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G = C6.D30order 360 = 23·32·5

3rd non-split extension by C6 of D30 acting via D30/D15=C2

metabelian, supersoluble, monomial, A-group

Aliases: C6.3D30, C30.23D6, Dic32D15, Dic155S3, C10.3S32, C3⋊D156C4, C31(C4×D15), C6.3(S3×D5), C1510(C4×S3), C323(C4×D5), C2.3(S3×D15), (C3×C6).3D10, (C5×Dic3)⋊2S3, (C3×Dic3)⋊3D5, C52(C6.D6), C31(D30.C2), (C3×Dic15)⋊6C2, (Dic3×C15)⋊3C2, (C3×C30).17C22, (C3×C15)⋊21(C2×C4), (C2×C3⋊D15).3C2, SmallGroup(360,79)

Series: Derived Chief Lower central Upper central

C1C3×C15 — C6.D30
C1C5C15C3×C15C3×C30C3×Dic15 — C6.D30
C3×C15 — C6.D30
C1C2

Generators and relations for C6.D30
 G = < a,b,c | a6=c2=1, b30=a3, bab-1=cac=a-1, cbc=b29 >

Subgroups: 580 in 74 conjugacy classes, 26 normal (24 characteristic)
C1, C2, C2 [×2], C3 [×2], C3, C4 [×2], C22, C5, S3 [×6], C6 [×2], C6, C2×C4, C32, D5 [×2], C10, Dic3, Dic3, C12 [×2], D6 [×3], C15 [×2], C15, C3⋊S3 [×2], C3×C6, Dic5, C20, D10, C4×S3 [×2], D15 [×6], C30 [×2], C30, C3×Dic3, C3×Dic3, C2×C3⋊S3, C4×D5, C3×C15, C5×Dic3, C3×Dic5, Dic15, C60, D30 [×3], C6.D6, C3⋊D15 [×2], C3×C30, D30.C2, C4×D15, Dic3×C15, C3×Dic15, C2×C3⋊D15, C6.D30
Quotients: C1, C2 [×3], C4 [×2], C22, S3 [×2], C2×C4, D5, D6 [×2], D10, C4×S3 [×2], D15, S32, C4×D5, S3×D5, D30, C6.D6, D30.C2, C4×D15, S3×D15, C6.D30

Smallest permutation representation of C6.D30
On 60 points
Generators in S60
(1 11 21 31 41 51)(2 52 42 32 22 12)(3 13 23 33 43 53)(4 54 44 34 24 14)(5 15 25 35 45 55)(6 56 46 36 26 16)(7 17 27 37 47 57)(8 58 48 38 28 18)(9 19 29 39 49 59)(10 60 50 40 30 20)
(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 45)(2 14)(3 43)(4 12)(5 41)(6 10)(7 39)(9 37)(11 35)(13 33)(15 31)(16 60)(17 29)(18 58)(19 27)(20 56)(21 25)(22 54)(24 52)(26 50)(28 48)(30 46)(32 44)(34 42)(36 40)(47 59)(49 57)(51 55)

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

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

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

54 conjugacy classes

class 1 2A2B2C3A3B3C4A4B4C4D5A5B6A6B6C10A10B12A12B12C12D15A15B15C15D15E···15J20A20B20C20D30A30B30C30D30E···30J60A···60H
order12223334444556661010121212121515151515···15202020203030303030···3060···60
size114545224331515222242266303022224···4666622224···46···6

54 irreducible representations

dim111112222222222444444
type+++++++++++++++++
imageC1C2C2C2C4S3S3D5D6D10C4×S3D15C4×D5D30C4×D15S32S3×D5C6.D6D30.C2S3×D15C6.D30
kernelC6.D30Dic3×C15C3×Dic15C2×C3⋊D15C3⋊D15C5×Dic3Dic15C3×Dic3C30C3×C6C15Dic3C32C6C3C10C6C5C3C2C1
# reps111141122244448121244

Matrix representation of C6.D30 in GL6(𝔽61)

100000
010000
0060000
0006000
0000601
0000600
,
17600000
100000
0005000
00115000
000001
000010
,
17600000
44440000
0016000
0006000
000001
000010

G:=sub<GL(6,GF(61))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,60,0,0,0,0,1,0],[17,1,0,0,0,0,60,0,0,0,0,0,0,0,0,11,0,0,0,0,50,50,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[17,44,0,0,0,0,60,44,0,0,0,0,0,0,1,0,0,0,0,0,60,60,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

C6.D30 in GAP, Magma, Sage, TeX

C_6.D_{30}
% in TeX

G:=Group("C6.D30");
// GroupNames label

G:=SmallGroup(360,79);
// by ID

G=gap.SmallGroup(360,79);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-3,-5,24,31,201,1444,10373]);
// Polycyclic

G:=Group<a,b,c|a^6=c^2=1,b^30=a^3,b*a*b^-1=c*a*c=a^-1,c*b*c=b^29>;
// generators/relations

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