metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C10.7(C4×D4), C22⋊C4⋊5F5, (C2×F5).1D4, C2.10(D4×F5), D10⋊C4⋊2C4, D10.60(C2×D4), C23.D5⋊10C4, C23.10(C2×F5), D5.1(C4⋊D4), D10.3Q8⋊8C2, C5⋊(C24.C22), D10.44(C4○D4), D5.2(C4.4D4), C22.75(C22×F5), D5.2(C42⋊2C2), (C23×D5).85C22, C10.10(C42⋊C2), (C22×F5).14C22, D5.2(C22.D4), (C22×D5).270C23, C2.13(D10.C23), (C2×C4×F5)⋊8C2, (C2×C4⋊F5)⋊7C2, (C5×C22⋊C4)⋊5C4, (C2×C4).24(C2×F5), (C2×C20).91(C2×C4), (D5×C22⋊C4).6C2, (C2×C22⋊F5).5C2, (C2×C4×D5).276C22, (C2×C10).38(C22×C4), (C22×C10).21(C2×C4), (C2×Dic5).53(C2×C4), (C22×D5).45(C2×C4), SmallGroup(320,1038)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 874 in 190 conjugacy classes, 54 normal (50 characteristic)
C1, C2 [×3], C2 [×6], C4 [×10], C22, C22 [×16], C5, C2×C4 [×2], C2×C4 [×20], C23, C23 [×8], D5 [×4], D5, C10 [×3], C10, C42 [×2], C22⋊C4, C22⋊C4 [×7], C4⋊C4 [×2], C22×C4 [×6], C24, Dic5 [×2], C20 [×2], F5 [×6], D10 [×6], D10 [×7], C2×C10, C2×C10 [×3], C2.C42 [×2], C2×C42, C2×C22⋊C4 [×3], C2×C4⋊C4, C4×D5 [×4], C2×Dic5 [×2], C2×C20 [×2], C2×F5 [×4], C2×F5 [×10], C22×D5 [×2], C22×D5 [×6], C22×C10, C24.C22, D10⋊C4 [×2], C23.D5, C5×C22⋊C4, C4×F5 [×2], C4⋊F5 [×2], C22⋊F5 [×4], C2×C4×D5 [×2], C22×F5 [×4], C23×D5, D10.3Q8 [×2], D5×C22⋊C4, C2×C4×F5, C2×C4⋊F5, C2×C22⋊F5 [×2], C10.(C4×D4)
Quotients:
C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22×C4, C2×D4 [×2], C4○D4 [×4], F5, C42⋊C2, C4×D4 [×2], C4⋊D4, C22.D4, C4.4D4, C42⋊2C2, C2×F5 [×3], C24.C22, C22×F5, D10.C23, D4×F5 [×2], C10.(C4×D4)
Generators and relations
G = < a,b,c,d | a10=b4=c4=d2=1, bab-1=a3, ac=ca, ad=da, bc=cb, dbd=a5b, dcd=a5c-1 >
(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 73)(2 80 10 76)(3 77 9 79)(4 74 8 72)(5 71 7 75)(6 78)(11 24)(12 21 20 27)(13 28 19 30)(14 25 18 23)(15 22 17 26)(16 29)(31 59 35 57)(32 56 34 60)(33 53)(36 54 40 52)(37 51 39 55)(38 58)(41 67 49 63)(42 64 48 66)(43 61 47 69)(44 68 46 62)(45 65)(50 70)
(1 53 24 65)(2 54 25 66)(3 55 26 67)(4 56 27 68)(5 57 28 69)(6 58 29 70)(7 59 30 61)(8 60 21 62)(9 51 22 63)(10 52 23 64)(11 45 73 33)(12 46 74 34)(13 47 75 35)(14 48 76 36)(15 49 77 37)(16 50 78 38)(17 41 79 39)(18 42 80 40)(19 43 71 31)(20 44 72 32)
(1 45)(2 46)(3 47)(4 48)(5 49)(6 50)(7 41)(8 42)(9 43)(10 44)(11 58)(12 59)(13 60)(14 51)(15 52)(16 53)(17 54)(18 55)(19 56)(20 57)(21 40)(22 31)(23 32)(24 33)(25 34)(26 35)(27 36)(28 37)(29 38)(30 39)(61 74)(62 75)(63 76)(64 77)(65 78)(66 79)(67 80)(68 71)(69 72)(70 73)
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), (1,73)(2,80,10,76)(3,77,9,79)(4,74,8,72)(5,71,7,75)(6,78)(11,24)(12,21,20,27)(13,28,19,30)(14,25,18,23)(15,22,17,26)(16,29)(31,59,35,57)(32,56,34,60)(33,53)(36,54,40,52)(37,51,39,55)(38,58)(41,67,49,63)(42,64,48,66)(43,61,47,69)(44,68,46,62)(45,65)(50,70), (1,53,24,65)(2,54,25,66)(3,55,26,67)(4,56,27,68)(5,57,28,69)(6,58,29,70)(7,59,30,61)(8,60,21,62)(9,51,22,63)(10,52,23,64)(11,45,73,33)(12,46,74,34)(13,47,75,35)(14,48,76,36)(15,49,77,37)(16,50,78,38)(17,41,79,39)(18,42,80,40)(19,43,71,31)(20,44,72,32), (1,45)(2,46)(3,47)(4,48)(5,49)(6,50)(7,41)(8,42)(9,43)(10,44)(11,58)(12,59)(13,60)(14,51)(15,52)(16,53)(17,54)(18,55)(19,56)(20,57)(21,40)(22,31)(23,32)(24,33)(25,34)(26,35)(27,36)(28,37)(29,38)(30,39)(61,74)(62,75)(63,76)(64,77)(65,78)(66,79)(67,80)(68,71)(69,72)(70,73)>;
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), (1,73)(2,80,10,76)(3,77,9,79)(4,74,8,72)(5,71,7,75)(6,78)(11,24)(12,21,20,27)(13,28,19,30)(14,25,18,23)(15,22,17,26)(16,29)(31,59,35,57)(32,56,34,60)(33,53)(36,54,40,52)(37,51,39,55)(38,58)(41,67,49,63)(42,64,48,66)(43,61,47,69)(44,68,46,62)(45,65)(50,70), (1,53,24,65)(2,54,25,66)(3,55,26,67)(4,56,27,68)(5,57,28,69)(6,58,29,70)(7,59,30,61)(8,60,21,62)(9,51,22,63)(10,52,23,64)(11,45,73,33)(12,46,74,34)(13,47,75,35)(14,48,76,36)(15,49,77,37)(16,50,78,38)(17,41,79,39)(18,42,80,40)(19,43,71,31)(20,44,72,32), (1,45)(2,46)(3,47)(4,48)(5,49)(6,50)(7,41)(8,42)(9,43)(10,44)(11,58)(12,59)(13,60)(14,51)(15,52)(16,53)(17,54)(18,55)(19,56)(20,57)(21,40)(22,31)(23,32)(24,33)(25,34)(26,35)(27,36)(28,37)(29,38)(30,39)(61,74)(62,75)(63,76)(64,77)(65,78)(66,79)(67,80)(68,71)(69,72)(70,73) );
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)], [(1,73),(2,80,10,76),(3,77,9,79),(4,74,8,72),(5,71,7,75),(6,78),(11,24),(12,21,20,27),(13,28,19,30),(14,25,18,23),(15,22,17,26),(16,29),(31,59,35,57),(32,56,34,60),(33,53),(36,54,40,52),(37,51,39,55),(38,58),(41,67,49,63),(42,64,48,66),(43,61,47,69),(44,68,46,62),(45,65),(50,70)], [(1,53,24,65),(2,54,25,66),(3,55,26,67),(4,56,27,68),(5,57,28,69),(6,58,29,70),(7,59,30,61),(8,60,21,62),(9,51,22,63),(10,52,23,64),(11,45,73,33),(12,46,74,34),(13,47,75,35),(14,48,76,36),(15,49,77,37),(16,50,78,38),(17,41,79,39),(18,42,80,40),(19,43,71,31),(20,44,72,32)], [(1,45),(2,46),(3,47),(4,48),(5,49),(6,50),(7,41),(8,42),(9,43),(10,44),(11,58),(12,59),(13,60),(14,51),(15,52),(16,53),(17,54),(18,55),(19,56),(20,57),(21,40),(22,31),(23,32),(24,33),(25,34),(26,35),(27,36),(28,37),(29,38),(30,39),(61,74),(62,75),(63,76),(64,77),(65,78),(66,79),(67,80),(68,71),(69,72),(70,73)])
Matrix representation ►G ⊆ GL8(𝔽41)
40 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 40 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 40 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 40 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 40 | 1 |
0 | 0 | 0 | 0 | 0 | 0 | 40 | 0 |
0 | 0 | 0 | 0 | 1 | 0 | 40 | 0 |
0 | 0 | 0 | 0 | 0 | 1 | 40 | 0 |
0 | 9 | 0 | 0 | 0 | 0 | 0 | 0 |
9 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 40 | 39 | 0 | 0 | 0 | 0 |
0 | 0 | 1 | 1 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 32 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 32 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 40 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 40 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 40 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 40 |
1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 40 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 1 | 2 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 40 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 40 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 40 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 40 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 40 |
G:=sub<GL(8,GF(41))| [40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,40,40,40,40,0,0,0,0,1,0,0,0],[0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,0,40,1,0,0,0,0,0,0,39,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,2,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40] >;
38 conjugacy classes
class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | ··· | 4M | 4N | ··· | 4R | 5 | 10A | 10B | 10C | 10D | 10E | 20A | 20B | 20C | 20D |
order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 5 | 10 | 10 | 10 | 10 | 10 | 20 | 20 | 20 | 20 |
size | 1 | 1 | 1 | 1 | 4 | 5 | 5 | 5 | 5 | 20 | 2 | 2 | 4 | 10 | ··· | 10 | 20 | ··· | 20 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 8 |
38 irreducible representations
dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 8 |
type | + | + | + | + | + | + | + | + | + | + | + | |||||
image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | D4 | C4○D4 | F5 | C2×F5 | C2×F5 | D10.C23 | D4×F5 |
kernel | C10.(C4×D4) | D10.3Q8 | D5×C22⋊C4 | C2×C4×F5 | C2×C4⋊F5 | C2×C22⋊F5 | D10⋊C4 | C23.D5 | C5×C22⋊C4 | C2×F5 | D10 | C22⋊C4 | C2×C4 | C23 | C2 | C2 |
# reps | 1 | 2 | 1 | 1 | 1 | 2 | 4 | 2 | 2 | 4 | 8 | 1 | 2 | 1 | 4 | 2 |
In GAP, Magma, Sage, TeX
C_{10}.(C_4\times D_4)
% in TeX
G:=Group("C10.(C4xD4)");
// GroupNames label
G:=SmallGroup(320,1038);
// by ID
G=gap.SmallGroup(320,1038);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,120,422,387,6278,1595]);
// Polycyclic
G:=Group<a,b,c,d|a^10=b^4=c^4=d^2=1,b*a*b^-1=a^3,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=a^5*b,d*c*d=a^5*c^-1>;
// generators/relations