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## G = D5×C3⋊D4order 240 = 24·3·5

### Direct product of D5 and C3⋊D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — D5×C3⋊D4
 Chief series C1 — C5 — C15 — C30 — C6×D5 — C2×S3×D5 — D5×C3⋊D4
 Lower central C15 — C30 — D5×C3⋊D4
 Upper central C1 — C2 — C22

Generators and relations for D5×C3⋊D4
G = < a,b,c,d,e | a5=b2=c3=d4=e2=1, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=ece=c-1, ede=d-1 >

Subgroups: 528 in 108 conjugacy classes, 36 normal (32 characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, S3, C6, C6, C2×C4, D4, C23, D5, D5, C10, C10, Dic3, Dic3, D6, D6, C2×C6, C2×C6, C15, C2×D4, Dic5, C20, D10, D10, C2×C10, C2×C10, C2×Dic3, C3⋊D4, C3⋊D4, C22×S3, C22×C6, C5×S3, C3×D5, C3×D5, D15, C30, C30, C4×D5, D20, C5⋊D4, C5×D4, C22×D5, C22×D5, C2×C3⋊D4, C5×Dic3, Dic15, S3×D5, C6×D5, C6×D5, S3×C10, D30, C2×C30, D4×D5, D5×Dic3, C15⋊D4, C3⋊D20, C5×C3⋊D4, C157D4, C2×S3×D5, D5×C2×C6, D5×C3⋊D4
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, D10, C3⋊D4, C22×S3, C22×D5, C2×C3⋊D4, S3×D5, D4×D5, C2×S3×D5, D5×C3⋊D4

Smallest permutation representation of D5×C3⋊D4
On 60 points
Generators in S60
(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 18)(2 17)(3 16)(4 20)(5 19)(6 21)(7 25)(8 24)(9 23)(10 22)(11 26)(12 30)(13 29)(14 28)(15 27)(31 46)(32 50)(33 49)(34 48)(35 47)(36 51)(37 55)(38 54)(39 53)(40 52)(41 56)(42 60)(43 59)(44 58)(45 57)
(1 9 14)(2 10 15)(3 6 11)(4 7 12)(5 8 13)(16 21 26)(17 22 27)(18 23 28)(19 24 29)(20 25 30)(31 36 41)(32 37 42)(33 38 43)(34 39 44)(35 40 45)(46 51 56)(47 52 57)(48 53 58)(49 54 59)(50 55 60)
(1 34 19 49)(2 35 20 50)(3 31 16 46)(4 32 17 47)(5 33 18 48)(6 41 21 56)(7 42 22 57)(8 43 23 58)(9 44 24 59)(10 45 25 60)(11 36 26 51)(12 37 27 52)(13 38 28 53)(14 39 29 54)(15 40 30 55)
(6 11)(7 12)(8 13)(9 14)(10 15)(21 26)(22 27)(23 28)(24 29)(25 30)(31 46)(32 47)(33 48)(34 49)(35 50)(36 56)(37 57)(38 58)(39 59)(40 60)(41 51)(42 52)(43 53)(44 54)(45 55)

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

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

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

36 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3 4A 4B 5A 5B 6A 6B 6C 6D 6E 6F 6G 10A 10B 10C 10D 10E 10F 15A 15B 20A 20B 30A ··· 30F order 1 2 2 2 2 2 2 2 3 4 4 5 5 6 6 6 6 6 6 6 10 10 10 10 10 10 15 15 20 20 30 ··· 30 size 1 1 2 5 5 6 10 30 2 6 30 2 2 2 2 2 10 10 10 10 2 2 4 4 12 12 4 4 12 12 4 ··· 4

36 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 S3 D4 D5 D6 D6 D10 D10 D10 C3⋊D4 S3×D5 D4×D5 C2×S3×D5 D5×C3⋊D4 kernel D5×C3⋊D4 D5×Dic3 C15⋊D4 C3⋊D20 C5×C3⋊D4 C15⋊7D4 C2×S3×D5 D5×C2×C6 C22×D5 C3×D5 C3⋊D4 D10 C2×C10 Dic3 D6 C2×C6 D5 C22 C3 C2 C1 # reps 1 1 1 1 1 1 1 1 1 2 2 2 1 2 2 2 4 2 2 2 4

Matrix representation of D5×C3⋊D4 in GL4(𝔽61) generated by

 1 0 0 0 0 1 0 0 0 0 0 1 0 0 60 17
,
 60 0 0 0 0 60 0 0 0 0 0 1 0 0 1 0
,
 0 60 0 0 1 60 0 0 0 0 1 0 0 0 0 1
,
 9 9 0 0 18 52 0 0 0 0 60 0 0 0 0 60
,
 1 60 0 0 0 60 0 0 0 0 1 0 0 0 0 1
G:=sub<GL(4,GF(61))| [1,0,0,0,0,1,0,0,0,0,0,60,0,0,1,17],[60,0,0,0,0,60,0,0,0,0,0,1,0,0,1,0],[0,1,0,0,60,60,0,0,0,0,1,0,0,0,0,1],[9,18,0,0,9,52,0,0,0,0,60,0,0,0,0,60],[1,0,0,0,60,60,0,0,0,0,1,0,0,0,0,1] >;

D5×C3⋊D4 in GAP, Magma, Sage, TeX

D_5\times C_3\rtimes D_4
% in TeX

G:=Group("D5xC3:D4");
// GroupNames label

G:=SmallGroup(240,149);
// by ID

G=gap.SmallGroup(240,149);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-5,116,490,6917]);
// Polycyclic

G:=Group<a,b,c,d,e|a^5=b^2=c^3=d^4=e^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d^-1=e*c*e=c^-1,e*d*e=d^-1>;
// generators/relations

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