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## G = C22×C5⋊S4order 480 = 25·3·5

### Direct product of C22 and C5⋊S4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C5×A4 — C22×C5⋊S4
 Chief series C1 — C22 — C2×C10 — C5×A4 — C5⋊S4 — C2×C5⋊S4 — C22×C5⋊S4
 Lower central C5×A4 — C22×C5⋊S4
 Upper central C1 — C22

Generators and relations for C22×C5⋊S4
G = < a,b,c,d,e,f,g | a2=b2=c5=d2=e2=f3=g2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, bd=db, be=eb, bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, gcg=c-1, fdf-1=gdg=de=ed, fef-1=d, eg=ge, gfg=f-1 >

Subgroups: 2152 in 262 conjugacy classes, 41 normal (13 characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, S3, C6, C2×C4, D4, C23, C23, D5, C10, C10, A4, D6, C2×C6, C15, C22×C4, C2×D4, C24, C24, Dic5, D10, C2×C10, C2×C10, S4, C2×A4, C22×S3, D15, C30, C22×D4, C2×Dic5, C5⋊D4, C22×D5, C22×C10, C22×C10, C2×S4, C22×A4, C5×A4, D30, C2×C30, C22×Dic5, C2×C5⋊D4, C23×D5, C23×C10, C22×S4, C5⋊S4, C10×A4, C22×D15, C22×C5⋊D4, C2×C5⋊S4, A4×C2×C10, C22×C5⋊S4
Quotients: C1, C2, C22, S3, C23, D5, D6, D10, S4, C22×S3, D15, C22×D5, C2×S4, D30, C22×S4, C5⋊S4, C22×D15, C2×C5⋊S4, C22×C5⋊S4

Smallest permutation representation of C22×C5⋊S4
On 60 points
Generators in S60
(1 43)(2 44)(3 45)(4 41)(5 42)(6 29)(7 30)(8 26)(9 27)(10 28)(11 36)(12 37)(13 38)(14 39)(15 40)(16 51)(17 52)(18 53)(19 54)(20 55)(21 46)(22 47)(23 48)(24 49)(25 50)(31 56)(32 57)(33 58)(34 59)(35 60)
(1 13)(2 14)(3 15)(4 11)(5 12)(6 59)(7 60)(8 56)(9 57)(10 58)(16 21)(17 22)(18 23)(19 24)(20 25)(26 31)(27 32)(28 33)(29 34)(30 35)(36 41)(37 42)(38 43)(39 44)(40 45)(46 51)(47 52)(48 53)(49 54)(50 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 13)(2 14)(3 15)(4 11)(5 12)(6 59)(7 60)(8 56)(9 57)(10 58)(26 31)(27 32)(28 33)(29 34)(30 35)(36 41)(37 42)(38 43)(39 44)(40 45)
(6 59)(7 60)(8 56)(9 57)(10 58)(16 21)(17 22)(18 23)(19 24)(20 25)(26 31)(27 32)(28 33)(29 34)(30 35)(46 51)(47 52)(48 53)(49 54)(50 55)
(1 28 18)(2 29 19)(3 30 20)(4 26 16)(5 27 17)(6 54 44)(7 55 45)(8 51 41)(9 52 42)(10 53 43)(11 31 21)(12 32 22)(13 33 23)(14 34 24)(15 35 25)(36 56 46)(37 57 47)(38 58 48)(39 59 49)(40 60 50)
(1 13)(2 12)(3 11)(4 15)(5 14)(6 47)(7 46)(8 50)(9 49)(10 48)(16 35)(17 34)(18 33)(19 32)(20 31)(21 30)(22 29)(23 28)(24 27)(25 26)(36 45)(37 44)(38 43)(39 42)(40 41)(51 60)(52 59)(53 58)(54 57)(55 56)

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

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

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

52 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 3 4A 4B 4C 4D 5A 5B 6A 6B 6C 10A ··· 10F 10G ··· 10N 15A 15B 15C 15D 30A ··· 30L order 1 2 2 2 2 2 2 2 2 2 2 2 3 4 4 4 4 5 5 6 6 6 10 ··· 10 10 ··· 10 15 15 15 15 30 ··· 30 size 1 1 1 1 3 3 3 3 30 30 30 30 8 30 30 30 30 2 2 8 8 8 2 ··· 2 6 ··· 6 8 8 8 8 8 ··· 8

52 irreducible representations

 dim 1 1 1 2 2 2 2 2 2 3 3 6 6 type + + + + + + + + + + + + + image C1 C2 C2 S3 D5 D6 D10 D15 D30 S4 C2×S4 C5⋊S4 C2×C5⋊S4 kernel C22×C5⋊S4 C2×C5⋊S4 A4×C2×C10 C23×C10 C22×A4 C22×C10 C2×A4 C24 C23 C2×C10 C10 C22 C2 # reps 1 6 1 1 2 3 6 4 12 2 6 2 6

Matrix representation of C22×C5⋊S4 in GL5(𝔽61)

 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
,
 60 0 0 0 0 0 60 0 0 0 0 0 60 0 0 0 0 0 60 0 0 0 0 0 60
,
 60 60 0 0 0 45 44 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 0 0 0 0 0 1 0 0 0 0 60 60
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 60 0 0 0 2 0 60
,
 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 2 60 59 0 0 29 32 1
,
 31 23 0 0 0 6 30 0 0 0 0 0 60 0 0 0 0 59 1 2 0 0 0 0 60

G:=sub<GL(5,GF(61))| [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],[60,0,0,0,0,0,60,0,0,0,0,0,60,0,0,0,0,0,60,0,0,0,0,0,60],[60,45,0,0,0,60,44,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,60,0,0,0,0,0,1,60,0,0,0,0,60],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,2,0,0,0,60,0,0,0,0,0,60],[1,0,0,0,0,0,1,0,0,0,0,0,0,2,29,0,0,1,60,32,0,0,0,59,1],[31,6,0,0,0,23,30,0,0,0,0,0,60,59,0,0,0,0,1,0,0,0,0,2,60] >;

C22×C5⋊S4 in GAP, Magma, Sage, TeX

C_2^2\times C_5\rtimes S_4
% in TeX

G:=Group("C2^2xC5:S4");
// GroupNames label

G:=SmallGroup(480,1199);
// by ID

G=gap.SmallGroup(480,1199);
# by ID

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

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

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