direct product, non-abelian, soluble
Aliases: C5×D4.A4, 2- (1+4)⋊2C15, C4○D4⋊C30, (C2×Q8)⋊C30, D4.(C5×A4), (C5×D4).A4, C4.A4⋊4C10, C4.3(C10×A4), C20.9(C2×A4), (Q8×C10)⋊3C6, Q8.3(C2×C30), (C5×2- (1+4))⋊C3, C22.5(C10×A4), C10.18(C22×A4), (C2×SL2(𝔽3))⋊1C10, (C10×SL2(𝔽3))⋊1C2, SL2(𝔽3).5(C2×C10), (C5×SL2(𝔽3)).17C22, C2.7(A4×C2×C10), (C5×C4○D4)⋊3C6, (C5×C4.A4)⋊9C2, (C2×C10).9(C2×A4), (C5×Q8).13(C2×C6), SmallGroup(480,1132)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2 — Q8 — C5×Q8 — C5×SL2(𝔽3) — C10×SL2(𝔽3) — C5×D4.A4 |
| Q8 — C5×D4.A4 |
Subgroups: 246 in 92 conjugacy classes, 32 normal (20 characteristic)
C1, C2, C2 [×3], C3, C4, C4 [×3], C22 [×2], C22, C5, C6 [×3], C2×C4 [×5], D4, D4 [×3], Q8, Q8 [×3], C10, C10 [×3], C12, C2×C6 [×2], C15, C2×Q8 [×2], C2×Q8, C4○D4, C4○D4 [×3], C20, C20 [×3], C2×C10 [×2], C2×C10, SL2(𝔽3), C3×D4, C30 [×3], 2- (1+4), C2×C20 [×5], C5×D4, C5×D4 [×3], C5×Q8, C5×Q8 [×3], C2×SL2(𝔽3) [×2], C4.A4, C60, C2×C30 [×2], Q8×C10 [×2], Q8×C10, C5×C4○D4, C5×C4○D4 [×3], D4.A4, C5×SL2(𝔽3), D4×C15, C5×2- (1+4), C10×SL2(𝔽3) [×2], C5×C4.A4, C5×D4.A4
Quotients:
C1, C2 [×3], C3, C22, C5, C6 [×3], C10 [×3], A4, C2×C6, C15, C2×C10, C2×A4 [×3], C30 [×3], C22×A4, C5×A4, C2×C30, D4.A4, C10×A4 [×3], A4×C2×C10, C5×D4.A4
Generators and relations
G = < a,b,c,d,e,f | a5=b4=c2=f3=1, d2=e2=b2, ab=ba, ac=ca, ad=da, ae=ea, af=fa, cbc=b-1, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, ede-1=b2d, fdf-1=b2de, fef-1=d >
►On 80 points
Generators in S80
(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)(61 62 63 64 65)(66 67 68 69 70)(71 72 73 74 75)(76 77 78 79 80)
(1 57 29 7)(2 58 30 8)(3 59 26 9)(4 60 27 10)(5 56 28 6)(11 22 53 63)(12 23 54 64)(13 24 55 65)(14 25 51 61)(15 21 52 62)(16 66 77 37)(17 67 78 38)(18 68 79 39)(19 69 80 40)(20 70 76 36)(31 46 72 41)(32 47 73 42)(33 48 74 43)(34 49 75 44)(35 50 71 45)
(1 7)(2 8)(3 9)(4 10)(5 6)(11 63)(12 64)(13 65)(14 61)(15 62)(16 66)(17 67)(18 68)(19 69)(20 70)(21 52)(22 53)(23 54)(24 55)(25 51)(26 59)(27 60)(28 56)(29 57)(30 58)(31 46)(32 47)(33 48)(34 49)(35 50)(36 76)(37 77)(38 78)(39 79)(40 80)(41 72)(42 73)(43 74)(44 75)(45 71)
(1 47 29 42)(2 48 30 43)(3 49 26 44)(4 50 27 45)(5 46 28 41)(6 31 56 72)(7 32 57 73)(8 33 58 74)(9 34 59 75)(10 35 60 71)(11 36 53 70)(12 37 54 66)(13 38 55 67)(14 39 51 68)(15 40 52 69)(16 64 77 23)(17 65 78 24)(18 61 79 25)(19 62 80 21)(20 63 76 22)
(1 37 29 66)(2 38 30 67)(3 39 26 68)(4 40 27 69)(5 36 28 70)(6 76 56 20)(7 77 57 16)(8 78 58 17)(9 79 59 18)(10 80 60 19)(11 41 53 46)(12 42 54 47)(13 43 55 48)(14 44 51 49)(15 45 52 50)(21 35 62 71)(22 31 63 72)(23 32 64 73)(24 33 65 74)(25 34 61 75)
(11 41 70)(12 42 66)(13 43 67)(14 44 68)(15 45 69)(16 64 73)(17 65 74)(18 61 75)(19 62 71)(20 63 72)(21 35 80)(22 31 76)(23 32 77)(24 33 78)(25 34 79)(36 53 46)(37 54 47)(38 55 48)(39 51 49)(40 52 50)
G:=sub<Sym(80)| (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)(61,62,63,64,65)(66,67,68,69,70)(71,72,73,74,75)(76,77,78,79,80), (1,57,29,7)(2,58,30,8)(3,59,26,9)(4,60,27,10)(5,56,28,6)(11,22,53,63)(12,23,54,64)(13,24,55,65)(14,25,51,61)(15,21,52,62)(16,66,77,37)(17,67,78,38)(18,68,79,39)(19,69,80,40)(20,70,76,36)(31,46,72,41)(32,47,73,42)(33,48,74,43)(34,49,75,44)(35,50,71,45), (1,7)(2,8)(3,9)(4,10)(5,6)(11,63)(12,64)(13,65)(14,61)(15,62)(16,66)(17,67)(18,68)(19,69)(20,70)(21,52)(22,53)(23,54)(24,55)(25,51)(26,59)(27,60)(28,56)(29,57)(30,58)(31,46)(32,47)(33,48)(34,49)(35,50)(36,76)(37,77)(38,78)(39,79)(40,80)(41,72)(42,73)(43,74)(44,75)(45,71), (1,47,29,42)(2,48,30,43)(3,49,26,44)(4,50,27,45)(5,46,28,41)(6,31,56,72)(7,32,57,73)(8,33,58,74)(9,34,59,75)(10,35,60,71)(11,36,53,70)(12,37,54,66)(13,38,55,67)(14,39,51,68)(15,40,52,69)(16,64,77,23)(17,65,78,24)(18,61,79,25)(19,62,80,21)(20,63,76,22), (1,37,29,66)(2,38,30,67)(3,39,26,68)(4,40,27,69)(5,36,28,70)(6,76,56,20)(7,77,57,16)(8,78,58,17)(9,79,59,18)(10,80,60,19)(11,41,53,46)(12,42,54,47)(13,43,55,48)(14,44,51,49)(15,45,52,50)(21,35,62,71)(22,31,63,72)(23,32,64,73)(24,33,65,74)(25,34,61,75), (11,41,70)(12,42,66)(13,43,67)(14,44,68)(15,45,69)(16,64,73)(17,65,74)(18,61,75)(19,62,71)(20,63,72)(21,35,80)(22,31,76)(23,32,77)(24,33,78)(25,34,79)(36,53,46)(37,54,47)(38,55,48)(39,51,49)(40,52,50)>;
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)(61,62,63,64,65)(66,67,68,69,70)(71,72,73,74,75)(76,77,78,79,80), (1,57,29,7)(2,58,30,8)(3,59,26,9)(4,60,27,10)(5,56,28,6)(11,22,53,63)(12,23,54,64)(13,24,55,65)(14,25,51,61)(15,21,52,62)(16,66,77,37)(17,67,78,38)(18,68,79,39)(19,69,80,40)(20,70,76,36)(31,46,72,41)(32,47,73,42)(33,48,74,43)(34,49,75,44)(35,50,71,45), (1,7)(2,8)(3,9)(4,10)(5,6)(11,63)(12,64)(13,65)(14,61)(15,62)(16,66)(17,67)(18,68)(19,69)(20,70)(21,52)(22,53)(23,54)(24,55)(25,51)(26,59)(27,60)(28,56)(29,57)(30,58)(31,46)(32,47)(33,48)(34,49)(35,50)(36,76)(37,77)(38,78)(39,79)(40,80)(41,72)(42,73)(43,74)(44,75)(45,71), (1,47,29,42)(2,48,30,43)(3,49,26,44)(4,50,27,45)(5,46,28,41)(6,31,56,72)(7,32,57,73)(8,33,58,74)(9,34,59,75)(10,35,60,71)(11,36,53,70)(12,37,54,66)(13,38,55,67)(14,39,51,68)(15,40,52,69)(16,64,77,23)(17,65,78,24)(18,61,79,25)(19,62,80,21)(20,63,76,22), (1,37,29,66)(2,38,30,67)(3,39,26,68)(4,40,27,69)(5,36,28,70)(6,76,56,20)(7,77,57,16)(8,78,58,17)(9,79,59,18)(10,80,60,19)(11,41,53,46)(12,42,54,47)(13,43,55,48)(14,44,51,49)(15,45,52,50)(21,35,62,71)(22,31,63,72)(23,32,64,73)(24,33,65,74)(25,34,61,75), (11,41,70)(12,42,66)(13,43,67)(14,44,68)(15,45,69)(16,64,73)(17,65,74)(18,61,75)(19,62,71)(20,63,72)(21,35,80)(22,31,76)(23,32,77)(24,33,78)(25,34,79)(36,53,46)(37,54,47)(38,55,48)(39,51,49)(40,52,50) );
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),(61,62,63,64,65),(66,67,68,69,70),(71,72,73,74,75),(76,77,78,79,80)], [(1,57,29,7),(2,58,30,8),(3,59,26,9),(4,60,27,10),(5,56,28,6),(11,22,53,63),(12,23,54,64),(13,24,55,65),(14,25,51,61),(15,21,52,62),(16,66,77,37),(17,67,78,38),(18,68,79,39),(19,69,80,40),(20,70,76,36),(31,46,72,41),(32,47,73,42),(33,48,74,43),(34,49,75,44),(35,50,71,45)], [(1,7),(2,8),(3,9),(4,10),(5,6),(11,63),(12,64),(13,65),(14,61),(15,62),(16,66),(17,67),(18,68),(19,69),(20,70),(21,52),(22,53),(23,54),(24,55),(25,51),(26,59),(27,60),(28,56),(29,57),(30,58),(31,46),(32,47),(33,48),(34,49),(35,50),(36,76),(37,77),(38,78),(39,79),(40,80),(41,72),(42,73),(43,74),(44,75),(45,71)], [(1,47,29,42),(2,48,30,43),(3,49,26,44),(4,50,27,45),(5,46,28,41),(6,31,56,72),(7,32,57,73),(8,33,58,74),(9,34,59,75),(10,35,60,71),(11,36,53,70),(12,37,54,66),(13,38,55,67),(14,39,51,68),(15,40,52,69),(16,64,77,23),(17,65,78,24),(18,61,79,25),(19,62,80,21),(20,63,76,22)], [(1,37,29,66),(2,38,30,67),(3,39,26,68),(4,40,27,69),(5,36,28,70),(6,76,56,20),(7,77,57,16),(8,78,58,17),(9,79,59,18),(10,80,60,19),(11,41,53,46),(12,42,54,47),(13,43,55,48),(14,44,51,49),(15,45,52,50),(21,35,62,71),(22,31,63,72),(23,32,64,73),(24,33,65,74),(25,34,61,75)], [(11,41,70),(12,42,66),(13,43,67),(14,44,68),(15,45,69),(16,64,73),(17,65,74),(18,61,75),(19,62,71),(20,63,72),(21,35,80),(22,31,76),(23,32,77),(24,33,78),(25,34,79),(36,53,46),(37,54,47),(38,55,48),(39,51,49),(40,52,50)])
Matrix representation ►G ⊆ GL4(𝔽61) generated by
| 34 | 0 | 0 | 0 |
| 0 | 34 | 0 | 0 |
| 0 | 0 | 34 | 0 |
| 0 | 0 | 0 | 34 |
| 34 | 42 | 1 | 3 |
| 0 | 53 | 0 | 59 |
| 2 | 0 | 27 | 19 |
| 0 | 2 | 0 | 8 |
| 27 | 19 | 59 | 58 |
| 0 | 8 | 0 | 1 |
| 59 | 0 | 34 | 42 |
| 0 | 59 | 0 | 53 |
| 48 | 14 | 50 | 47 |
| 14 | 13 | 11 | 11 |
| 0 | 0 | 48 | 14 |
| 0 | 0 | 14 | 13 |
| 0 | 1 | 40 | 21 |
| 60 | 0 | 21 | 21 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 60 | 0 |
| 1 | 14 | 0 | 58 |
| 0 | 13 | 0 | 0 |
| 0 | 0 | 1 | 14 |
| 0 | 0 | 0 | 13 |
G:=sub<GL(4,GF(61))| [34,0,0,0,0,34,0,0,0,0,34,0,0,0,0,34],[34,0,2,0,42,53,0,2,1,0,27,0,3,59,19,8],[27,0,59,0,19,8,0,59,59,0,34,0,58,1,42,53],[48,14,0,0,14,13,0,0,50,11,48,14,47,11,14,13],[0,60,0,0,1,0,0,0,40,21,0,60,21,21,1,0],[1,0,0,0,14,13,0,0,0,0,1,0,58,0,14,13] >;
95 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3A | 3B | 4A | 4B | 4C | 4D | 5A | 5B | 5C | 5D | 6A | 6B | 6C | 6D | 6E | 6F | 10A | 10B | 10C | 10D | 10E | ··· | 10L | 10M | 10N | 10O | 10P | 12A | 12B | 15A | ··· | 15H | 20A | 20B | 20C | 20D | 20E | ··· | 20P | 30A | ··· | 30H | 30I | ··· | 30X | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 6 | 10 | 10 | 10 | 10 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 12 | 12 | 15 | ··· | 15 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 30 | ··· | 30 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 2 | 2 | 6 | 4 | 4 | 2 | 6 | 6 | 6 | 1 | 1 | 1 | 1 | 4 | 4 | 8 | 8 | 8 | 8 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 6 | 6 | 6 | 6 | 8 | 8 | 4 | ··· | 4 | 2 | 2 | 2 | 2 | 6 | ··· | 6 | 4 | ··· | 4 | 8 | ··· | 8 | 8 | ··· | 8 |
95 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | - | ||||||||||||||
| image | C1 | C2 | C2 | C3 | C5 | C6 | C6 | C10 | C10 | C15 | C30 | C30 | A4 | C2×A4 | C2×A4 | C5×A4 | C10×A4 | C10×A4 | D4.A4 | D4.A4 | C5×D4.A4 |
| kernel | C5×D4.A4 | C10×SL2(𝔽3) | C5×C4.A4 | C5×2- (1+4) | D4.A4 | Q8×C10 | C5×C4○D4 | C2×SL2(𝔽3) | C4.A4 | 2- (1+4) | C2×Q8 | C4○D4 | C5×D4 | C20 | C2×C10 | D4 | C4 | C22 | C5 | C5 | C1 |
| # reps | 1 | 2 | 1 | 2 | 4 | 4 | 2 | 8 | 4 | 8 | 16 | 8 | 1 | 1 | 2 | 4 | 4 | 8 | 1 | 2 | 12 |
In GAP, Magma, Sage, TeX
C_5\times D_4.A_4
% in TeX
G:=Group("C5xD4.A4"); // GroupNames label
G:=SmallGroup(480,1132);
// by ID
G=gap.SmallGroup(480,1132);
# by ID
G:=PCGroup([7,-2,-2,-3,-5,-2,2,-2,3389,1068,172,1909,285,124]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^5=b^4=c^2=f^3=1,d^2=e^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,b*f=f*b,c*d=d*c,c*e=e*c,c*f=f*c,e*d*e^-1=b^2*d,f*d*f^-1=b^2*d*e,f*e*f^-1=d>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀