direct product, metabelian, soluble, monomial, A-group
Aliases: D5×C22⋊A4, C22⋊(D5×A4), (D5×C24)⋊2C3, C24⋊6(C3×D5), (C23×C10)⋊6C6, (C22×D5)⋊2A4, C5⋊(C2×C22⋊A4), (C2×C10)⋊(C2×A4), (C5×C22⋊A4)⋊4C2, SmallGroup(480,1203)
Series: Derived ►Chief ►Lower central ►Upper central
| C23×C10 — D5×C22⋊A4 |
Subgroups: 2272 in 296 conjugacy classes, 24 normal (9 characteristic)
C1, C2 [×11], C3, C22 [×5], C22 [×50], C5, C6, C23 [×55], D5, D5 [×5], C10 [×5], A4 [×5], C15, C24, C24 [×10], D10 [×40], C2×C10 [×5], C2×C10 [×10], C2×A4 [×5], C3×D5, C25, C22×D5 [×5], C22×D5 [×45], C22×C10 [×5], C22⋊A4, C5×A4 [×5], C23×D5 [×10], C23×C10, C2×C22⋊A4, D5×A4 [×5], D5×C24, C5×C22⋊A4, D5×C22⋊A4
Quotients:
C1, C2, C3, C6, D5, A4 [×5], C2×A4 [×5], C3×D5, C22⋊A4, C2×C22⋊A4, D5×A4 [×5], D5×C22⋊A4
Generators and relations
G = < a,b,c,d,e,f,g | a5=b2=c2=d2=e2=f2=g3=1, bab=a-1, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, bd=db, be=eb, bf=fb, bg=gb, gcg-1=cd=dc, ce=ec, cf=fc, de=ed, df=fd, gdg-1=c, geg-1=ef=fe, gfg-1=e >
►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 8)(2 7)(3 6)(4 10)(5 9)(11 16)(12 20)(13 19)(14 18)(15 17)(21 26)(22 30)(23 29)(24 28)(25 27)(31 36)(32 40)(33 39)(34 38)(35 37)(41 46)(42 50)(43 49)(44 48)(45 47)(51 56)(52 60)(53 59)(54 58)(55 57)
(21 26)(22 27)(23 28)(24 29)(25 30)(31 36)(32 37)(33 38)(34 39)(35 40)(41 46)(42 47)(43 48)(44 49)(45 50)(51 56)(52 57)(53 58)(54 59)(55 60)
(1 9)(2 10)(3 6)(4 7)(5 8)(11 16)(12 17)(13 18)(14 19)(15 20)(21 26)(22 27)(23 28)(24 29)(25 30)(31 36)(32 37)(33 38)(34 39)(35 40)
(1 14)(2 15)(3 11)(4 12)(5 13)(6 16)(7 17)(8 18)(9 19)(10 20)(41 51)(42 52)(43 53)(44 54)(45 55)(46 56)(47 57)(48 58)(49 59)(50 60)
(21 31)(22 32)(23 33)(24 34)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)(41 51)(42 52)(43 53)(44 54)(45 55)(46 56)(47 57)(48 58)(49 59)(50 60)
(1 44 24)(2 45 25)(3 41 21)(4 42 22)(5 43 23)(6 46 26)(7 47 27)(8 48 28)(9 49 29)(10 50 30)(11 51 31)(12 52 32)(13 53 33)(14 54 34)(15 55 35)(16 56 36)(17 57 37)(18 58 38)(19 59 39)(20 60 40)
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), (1,8)(2,7)(3,6)(4,10)(5,9)(11,16)(12,20)(13,19)(14,18)(15,17)(21,26)(22,30)(23,29)(24,28)(25,27)(31,36)(32,40)(33,39)(34,38)(35,37)(41,46)(42,50)(43,49)(44,48)(45,47)(51,56)(52,60)(53,59)(54,58)(55,57), (21,26)(22,27)(23,28)(24,29)(25,30)(31,36)(32,37)(33,38)(34,39)(35,40)(41,46)(42,47)(43,48)(44,49)(45,50)(51,56)(52,57)(53,58)(54,59)(55,60), (1,9)(2,10)(3,6)(4,7)(5,8)(11,16)(12,17)(13,18)(14,19)(15,20)(21,26)(22,27)(23,28)(24,29)(25,30)(31,36)(32,37)(33,38)(34,39)(35,40), (1,14)(2,15)(3,11)(4,12)(5,13)(6,16)(7,17)(8,18)(9,19)(10,20)(41,51)(42,52)(43,53)(44,54)(45,55)(46,56)(47,57)(48,58)(49,59)(50,60), (21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40)(41,51)(42,52)(43,53)(44,54)(45,55)(46,56)(47,57)(48,58)(49,59)(50,60), (1,44,24)(2,45,25)(3,41,21)(4,42,22)(5,43,23)(6,46,26)(7,47,27)(8,48,28)(9,49,29)(10,50,30)(11,51,31)(12,52,32)(13,53,33)(14,54,34)(15,55,35)(16,56,36)(17,57,37)(18,58,38)(19,59,39)(20,60,40)>;
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), (1,8)(2,7)(3,6)(4,10)(5,9)(11,16)(12,20)(13,19)(14,18)(15,17)(21,26)(22,30)(23,29)(24,28)(25,27)(31,36)(32,40)(33,39)(34,38)(35,37)(41,46)(42,50)(43,49)(44,48)(45,47)(51,56)(52,60)(53,59)(54,58)(55,57), (21,26)(22,27)(23,28)(24,29)(25,30)(31,36)(32,37)(33,38)(34,39)(35,40)(41,46)(42,47)(43,48)(44,49)(45,50)(51,56)(52,57)(53,58)(54,59)(55,60), (1,9)(2,10)(3,6)(4,7)(5,8)(11,16)(12,17)(13,18)(14,19)(15,20)(21,26)(22,27)(23,28)(24,29)(25,30)(31,36)(32,37)(33,38)(34,39)(35,40), (1,14)(2,15)(3,11)(4,12)(5,13)(6,16)(7,17)(8,18)(9,19)(10,20)(41,51)(42,52)(43,53)(44,54)(45,55)(46,56)(47,57)(48,58)(49,59)(50,60), (21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40)(41,51)(42,52)(43,53)(44,54)(45,55)(46,56)(47,57)(48,58)(49,59)(50,60), (1,44,24)(2,45,25)(3,41,21)(4,42,22)(5,43,23)(6,46,26)(7,47,27)(8,48,28)(9,49,29)(10,50,30)(11,51,31)(12,52,32)(13,53,33)(14,54,34)(15,55,35)(16,56,36)(17,57,37)(18,58,38)(19,59,39)(20,60,40) );
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)], [(1,8),(2,7),(3,6),(4,10),(5,9),(11,16),(12,20),(13,19),(14,18),(15,17),(21,26),(22,30),(23,29),(24,28),(25,27),(31,36),(32,40),(33,39),(34,38),(35,37),(41,46),(42,50),(43,49),(44,48),(45,47),(51,56),(52,60),(53,59),(54,58),(55,57)], [(21,26),(22,27),(23,28),(24,29),(25,30),(31,36),(32,37),(33,38),(34,39),(35,40),(41,46),(42,47),(43,48),(44,49),(45,50),(51,56),(52,57),(53,58),(54,59),(55,60)], [(1,9),(2,10),(3,6),(4,7),(5,8),(11,16),(12,17),(13,18),(14,19),(15,20),(21,26),(22,27),(23,28),(24,29),(25,30),(31,36),(32,37),(33,38),(34,39),(35,40)], [(1,14),(2,15),(3,11),(4,12),(5,13),(6,16),(7,17),(8,18),(9,19),(10,20),(41,51),(42,52),(43,53),(44,54),(45,55),(46,56),(47,57),(48,58),(49,59),(50,60)], [(21,31),(22,32),(23,33),(24,34),(25,35),(26,36),(27,37),(28,38),(29,39),(30,40),(41,51),(42,52),(43,53),(44,54),(45,55),(46,56),(47,57),(48,58),(49,59),(50,60)], [(1,44,24),(2,45,25),(3,41,21),(4,42,22),(5,43,23),(6,46,26),(7,47,27),(8,48,28),(9,49,29),(10,50,30),(11,51,31),(12,52,32),(13,53,33),(14,54,34),(15,55,35),(16,56,36),(17,57,37),(18,58,38),(19,59,39),(20,60,40)])
Matrix representation ►G ⊆ GL8(𝔽31)
| 12 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 30 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 12 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 30 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 30 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 30 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 30 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 30 | 0 | 0 | 0 | 0 |
| 0 | 0 | 26 | 0 | 30 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 30 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 30 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 30 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 29 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 30 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 30 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 30 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 30 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 2 | 30 | 0 |
| 0 | 0 | 0 | 0 | 0 | 2 | 0 | 30 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 5 | 21 | 0 | 0 | 0 |
| 0 | 0 | 30 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 25 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 30 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 30 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 30 | 0 |
G:=sub<GL(8,GF(31))| [12,30,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,12,30,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,30,0,5,0,0,0,0,0,0,30,30,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,26,0,0,0,0,0,0,30,0,0,0,0,0,0,0,0,30,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,30,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,30,0,0,0,0,0,0,0,0,30,29,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,30],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,30,0,5,0,0,0,0,0,0,30,30,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,2,2,0,0,0,0,0,0,30,0,0,0,0,0,0,0,0,30],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,6,30,0,0,0,0,0,0,5,0,0,0,0,0,0,0,21,0,25,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,30,30,30,0,0,0,0,0,0,1,0] >;
32 conjugacy classes
| class | 1 | 2A | ··· | 2E | 2F | 2G | ··· | 2K | 3A | 3B | 5A | 5B | 6A | 6B | 10A | ··· | 10J | 15A | 15B | 15C | 15D |
| order | 1 | 2 | ··· | 2 | 2 | 2 | ··· | 2 | 3 | 3 | 5 | 5 | 6 | 6 | 10 | ··· | 10 | 15 | 15 | 15 | 15 |
| size | 1 | 3 | ··· | 3 | 5 | 15 | ··· | 15 | 16 | 16 | 2 | 2 | 80 | 80 | 6 | ··· | 6 | 32 | 32 | 32 | 32 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 3 | 3 | 6 |
| type | + | + | + | + | + | + | |||
| image | C1 | C2 | C3 | C6 | D5 | C3×D5 | A4 | C2×A4 | D5×A4 |
| kernel | D5×C22⋊A4 | C5×C22⋊A4 | D5×C24 | C23×C10 | C22⋊A4 | C24 | C22×D5 | C2×C10 | C22 |
| # reps | 1 | 1 | 2 | 2 | 2 | 4 | 5 | 5 | 10 |
In GAP, Magma, Sage, TeX
D_5\times C_2^2\rtimes A_4
% in TeX
G:=Group("D5xC2^2:A4"); // GroupNames label
G:=SmallGroup(480,1203);
// by ID
G=gap.SmallGroup(480,1203);
# by ID
G:=PCGroup([7,-2,-3,-2,2,-2,2,-5,198,94,1271,516,18822]);
// Polycyclic
G:=Group<a,b,c,d,e,f,g|a^5=b^2=c^2=d^2=e^2=f^2=g^3=1,b*a*b=a^-1,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,g*c*g^-1=c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,g*d*g^-1=c,g*e*g^-1=e*f=f*e,g*f*g^-1=e>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀