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G = C3×C157D4order 360 = 23·32·5

Direct product of C3 and C157D4

direct product, metabelian, supersoluble, monomial

Aliases: C3×C157D4, D302C6, C622D5, C30.55D6, C6.23D30, Dic151C6, (C6×C30)⋊4C2, (C2×C30)⋊8S3, (C2×C30)⋊6C6, (C2×C6)⋊3D15, C157(C3×D4), (C3×C15)⋊23D4, (C6×D15)⋊2C2, C6.12(C6×D5), C2.5(C6×D15), C10.12(S3×C6), C30.12(C2×C6), (C3×C6).31D10, C223(C3×D15), C329(C5⋊D4), C1513(C3⋊D4), (C3×Dic15)⋊1C2, (C3×C30).41C22, C53(C3×C3⋊D4), (C2×C6)⋊4(C3×D5), C33(C3×C5⋊D4), (C2×C10)⋊6(C3×S3), SmallGroup(360,104)

Series: Derived Chief Lower central Upper central

C1C30 — C3×C157D4
C1C5C15C30C3×C30C6×D15 — C3×C157D4
C15C30 — C3×C157D4
C1C6C2×C6

Generators and relations for C3×C157D4
 G = < a,b,c,d | a3=b15=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >

Subgroups: 300 in 74 conjugacy classes, 30 normal (all characteristic)
C1, C2, C2 [×2], C3 [×2], C3, C4, C22, C22, C5, S3, C6 [×2], C6 [×6], D4, C32, D5, C10, C10, Dic3, C12, D6, C2×C6 [×2], C2×C6 [×2], C15 [×2], C15, C3×S3, C3×C6, C3×C6, Dic5, D10, C2×C10, C3⋊D4, C3×D4, C3×D5, D15, C30 [×2], C30 [×5], C3×Dic3, S3×C6, C62, C5⋊D4, C3×C15, C3×Dic5, Dic15, C6×D5, D30, C2×C30 [×2], C2×C30, C3×C3⋊D4, C3×D15, C3×C30, C3×C30, C3×C5⋊D4, C157D4, C3×Dic15, C6×D15, C6×C30, C3×C157D4
Quotients: C1, C2 [×3], C3, C22, S3, C6 [×3], D4, D5, D6, C2×C6, C3×S3, D10, C3⋊D4, C3×D4, C3×D5, D15, S3×C6, C5⋊D4, C6×D5, D30, C3×C3⋊D4, C3×D15, C3×C5⋊D4, C157D4, C6×D15, C3×C157D4

Smallest permutation representation of C3×C157D4
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 58 30 31)(2 57 16 45)(3 56 17 44)(4 55 18 43)(5 54 19 42)(6 53 20 41)(7 52 21 40)(8 51 22 39)(9 50 23 38)(10 49 24 37)(11 48 25 36)(12 47 26 35)(13 46 27 34)(14 60 28 33)(15 59 29 32)
(1 31)(2 45)(3 44)(4 43)(5 42)(6 41)(7 40)(8 39)(9 38)(10 37)(11 36)(12 35)(13 34)(14 33)(15 32)(16 57)(17 56)(18 55)(19 54)(20 53)(21 52)(22 51)(23 50)(24 49)(25 48)(26 47)(27 46)(28 60)(29 59)(30 58)

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

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

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

99 conjugacy classes

class 1 2A2B2C3A3B3C3D3E 4 5A5B6A6B6C···6M6N6O10A···10F12A12B15A···15P30A···30AV
order122233333455666···66610···10121215···1530···30
size11230112223022112···230302···230302···22···2

99 irreducible representations

dim1111111122222222222222222222
type+++++++++++
imageC1C2C2C2C3C6C6C6S3D4D5D6C3×S3D10C3⋊D4C3×D4C3×D5D15S3×C6C5⋊D4C6×D5D30C3×C3⋊D4C3×D15C3×C5⋊D4C157D4C6×D15C3×C157D4
kernelC3×C157D4C3×Dic15C6×D15C6×C30C157D4Dic15D30C2×C30C2×C30C3×C15C62C30C2×C10C3×C6C15C15C2×C6C2×C6C10C32C6C6C5C22C3C3C2C1
# reps11112222112122224424444888816

Matrix representation of C3×C157D4 in GL2(𝔽31) generated by

250
025
,
183
1316
,
06
50
,
2629
125
G:=sub<GL(2,GF(31))| [25,0,0,25],[18,13,3,16],[0,5,6,0],[26,12,29,5] >;

C3×C157D4 in GAP, Magma, Sage, TeX

C_3\times C_{15}\rtimes_7D_4
% in TeX

G:=Group("C3xC15:7D4");
// GroupNames label

G:=SmallGroup(360,104);
// by ID

G=gap.SmallGroup(360,104);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-3,-5,169,1444,10373]);
// 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=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations

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