direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C5×C22.45C24, C10.1632+ (1+4), (C4×D4)⋊19C10, (D4×C20)⋊48C2, (C4×C20)⋊44C22, C42⋊10(C2×C10), C22⋊Q8⋊15C10, C42⋊2C2⋊4C10, C22≀C2.2C10, C4.4D4⋊12C10, C24.20(C2×C10), (C22×C20)⋊6C22, (Q8×C10)⋊30C22, C42⋊C2⋊14C10, (C2×C20).678C23, (C2×C10).371C24, (D4×C10).323C22, C22.D4⋊10C10, (C23×C10).20C22, C23.18(C22×C10), C22.45(C23×C10), C2.15(C5×2+ (1+4)), (C22×C10).266C23, C4⋊C4⋊17(C2×C10), (C2×Q8)⋊5(C2×C10), (C5×C4⋊C4)⋊74C22, C22⋊C4⋊6(C2×C10), (C22×C4)⋊4(C2×C10), C2.24(C10×C4○D4), (C5×C22⋊Q8)⋊42C2, C22.9(C5×C4○D4), (C10×C22⋊C4)⋊35C2, (C2×C22⋊C4)⋊15C10, (C5×C22≀C2).4C2, (C2×D4).69(C2×C10), C10.243(C2×C4○D4), (C5×C4.4D4)⋊32C2, (C5×C42⋊2C2)⋊15C2, (C5×C42⋊C2)⋊35C2, (C5×C22⋊C4)⋊41C22, (C2×C4).61(C22×C10), (C2×C10).118(C4○D4), (C5×C22.D4)⋊29C2, SmallGroup(320,1553)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 394 in 248 conjugacy classes, 150 normal (34 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×11], C22, C22 [×4], C22 [×14], C5, C2×C4, C2×C4 [×10], C2×C4 [×7], D4 [×5], Q8, C23 [×2], C23 [×2], C23 [×5], C10, C10 [×2], C10 [×6], C42, C42 [×2], C22⋊C4 [×2], C22⋊C4 [×12], C4⋊C4 [×8], C22×C4, C22×C4 [×4], C2×D4, C2×D4 [×2], C2×Q8, C24, C20 [×11], C2×C10, C2×C10 [×4], C2×C10 [×14], C2×C22⋊C4 [×2], C42⋊C2 [×2], C4×D4 [×2], C22≀C2, C22⋊Q8 [×2], C22.D4, C22.D4 [×2], C4.4D4, C42⋊2C2 [×2], C2×C20, C2×C20 [×10], C2×C20 [×7], C5×D4 [×5], C5×Q8, C22×C10 [×2], C22×C10 [×2], C22×C10 [×5], C22.45C24, C4×C20, C4×C20 [×2], C5×C22⋊C4 [×2], C5×C22⋊C4 [×12], C5×C4⋊C4 [×8], C22×C20, C22×C20 [×4], D4×C10, D4×C10 [×2], Q8×C10, C23×C10, C10×C22⋊C4 [×2], C5×C42⋊C2 [×2], D4×C20 [×2], C5×C22≀C2, C5×C22⋊Q8 [×2], C5×C22.D4, C5×C22.D4 [×2], C5×C4.4D4, C5×C42⋊2C2 [×2], C5×C22.45C24
Quotients:
C1, C2 [×15], C22 [×35], C5, C23 [×15], C10 [×15], C4○D4 [×4], C24, C2×C10 [×35], C2×C4○D4 [×2], 2+ (1+4), C22×C10 [×15], C22.45C24, C5×C4○D4 [×4], C23×C10, C10×C4○D4 [×2], C5×2+ (1+4), C5×C22.45C24
Generators and relations
G = < a,b,c,d,e,f,g | a5=b2=c2=f2=g2=1, d2=b, e2=c, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, ede-1=bd=db, geg=be=eb, bf=fb, bg=gb, fdf=cd=dc, ce=ec, cf=fc, cg=gc, dg=gd, ef=fe, fg=gf >
►On 80 points
(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 35)(2 31)(3 32)(4 33)(5 34)(6 16)(7 17)(8 18)(9 19)(10 20)(11 76)(12 77)(13 78)(14 79)(15 80)(21 28)(22 29)(23 30)(24 26)(25 27)(36 55)(37 51)(38 52)(39 53)(40 54)(41 48)(42 49)(43 50)(44 46)(45 47)(56 75)(57 71)(58 72)(59 73)(60 74)(61 68)(62 69)(63 70)(64 66)(65 67)
(1 30)(2 26)(3 27)(4 28)(5 29)(6 76)(7 77)(8 78)(9 79)(10 80)(11 16)(12 17)(13 18)(14 19)(15 20)(21 33)(22 34)(23 35)(24 31)(25 32)(36 50)(37 46)(38 47)(39 48)(40 49)(41 53)(42 54)(43 55)(44 51)(45 52)(56 70)(57 66)(58 67)(59 68)(60 69)(61 73)(62 74)(63 75)(64 71)(65 72)
(1 75 35 56)(2 71 31 57)(3 72 32 58)(4 73 33 59)(5 74 34 60)(6 50 16 43)(7 46 17 44)(8 47 18 45)(9 48 19 41)(10 49 20 42)(11 55 76 36)(12 51 77 37)(13 52 78 38)(14 53 79 39)(15 54 80 40)(21 68 28 61)(22 69 29 62)(23 70 30 63)(24 66 26 64)(25 67 27 65)
(1 50 30 36)(2 46 26 37)(3 47 27 38)(4 48 28 39)(5 49 29 40)(6 63 76 75)(7 64 77 71)(8 65 78 72)(9 61 79 73)(10 62 80 74)(11 56 16 70)(12 57 17 66)(13 58 18 67)(14 59 19 68)(15 60 20 69)(21 53 33 41)(22 54 34 42)(23 55 35 43)(24 51 31 44)(25 52 32 45)
(6 76)(7 77)(8 78)(9 79)(10 80)(11 16)(12 17)(13 18)(14 19)(15 20)(56 70)(57 66)(58 67)(59 68)(60 69)(61 73)(62 74)(63 75)(64 71)(65 72)
(1 30)(2 26)(3 27)(4 28)(5 29)(6 11)(7 12)(8 13)(9 14)(10 15)(16 76)(17 77)(18 78)(19 79)(20 80)(21 33)(22 34)(23 35)(24 31)(25 32)(36 43)(37 44)(38 45)(39 41)(40 42)(46 51)(47 52)(48 53)(49 54)(50 55)(56 70)(57 66)(58 67)(59 68)(60 69)(61 73)(62 74)(63 75)(64 71)(65 72)
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), (1,35)(2,31)(3,32)(4,33)(5,34)(6,16)(7,17)(8,18)(9,19)(10,20)(11,76)(12,77)(13,78)(14,79)(15,80)(21,28)(22,29)(23,30)(24,26)(25,27)(36,55)(37,51)(38,52)(39,53)(40,54)(41,48)(42,49)(43,50)(44,46)(45,47)(56,75)(57,71)(58,72)(59,73)(60,74)(61,68)(62,69)(63,70)(64,66)(65,67), (1,30)(2,26)(3,27)(4,28)(5,29)(6,76)(7,77)(8,78)(9,79)(10,80)(11,16)(12,17)(13,18)(14,19)(15,20)(21,33)(22,34)(23,35)(24,31)(25,32)(36,50)(37,46)(38,47)(39,48)(40,49)(41,53)(42,54)(43,55)(44,51)(45,52)(56,70)(57,66)(58,67)(59,68)(60,69)(61,73)(62,74)(63,75)(64,71)(65,72), (1,75,35,56)(2,71,31,57)(3,72,32,58)(4,73,33,59)(5,74,34,60)(6,50,16,43)(7,46,17,44)(8,47,18,45)(9,48,19,41)(10,49,20,42)(11,55,76,36)(12,51,77,37)(13,52,78,38)(14,53,79,39)(15,54,80,40)(21,68,28,61)(22,69,29,62)(23,70,30,63)(24,66,26,64)(25,67,27,65), (1,50,30,36)(2,46,26,37)(3,47,27,38)(4,48,28,39)(5,49,29,40)(6,63,76,75)(7,64,77,71)(8,65,78,72)(9,61,79,73)(10,62,80,74)(11,56,16,70)(12,57,17,66)(13,58,18,67)(14,59,19,68)(15,60,20,69)(21,53,33,41)(22,54,34,42)(23,55,35,43)(24,51,31,44)(25,52,32,45), (6,76)(7,77)(8,78)(9,79)(10,80)(11,16)(12,17)(13,18)(14,19)(15,20)(56,70)(57,66)(58,67)(59,68)(60,69)(61,73)(62,74)(63,75)(64,71)(65,72), (1,30)(2,26)(3,27)(4,28)(5,29)(6,11)(7,12)(8,13)(9,14)(10,15)(16,76)(17,77)(18,78)(19,79)(20,80)(21,33)(22,34)(23,35)(24,31)(25,32)(36,43)(37,44)(38,45)(39,41)(40,42)(46,51)(47,52)(48,53)(49,54)(50,55)(56,70)(57,66)(58,67)(59,68)(60,69)(61,73)(62,74)(63,75)(64,71)(65,72)>;
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), (1,35)(2,31)(3,32)(4,33)(5,34)(6,16)(7,17)(8,18)(9,19)(10,20)(11,76)(12,77)(13,78)(14,79)(15,80)(21,28)(22,29)(23,30)(24,26)(25,27)(36,55)(37,51)(38,52)(39,53)(40,54)(41,48)(42,49)(43,50)(44,46)(45,47)(56,75)(57,71)(58,72)(59,73)(60,74)(61,68)(62,69)(63,70)(64,66)(65,67), (1,30)(2,26)(3,27)(4,28)(5,29)(6,76)(7,77)(8,78)(9,79)(10,80)(11,16)(12,17)(13,18)(14,19)(15,20)(21,33)(22,34)(23,35)(24,31)(25,32)(36,50)(37,46)(38,47)(39,48)(40,49)(41,53)(42,54)(43,55)(44,51)(45,52)(56,70)(57,66)(58,67)(59,68)(60,69)(61,73)(62,74)(63,75)(64,71)(65,72), (1,75,35,56)(2,71,31,57)(3,72,32,58)(4,73,33,59)(5,74,34,60)(6,50,16,43)(7,46,17,44)(8,47,18,45)(9,48,19,41)(10,49,20,42)(11,55,76,36)(12,51,77,37)(13,52,78,38)(14,53,79,39)(15,54,80,40)(21,68,28,61)(22,69,29,62)(23,70,30,63)(24,66,26,64)(25,67,27,65), (1,50,30,36)(2,46,26,37)(3,47,27,38)(4,48,28,39)(5,49,29,40)(6,63,76,75)(7,64,77,71)(8,65,78,72)(9,61,79,73)(10,62,80,74)(11,56,16,70)(12,57,17,66)(13,58,18,67)(14,59,19,68)(15,60,20,69)(21,53,33,41)(22,54,34,42)(23,55,35,43)(24,51,31,44)(25,52,32,45), (6,76)(7,77)(8,78)(9,79)(10,80)(11,16)(12,17)(13,18)(14,19)(15,20)(56,70)(57,66)(58,67)(59,68)(60,69)(61,73)(62,74)(63,75)(64,71)(65,72), (1,30)(2,26)(3,27)(4,28)(5,29)(6,11)(7,12)(8,13)(9,14)(10,15)(16,76)(17,77)(18,78)(19,79)(20,80)(21,33)(22,34)(23,35)(24,31)(25,32)(36,43)(37,44)(38,45)(39,41)(40,42)(46,51)(47,52)(48,53)(49,54)(50,55)(56,70)(57,66)(58,67)(59,68)(60,69)(61,73)(62,74)(63,75)(64,71)(65,72) );
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)], [(1,35),(2,31),(3,32),(4,33),(5,34),(6,16),(7,17),(8,18),(9,19),(10,20),(11,76),(12,77),(13,78),(14,79),(15,80),(21,28),(22,29),(23,30),(24,26),(25,27),(36,55),(37,51),(38,52),(39,53),(40,54),(41,48),(42,49),(43,50),(44,46),(45,47),(56,75),(57,71),(58,72),(59,73),(60,74),(61,68),(62,69),(63,70),(64,66),(65,67)], [(1,30),(2,26),(3,27),(4,28),(5,29),(6,76),(7,77),(8,78),(9,79),(10,80),(11,16),(12,17),(13,18),(14,19),(15,20),(21,33),(22,34),(23,35),(24,31),(25,32),(36,50),(37,46),(38,47),(39,48),(40,49),(41,53),(42,54),(43,55),(44,51),(45,52),(56,70),(57,66),(58,67),(59,68),(60,69),(61,73),(62,74),(63,75),(64,71),(65,72)], [(1,75,35,56),(2,71,31,57),(3,72,32,58),(4,73,33,59),(5,74,34,60),(6,50,16,43),(7,46,17,44),(8,47,18,45),(9,48,19,41),(10,49,20,42),(11,55,76,36),(12,51,77,37),(13,52,78,38),(14,53,79,39),(15,54,80,40),(21,68,28,61),(22,69,29,62),(23,70,30,63),(24,66,26,64),(25,67,27,65)], [(1,50,30,36),(2,46,26,37),(3,47,27,38),(4,48,28,39),(5,49,29,40),(6,63,76,75),(7,64,77,71),(8,65,78,72),(9,61,79,73),(10,62,80,74),(11,56,16,70),(12,57,17,66),(13,58,18,67),(14,59,19,68),(15,60,20,69),(21,53,33,41),(22,54,34,42),(23,55,35,43),(24,51,31,44),(25,52,32,45)], [(6,76),(7,77),(8,78),(9,79),(10,80),(11,16),(12,17),(13,18),(14,19),(15,20),(56,70),(57,66),(58,67),(59,68),(60,69),(61,73),(62,74),(63,75),(64,71),(65,72)], [(1,30),(2,26),(3,27),(4,28),(5,29),(6,11),(7,12),(8,13),(9,14),(10,15),(16,76),(17,77),(18,78),(19,79),(20,80),(21,33),(22,34),(23,35),(24,31),(25,32),(36,43),(37,44),(38,45),(39,41),(40,42),(46,51),(47,52),(48,53),(49,54),(50,55),(56,70),(57,66),(58,67),(59,68),(60,69),(61,73),(62,74),(63,75),(64,71),(65,72)])
Matrix representation ►G ⊆ GL4(𝔽41) generated by
| 37 | 0 | 0 | 0 |
| 0 | 37 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 5 | 39 | 0 | 0 |
| 12 | 36 | 0 | 0 |
| 0 | 0 | 32 | 0 |
| 0 | 0 | 40 | 9 |
| 9 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 |
| 0 | 0 | 9 | 2 |
| 0 | 0 | 1 | 32 |
| 1 | 0 | 0 | 0 |
| 5 | 40 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 32 | 40 |
G:=sub<GL(4,GF(41))| [37,0,0,0,0,37,0,0,0,0,16,0,0,0,0,16],[1,0,0,0,0,1,0,0,0,0,40,0,0,0,0,40],[40,0,0,0,0,40,0,0,0,0,1,0,0,0,0,1],[5,12,0,0,39,36,0,0,0,0,32,40,0,0,0,9],[9,0,0,0,0,9,0,0,0,0,9,1,0,0,2,32],[1,5,0,0,0,40,0,0,0,0,1,0,0,0,0,1],[40,0,0,0,0,40,0,0,0,0,1,32,0,0,0,40] >;
125 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | ··· | 4H | 4I | ··· | 4O | 5A | 5B | 5C | 5D | 10A | ··· | 10L | 10M | ··· | 10AB | 10AC | ··· | 10AJ | 20A | ··· | 20AF | 20AG | ··· | 20BH |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 5 | 5 | 5 | 5 | 10 | ··· | 10 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
125 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | ||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C5 | C10 | C10 | C10 | C10 | C10 | C10 | C10 | C10 | C4○D4 | C5×C4○D4 | 2+ (1+4) | C5×2+ (1+4) |
| kernel | C5×C22.45C24 | C10×C22⋊C4 | C5×C42⋊C2 | D4×C20 | C5×C22≀C2 | C5×C22⋊Q8 | C5×C22.D4 | C5×C4.4D4 | C5×C42⋊2C2 | C22.45C24 | C2×C22⋊C4 | C42⋊C2 | C4×D4 | C22≀C2 | C22⋊Q8 | C22.D4 | C4.4D4 | C42⋊2C2 | C2×C10 | C22 | C10 | C2 |
| # reps | 1 | 2 | 2 | 2 | 1 | 2 | 3 | 1 | 2 | 4 | 8 | 8 | 8 | 4 | 8 | 12 | 4 | 8 | 8 | 32 | 1 | 4 |
In GAP, Magma, Sage, TeX
C_5\times C_2^2._{45}C_2^4 % in TeX
G:=Group("C5xC2^2.45C2^4"); // GroupNames label
G:=SmallGroup(320,1553);
// by ID
G=gap.SmallGroup(320,1553);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-5,-2,-2,560,1149,1128,3446,1242]);
// Polycyclic
G:=Group<a,b,c,d,e,f,g|a^5=b^2=c^2=f^2=g^2=1,d^2=b,e^2=c,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,e*d*e^-1=b*d=d*b,g*e*g=b*e=e*b,b*f=f*b,b*g=g*b,f*d*f=c*d=d*c,c*e=e*c,c*f=f*c,c*g=g*c,d*g=g*d,e*f=f*e,f*g=g*f>;
// generators/relations