direct product, non-abelian, soluble, monomial
Aliases: C22×C5⋊S4, C23⋊D30, C24⋊3D15, C10⋊2(C2×S4), (C2×C10)⋊5S4, C5⋊2(C22×S4), (C2×A4)⋊2D10, (C5×A4)⋊2C23, (C23×C10)⋊5S3, A4⋊2(C22×D5), (C22×A4)⋊3D5, (C22×C10)⋊3D6, C22⋊(C22×D15), (C10×A4)⋊2C22, (A4×C2×C10)⋊3C2, (C2×C10)⋊3(C22×S3), SmallGroup(480,1199)
Series: Derived ►Chief ►Lower central ►Upper central
| C5×A4 — C22×C5⋊S4 |
Subgroups: 2152 in 262 conjugacy classes, 41 normal (13 characteristic)
C1, C2 [×3], C2 [×8], C3, C4 [×4], C22 [×2], C22 [×26], C5, S3 [×4], C6 [×3], C2×C4 [×6], D4 [×16], C23 [×3], C23 [×14], D5 [×4], C10 [×3], C10 [×4], A4, D6 [×6], C2×C6, C15, C22×C4, C2×D4 [×12], C24, C24, Dic5 [×4], D10 [×16], C2×C10 [×2], C2×C10 [×10], S4 [×4], C2×A4 [×3], C22×S3, D15 [×4], C30 [×3], C22×D4, C2×Dic5 [×6], C5⋊D4 [×16], C22×D5 [×10], C22×C10 [×3], C22×C10 [×4], C2×S4 [×6], C22×A4, C5×A4, D30 [×6], C2×C30, C22×Dic5, C2×C5⋊D4 [×12], C23×D5, C23×C10, C22×S4, C5⋊S4 [×4], C10×A4 [×3], C22×D15, C22×C5⋊D4, C2×C5⋊S4 [×6], A4×C2×C10, C22×C5⋊S4
Quotients:
C1, C2 [×7], C22 [×7], S3, C23, D5, D6 [×3], D10 [×3], S4, C22×S3, D15, C22×D5, C2×S4 [×3], D30 [×3], C22×S4, C5⋊S4, C22×D15, C2×C5⋊S4 [×3], C22×C5⋊S4
Generators and relations
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 >
►On 60 points
Generators in S60
(1 43)(2 44)(3 45)(4 41)(5 42)(6 26)(7 27)(8 28)(9 29)(10 30)(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 56)(7 57)(8 58)(9 59)(10 60)(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 56)(7 57)(8 58)(9 59)(10 60)(26 31)(27 32)(28 33)(29 34)(30 35)(36 41)(37 42)(38 43)(39 44)(40 45)
(6 56)(7 57)(8 58)(9 59)(10 60)(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 51 41)(7 52 42)(8 53 43)(9 54 44)(10 55 45)(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 50)(7 49)(8 48)(9 47)(10 46)(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)(6,26)(7,27)(8,28)(9,29)(10,30)(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)(6,56)(7,57)(8,58)(9,59)(10,60)(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)(6,56)(7,57)(8,58)(9,59)(10,60)(26,31)(27,32)(28,33)(29,34)(30,35)(36,41)(37,42)(38,43)(39,44)(40,45), (6,56)(7,57)(8,58)(9,59)(10,60)(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)(6,51,41)(7,52,42)(8,53,43)(9,54,44)(10,55,45)(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)(6,50)(7,49)(8,48)(9,47)(10,46)(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)(6,26)(7,27)(8,28)(9,29)(10,30)(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)(6,56)(7,57)(8,58)(9,59)(10,60)(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)(6,56)(7,57)(8,58)(9,59)(10,60)(26,31)(27,32)(28,33)(29,34)(30,35)(36,41)(37,42)(38,43)(39,44)(40,45), (6,56)(7,57)(8,58)(9,59)(10,60)(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)(6,51,41)(7,52,42)(8,53,43)(9,54,44)(10,55,45)(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)(6,50)(7,49)(8,48)(9,47)(10,46)(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),(6,26),(7,27),(8,28),(9,29),(10,30),(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),(6,56),(7,57),(8,58),(9,59),(10,60),(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),(6,56),(7,57),(8,58),(9,59),(10,60),(26,31),(27,32),(28,33),(29,34),(30,35),(36,41),(37,42),(38,43),(39,44),(40,45)], [(6,56),(7,57),(8,58),(9,59),(10,60),(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),(6,51,41),(7,52,42),(8,53,43),(9,54,44),(10,55,45),(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),(6,50),(7,49),(8,48),(9,47),(10,46),(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)])
Matrix representation ►G ⊆ 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] >;
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 |
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
×
⋊
ℤ
𝔽
○
≀