10, C10, Dic3, C12, D6, C2xC6, C15, C15, C3xS3, C3xC6, C3xC6, Dic5, D10, C2xC10, C3:D4, C3xD4, C5xS3, C3xD5, C30, C30, C3xDic3, S3xC6, C62, C5:D4, C3xC15, C3xDic5, Dic15, C6xD5, C6xD5, S3xC10, C2xC30, C3xC3:D4, C32xD5, S3xC15, C3xC30, C15:D4, C3xC5:D4, C3xDic15, D5xC3xC6, S3xC30, C3xC15:D4
Quotients: C1, C2, C3, C22, S3, C6, D4, D5, D6, C2xC6, C3xS3, D10, C3:D4, C3xD4, C3xD5, S3xC6, C5:D4, S3xD5, C6xD5, C3xC3:D4, C15:D4, C3xC5:D4, C3xS3xD5, C3xC15:D4

Smallest permutation representation of C3xC15:D4
On 60 points
Generators in S60
(1 6 11)(2 7 12)(3 8 13)(4 9 14)(5 10 15)(16 21 26)(17 22 27)(18 23 28)(19 24 29)(20 25 30)(31 41 36)(32 42 37)(33 43 38)(34 44 39)(35 45 40)(46 56 51)(47 57 52)(48 58 53)(49 59 54)(50 60 55)
(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 38 16 55)(2 37 17 54)(3 36 18 53)(4 35 19 52)(5 34 20 51)(6 33 21 50)(7 32 22 49)(8 31 23 48)(9 45 24 47)(10 44 25 46)(11 43 26 60)(12 42 27 59)(13 41 28 58)(14 40 29 57)(15 39 30 56)
(2 5)(3 9)(4 13)(7 10)(8 14)(12 15)(17 20)(18 24)(19 28)(22 25)(23 29)(27 30)(31 57)(32 46)(33 50)(34 54)(35 58)(36 47)(37 51)(38 55)(39 59)(40 48)(41 52)(42 56)(43 60)(44 49)(45 53)

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

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

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

63 conjugacy classes

class 1 2A2B2C3A3B3C3D3E 4 5A5B6A6B6C6D6E6F6G6H···6O10A10B10C10D10E10F12A12B15A15B15C15D15E···15J30A30B30C30D30E···30J30K···30R
order12223333345566666666···610101010101012121515151515···153030303030···3030···30
size11610112223022112226610···10226666303022224···422224···46···6

63 irreducible representations

dim11111111222222222222224444
type++++++++++-
imageC1C2C2C2C3C6C6C6S3D4D5D6C3xS3D10C3:D4C3xD4C3xD5S3xC6C5:D4C6xD5C3xC3:D4C3xC5:D4S3xD5C15:D4C3xS3xD5C3xC15:D4
kernelC3xC15:D4C3xDic15D5xC3xC6S3xC30C15:D4Dic15C6xD5S3xC10C6xD5C3xC15S3xC6C30D10C3xC6C15C15D6C10C32C6C5C3C6C3C2C1
# reps11112222112122224244482244

Matrix representation of C3xC15:D4 in GL4(F61) generated by

1000
0100
00470
00047
,
444400
176000
00130
002047
,
144500
394700
006041
00551
,
1000
176000
0010
00660
G:=sub<GL(4,GF(61))| [1,0,0,0,0,1,0,0,0,0,47,0,0,0,0,47],[44,17,0,0,44,60,0,0,0,0,13,20,0,0,0,47],[14,39,0,0,45,47,0,0,0,0,60,55,0,0,41,1],[1,17,0,0,0,60,0,0,0,0,1,6,0,0,0,60] >;

C3xC15:D4 in GAP, Magma, Sage, TeX

C_3\times C_{15}\rtimes D_4
% in TeX

G:=Group("C3xC15:D4");
// GroupNames label

G:=SmallGroup(360,61);
// by ID

G=gap.SmallGroup(360,61);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-3,-5,169,730,10373]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^15=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,d*b*d=b^4,d*c*d=c^-1>;
// generators/relations

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