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G = C242D15order 480 = 25·3·5

1st semidirect product of C24 and D15 acting via D15/C5=S3

non-abelian, soluble, monomial

Aliases: C242D15, C23.5D30, (C2×C10)⋊4S4, (C5×A4)⋊7D4, C222(C5⋊S4), A43(C5⋊D4), C53(A4⋊D4), C10.26(C2×S4), A4⋊Dic51C2, (C23×C10)⋊4S3, C22⋊(C157D4), (C22×A4)⋊2D5, (C2×A4).12D10, (C22×C10).17D6, (C10×A4).12C22, (C2×C5⋊S4)⋊2C2, (A4×C2×C10)⋊2C2, C2.11(C2×C5⋊S4), (C2×C10)⋊4(C3⋊D4), SmallGroup(480,1034)

Series: Derived Chief Lower central Upper central

C1C22C10×A4 — C242D15
C1C22C2×C10C5×A4C10×A4C2×C5⋊S4 — C242D15
C5×A4C10×A4 — C242D15
C1C2C22

Generators and relations for C242D15
 G = < a,b,c,d,e,f | a2=b2=c2=d2=e15=f2=1, faf=ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, bf=fb, ece-1=fdf=cd=dc, cf=fc, ede-1=c, fef=e-1 >

Subgroups: 1036 in 124 conjugacy classes, 21 normal (all characteristic)
C1, C2, C2 [×5], C3, C4 [×3], C22 [×2], C22 [×11], C5, S3, C6 [×2], C2×C4 [×3], D4 [×6], C23, C23 [×5], D5, C10, C10 [×4], Dic3, A4, D6, C2×C6, C15, C22⋊C4 [×3], C2×D4 [×3], C24, Dic5 [×3], D10 [×3], C2×C10 [×2], C2×C10 [×8], C3⋊D4, S4, C2×A4, C2×A4, D15, C30 [×2], C22≀C2, C2×Dic5 [×3], C5⋊D4 [×6], C22×D5, C22×C10, C22×C10 [×4], A4⋊C4, C2×S4, C22×A4, Dic15, C5×A4, D30, C2×C30, C23.D5 [×3], C2×C5⋊D4 [×3], C23×C10, A4⋊D4, C157D4, C5⋊S4, C10×A4, C10×A4, C242D5, A4⋊Dic5, C2×C5⋊S4, A4×C2×C10, C242D15
Quotients: C1, C2 [×3], C22, S3, D4, D5, D6, D10, C3⋊D4, S4, D15, C5⋊D4, C2×S4, D30, A4⋊D4, C157D4, C5⋊S4, C2×C5⋊S4, C242D15

Smallest permutation representation of C242D15
On 60 points
Generators in S60
(16 41)(17 42)(18 43)(19 44)(20 45)(21 31)(22 32)(23 33)(24 34)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)
(1 49)(2 50)(3 51)(4 52)(5 53)(6 54)(7 55)(8 56)(9 57)(10 58)(11 59)(12 60)(13 46)(14 47)(15 48)(16 41)(17 42)(18 43)(19 44)(20 45)(21 31)(22 32)(23 33)(24 34)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)
(2 50)(3 51)(5 53)(6 54)(8 56)(9 57)(11 59)(12 60)(14 47)(15 48)(16 41)(17 42)(19 44)(20 45)(22 32)(23 33)(25 35)(26 36)(28 38)(29 39)
(1 49)(3 51)(4 52)(6 54)(7 55)(9 57)(10 58)(12 60)(13 46)(15 48)(17 42)(18 43)(20 45)(21 31)(23 33)(24 34)(26 36)(27 37)(29 39)(30 40)
(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 37)(2 36)(3 35)(4 34)(5 33)(6 32)(7 31)(8 45)(9 44)(10 43)(11 42)(12 41)(13 40)(14 39)(15 38)(16 60)(17 59)(18 58)(19 57)(20 56)(21 55)(22 54)(23 53)(24 52)(25 51)(26 50)(27 49)(28 48)(29 47)(30 46)

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

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

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

46 conjugacy classes

class 1 2A2B2C2D2E2F 3 4A4B4C5A5B6A6B6C10A···10F10G···10N15A15B15C15D30A···30L
order122222234445566610···1010···101515151530···30
size112336608606060228882···26···688888···8

46 irreducible representations

dim11112222222222336666
type++++++++++++++++
imageC1C2C2C2S3D4D5D6D10C3⋊D4D15C5⋊D4D30C157D4S4C2×S4A4⋊D4C5⋊S4C2×C5⋊S4C242D15
kernelC242D15A4⋊Dic5C2×C5⋊S4A4×C2×C10C23×C10C5×A4C22×A4C22×C10C2×A4C2×C10C24A4C23C22C2×C10C10C5C22C2C1
# reps11111121224448221224

Matrix representation of C242D15 in GL5(𝔽61)

01000
10000
00100
00010
00001
,
600000
060000
00100
00010
00001
,
10000
01000
006000
006001
006010
,
10000
01000
000601
000600
001600
,
3919000
1939000
00001
00100
00010
,
2219000
4239000
00100
00001
00010

G:=sub<GL(5,GF(61))| [0,1,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[60,0,0,0,0,0,60,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,60,60,60,0,0,0,0,1,0,0,0,1,0],[1,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,60,60,60,0,0,1,0,0],[39,19,0,0,0,19,39,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0],[22,42,0,0,0,19,39,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0] >;

C242D15 in GAP, Magma, Sage, TeX

C_2^4\rtimes_2D_{15}
% in TeX

G:=Group("C2^4:2D15");
// GroupNames label

G:=SmallGroup(480,1034);
// by ID

G=gap.SmallGroup(480,1034);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-5,-2,2,85,451,3364,10085,1286,5886,2232]);
// Polycyclic

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

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