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## G = C5×C3⋊D4order 120 = 23·3·5

### Direct product of C5 and C3⋊D4

Aliases: C5×C3⋊D4, C159D4, D62C10, Dic3⋊C10, C10.17D6, C30.22C22, C32(C5×D4), (C2×C10)⋊3S3, (C2×C6)⋊2C10, (C2×C30)⋊6C2, (S3×C10)⋊5C2, C2.5(S3×C10), C6.5(C2×C10), C222(C5×S3), (C5×Dic3)⋊4C2, SmallGroup(120,25)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C5×C3⋊D4
 Chief series C1 — C3 — C6 — C30 — S3×C10 — C5×C3⋊D4
 Lower central C3 — C6 — C5×C3⋊D4
 Upper central C1 — C10 — C2×C10

Generators and relations for C5×C3⋊D4
G = < a,b,c,d | a5=b3=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >

Smallest permutation representation of C5×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 45 33)(2 41 34)(3 42 35)(4 43 31)(5 44 32)(6 17 15)(7 18 11)(8 19 12)(9 20 13)(10 16 14)(21 58 51)(22 59 52)(23 60 53)(24 56 54)(25 57 55)(26 47 40)(27 48 36)(28 49 37)(29 50 38)(30 46 39)
(1 6 26 56)(2 7 27 57)(3 8 28 58)(4 9 29 59)(5 10 30 60)(11 48 25 41)(12 49 21 42)(13 50 22 43)(14 46 23 44)(15 47 24 45)(16 39 53 32)(17 40 54 33)(18 36 55 34)(19 37 51 35)(20 38 52 31)
(6 56)(7 57)(8 58)(9 59)(10 60)(11 55)(12 51)(13 52)(14 53)(15 54)(16 23)(17 24)(18 25)(19 21)(20 22)(31 43)(32 44)(33 45)(34 41)(35 42)(36 48)(37 49)(38 50)(39 46)(40 47)

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

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

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

C5×C3⋊D4 is a maximal subgroup of   C30.C23  Dic3.D10  D10⋊D6  C5×S3×D4

45 conjugacy classes

 class 1 2A 2B 2C 3 4 5A 5B 5C 5D 6A 6B 6C 10A 10B 10C 10D 10E 10F 10G 10H 10I 10J 10K 10L 15A 15B 15C 15D 20A 20B 20C 20D 30A ··· 30L order 1 2 2 2 3 4 5 5 5 5 6 6 6 10 10 10 10 10 10 10 10 10 10 10 10 15 15 15 15 20 20 20 20 30 ··· 30 size 1 1 2 6 2 6 1 1 1 1 2 2 2 1 1 1 1 2 2 2 2 6 6 6 6 2 2 2 2 6 6 6 6 2 ··· 2

45 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + + + + + image C1 C2 C2 C2 C5 C10 C10 C10 S3 D4 D6 C3⋊D4 C5×S3 C5×D4 S3×C10 C5×C3⋊D4 kernel C5×C3⋊D4 C5×Dic3 S3×C10 C2×C30 C3⋊D4 Dic3 D6 C2×C6 C2×C10 C15 C10 C5 C22 C3 C2 C1 # reps 1 1 1 1 4 4 4 4 1 1 1 2 4 4 4 8

Matrix representation of C5×C3⋊D4 in GL2(𝔽31) generated by

 4 0 0 4
,
 1 9 10 29
,
 20 29 30 11
,
 1 9 0 30
G:=sub<GL(2,GF(31))| [4,0,0,4],[1,10,9,29],[20,30,29,11],[1,0,9,30] >;

C5×C3⋊D4 in GAP, Magma, Sage, TeX

C_5\times C_3\rtimes D_4
% in TeX

G:=Group("C5xC3:D4");
// GroupNames label

G:=SmallGroup(120,25);
// by ID

G=gap.SmallGroup(120,25);
# by ID

G:=PCGroup([5,-2,-2,-5,-2,-3,221,2004]);
// Polycyclic

G:=Group<a,b,c,d|a^5=b^3=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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