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## G = C22×S3×D5order 240 = 24·3·5

### Direct product of C22, S3 and D5

Series: Derived Chief Lower central Upper central

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

Generators and relations for C22×S3×D5
G = < a,b,c,d,e,f | a2=b2=c3=d2=e5=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, dcd=c-1, ce=ec, cf=fc, de=ed, df=fd, fef=e-1 >

Subgroups: 1200 in 268 conjugacy classes, 104 normal (14 characteristic)
C1, C2 [×3], C2 [×12], C3, C22, C22 [×34], C5, S3 [×4], S3 [×4], C6 [×3], C6 [×4], C23 [×15], D5 [×4], D5 [×4], C10 [×3], C10 [×4], D6 [×6], D6 [×22], C2×C6, C2×C6 [×6], C15, C24, D10 [×6], D10 [×22], C2×C10, C2×C10 [×6], C22×S3, C22×S3 [×13], C22×C6, C5×S3 [×4], C3×D5 [×4], D15 [×4], C30 [×3], C22×D5, C22×D5 [×13], C22×C10, S3×C23, S3×D5 [×16], C6×D5 [×6], S3×C10 [×6], D30 [×6], C2×C30, C23×D5, C2×S3×D5 [×12], D5×C2×C6, S3×C2×C10, C22×D15, C22×S3×D5
Quotients: C1, C2 [×15], C22 [×35], S3, C23 [×15], D5, D6 [×7], C24, D10 [×7], C22×S3 [×7], C22×D5 [×7], S3×C23, S3×D5, C23×D5, C2×S3×D5 [×3], C22×S3×D5

Smallest permutation representation of C22×S3×D5
On 60 points
Generators in S60
(1 34)(2 35)(3 31)(4 32)(5 33)(6 36)(7 37)(8 38)(9 39)(10 40)(11 41)(12 42)(13 43)(14 44)(15 45)(16 46)(17 47)(18 48)(19 49)(20 50)(21 51)(22 52)(23 53)(24 54)(25 55)(26 56)(27 57)(28 58)(29 59)(30 60)
(1 19)(2 20)(3 16)(4 17)(5 18)(6 21)(7 22)(8 23)(9 24)(10 25)(11 26)(12 27)(13 28)(14 29)(15 30)(31 46)(32 47)(33 48)(34 49)(35 50)(36 51)(37 52)(38 53)(39 54)(40 55)(41 56)(42 57)(43 58)(44 59)(45 60)
(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)(2 35)(3 31)(4 32)(5 33)(6 41)(7 42)(8 43)(9 44)(10 45)(11 36)(12 37)(13 38)(14 39)(15 40)(16 46)(17 47)(18 48)(19 49)(20 50)(21 56)(22 57)(23 58)(24 59)(25 60)(26 51)(27 52)(28 53)(29 54)(30 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 33)(2 32)(3 31)(4 35)(5 34)(6 36)(7 40)(8 39)(9 38)(10 37)(11 41)(12 45)(13 44)(14 43)(15 42)(16 46)(17 50)(18 49)(19 48)(20 47)(21 51)(22 55)(23 54)(24 53)(25 52)(26 56)(27 60)(28 59)(29 58)(30 57)

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

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

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

C22×S3×D5 is a maximal subgroup of
D30.27D4  D64D20  D304D4  D305D4  D308D4
C22×S3×D5 is a maximal quotient of
D20.38D6  D20.39D6  C30.C24  D2024D6  D2025D6  D2026D6  D2029D6  C15⋊2- 1+4  D30.C23  D2013D6  D2014D6  D1214D10  D20.29D6  C30.33C24  D12.29D10  D2016D6  D2017D6  C15⋊2+ 1+4

48 conjugacy classes

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

48 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + + image C1 C2 C2 C2 C2 S3 D5 D6 D6 D10 D10 S3×D5 C2×S3×D5 kernel C22×S3×D5 C2×S3×D5 D5×C2×C6 S3×C2×C10 C22×D15 C22×D5 C22×S3 D10 C2×C10 D6 C2×C6 C22 C2 # reps 1 12 1 1 1 1 2 6 1 12 2 2 6

Matrix representation of C22×S3×D5 in GL4(𝔽31) generated by

 30 0 0 0 0 30 0 0 0 0 1 0 0 0 0 1
,
 30 0 0 0 0 30 0 0 0 0 30 0 0 0 0 30
,
 1 0 0 0 0 1 0 0 0 0 29 26 0 0 13 1
,
 30 0 0 0 0 30 0 0 0 0 1 0 0 0 18 30
,
 30 1 0 0 17 13 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 14 30 0 0 0 0 1 0 0 0 0 1
G:=sub<GL(4,GF(31))| [30,0,0,0,0,30,0,0,0,0,1,0,0,0,0,1],[30,0,0,0,0,30,0,0,0,0,30,0,0,0,0,30],[1,0,0,0,0,1,0,0,0,0,29,13,0,0,26,1],[30,0,0,0,0,30,0,0,0,0,1,18,0,0,0,30],[30,17,0,0,1,13,0,0,0,0,1,0,0,0,0,1],[1,14,0,0,0,30,0,0,0,0,1,0,0,0,0,1] >;

C22×S3×D5 in GAP, Magma, Sage, TeX

C_2^2\times S_3\times D_5
% in TeX

G:=Group("C2^2xS3xD5");
// GroupNames label

G:=SmallGroup(240,202);
// by ID

G=gap.SmallGroup(240,202);
# by ID

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

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

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