direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C5×D4⋊5D4, C10.1612+ (1+4), D4⋊5(C5×D4), (C5×D4)⋊23D4, (D4×C20)⋊44C2, (C4×D4)⋊15C10, C42⋊9(C2×C10), C4.40(D4×C10), C22≀C2⋊6C10, C4⋊D4⋊11C10, (C4×C20)⋊43C22, C20.401(C2×D4), (C22×D4)⋊9C10, C22⋊Q8⋊11C10, C22.5(D4×C10), C4.4D4⋊10C10, (D4×C10)⋊38C22, C24.19(C2×C10), (Q8×C10)⋊52C22, (C2×C20).713C23, (C2×C10).366C24, (C22×C20)⋊51C22, C10.194(C22×D4), C22.D4⋊8C10, (C22×C10).98C23, C22.40(C23×C10), C23.16(C22×C10), (C23×C10).19C22, C2.13(C5×2+ (1+4)), C4⋊C4⋊5(C2×C10), (D4×C2×C10)⋊24C2, C2.18(D4×C2×C10), (C2×C4○D4)⋊6C10, C22⋊2(C5×C4○D4), (C10×C4○D4)⋊22C2, (C2×D4)⋊13(C2×C10), (C5×C4⋊D4)⋊38C2, (C5×C4⋊C4)⋊73C22, (C2×Q8)⋊12(C2×C10), C2.20(C10×C4○D4), (C5×C22⋊Q8)⋊38C2, (C5×C22≀C2)⋊16C2, (C2×C10)⋊14(C4○D4), C22⋊C4⋊16(C2×C10), (C2×C22⋊C4)⋊14C10, (C10×C22⋊C4)⋊34C2, (C22×C4)⋊11(C2×C10), C10.239(C2×C4○D4), (C2×C10).182(C2×D4), (C5×C4.4D4)⋊30C2, (C5×C22⋊C4)⋊70C22, (C2×C4).59(C22×C10), (C5×C22.D4)⋊27C2, SmallGroup(320,1548)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 570 in 334 conjugacy classes, 166 normal (62 characteristic)
C1, C2 [×3], C2 [×9], C4 [×2], C4 [×8], C22, C22 [×6], C22 [×23], C5, C2×C4 [×5], C2×C4 [×4], C2×C4 [×10], D4 [×4], D4 [×14], Q8 [×2], C23 [×2], C23 [×4], C23 [×10], C10 [×3], C10 [×9], C42, C22⋊C4 [×2], C22⋊C4 [×10], C4⋊C4 [×2], C4⋊C4 [×2], C22×C4 [×2], C22×C4 [×4], C2×D4 [×3], C2×D4 [×6], C2×D4 [×4], C2×Q8, C4○D4 [×4], C24 [×2], C20 [×2], C20 [×8], C2×C10, C2×C10 [×6], C2×C10 [×23], C2×C22⋊C4 [×2], C4×D4 [×2], C22≀C2 [×2], C4⋊D4, C4⋊D4 [×2], C22⋊Q8, C22.D4 [×2], C4.4D4, C22×D4, C2×C4○D4, C2×C20 [×5], C2×C20 [×4], C2×C20 [×10], C5×D4 [×4], C5×D4 [×14], C5×Q8 [×2], C22×C10 [×2], C22×C10 [×4], C22×C10 [×10], D4⋊5D4, C4×C20, C5×C22⋊C4 [×2], C5×C22⋊C4 [×10], C5×C4⋊C4 [×2], C5×C4⋊C4 [×2], C22×C20 [×2], C22×C20 [×4], D4×C10 [×3], D4×C10 [×6], D4×C10 [×4], Q8×C10, C5×C4○D4 [×4], C23×C10 [×2], C10×C22⋊C4 [×2], D4×C20 [×2], C5×C22≀C2 [×2], C5×C4⋊D4, C5×C4⋊D4 [×2], C5×C22⋊Q8, C5×C22.D4 [×2], C5×C4.4D4, D4×C2×C10, C10×C4○D4, C5×D4⋊5D4
Quotients:
C1, C2 [×15], C22 [×35], C5, D4 [×4], C23 [×15], C10 [×15], C2×D4 [×6], C4○D4 [×2], C24, C2×C10 [×35], C22×D4, C2×C4○D4, 2+ (1+4), C5×D4 [×4], C22×C10 [×15], D4⋊5D4, D4×C10 [×6], C5×C4○D4 [×2], C23×C10, D4×C2×C10, C10×C4○D4, C5×2+ (1+4), C5×D4⋊5D4
Generators and relations
G = < a,b,c,d,e | a5=b4=c2=d4=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, dcd-1=ece=b2c, ede=d-1 >
►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 46 26 36)(2 47 27 37)(3 48 28 38)(4 49 29 39)(5 50 30 40)(6 71 16 61)(7 72 17 62)(8 73 18 63)(9 74 19 64)(10 75 20 65)(11 56 76 66)(12 57 77 67)(13 58 78 68)(14 59 79 69)(15 60 80 70)(21 51 31 41)(22 52 32 42)(23 53 33 43)(24 54 34 44)(25 55 35 45)
(1 36)(2 37)(3 38)(4 39)(5 40)(6 71)(7 72)(8 73)(9 74)(10 75)(11 56)(12 57)(13 58)(14 59)(15 60)(16 61)(17 62)(18 63)(19 64)(20 65)(21 41)(22 42)(23 43)(24 44)(25 45)(26 46)(27 47)(28 48)(29 49)(30 50)(31 51)(32 52)(33 53)(34 54)(35 55)(66 76)(67 77)(68 78)(69 79)(70 80)
(1 16 21 11)(2 17 22 12)(3 18 23 13)(4 19 24 14)(5 20 25 15)(6 31 76 26)(7 32 77 27)(8 33 78 28)(9 34 79 29)(10 35 80 30)(36 71 41 66)(37 72 42 67)(38 73 43 68)(39 74 44 69)(40 75 45 70)(46 61 51 56)(47 62 52 57)(48 63 53 58)(49 64 54 59)(50 65 55 60)
(1 76)(2 77)(3 78)(4 79)(5 80)(6 21)(7 22)(8 23)(9 24)(10 25)(11 26)(12 27)(13 28)(14 29)(15 30)(16 31)(17 32)(18 33)(19 34)(20 35)(36 56)(37 57)(38 58)(39 59)(40 60)(41 61)(42 62)(43 63)(44 64)(45 65)(46 66)(47 67)(48 68)(49 69)(50 70)(51 71)(52 72)(53 73)(54 74)(55 75)
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), (1,46,26,36)(2,47,27,37)(3,48,28,38)(4,49,29,39)(5,50,30,40)(6,71,16,61)(7,72,17,62)(8,73,18,63)(9,74,19,64)(10,75,20,65)(11,56,76,66)(12,57,77,67)(13,58,78,68)(14,59,79,69)(15,60,80,70)(21,51,31,41)(22,52,32,42)(23,53,33,43)(24,54,34,44)(25,55,35,45), (1,36)(2,37)(3,38)(4,39)(5,40)(6,71)(7,72)(8,73)(9,74)(10,75)(11,56)(12,57)(13,58)(14,59)(15,60)(16,61)(17,62)(18,63)(19,64)(20,65)(21,41)(22,42)(23,43)(24,44)(25,45)(26,46)(27,47)(28,48)(29,49)(30,50)(31,51)(32,52)(33,53)(34,54)(35,55)(66,76)(67,77)(68,78)(69,79)(70,80), (1,16,21,11)(2,17,22,12)(3,18,23,13)(4,19,24,14)(5,20,25,15)(6,31,76,26)(7,32,77,27)(8,33,78,28)(9,34,79,29)(10,35,80,30)(36,71,41,66)(37,72,42,67)(38,73,43,68)(39,74,44,69)(40,75,45,70)(46,61,51,56)(47,62,52,57)(48,63,53,58)(49,64,54,59)(50,65,55,60), (1,76)(2,77)(3,78)(4,79)(5,80)(6,21)(7,22)(8,23)(9,24)(10,25)(11,26)(12,27)(13,28)(14,29)(15,30)(16,31)(17,32)(18,33)(19,34)(20,35)(36,56)(37,57)(38,58)(39,59)(40,60)(41,61)(42,62)(43,63)(44,64)(45,65)(46,66)(47,67)(48,68)(49,69)(50,70)(51,71)(52,72)(53,73)(54,74)(55,75)>;
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), (1,46,26,36)(2,47,27,37)(3,48,28,38)(4,49,29,39)(5,50,30,40)(6,71,16,61)(7,72,17,62)(8,73,18,63)(9,74,19,64)(10,75,20,65)(11,56,76,66)(12,57,77,67)(13,58,78,68)(14,59,79,69)(15,60,80,70)(21,51,31,41)(22,52,32,42)(23,53,33,43)(24,54,34,44)(25,55,35,45), (1,36)(2,37)(3,38)(4,39)(5,40)(6,71)(7,72)(8,73)(9,74)(10,75)(11,56)(12,57)(13,58)(14,59)(15,60)(16,61)(17,62)(18,63)(19,64)(20,65)(21,41)(22,42)(23,43)(24,44)(25,45)(26,46)(27,47)(28,48)(29,49)(30,50)(31,51)(32,52)(33,53)(34,54)(35,55)(66,76)(67,77)(68,78)(69,79)(70,80), (1,16,21,11)(2,17,22,12)(3,18,23,13)(4,19,24,14)(5,20,25,15)(6,31,76,26)(7,32,77,27)(8,33,78,28)(9,34,79,29)(10,35,80,30)(36,71,41,66)(37,72,42,67)(38,73,43,68)(39,74,44,69)(40,75,45,70)(46,61,51,56)(47,62,52,57)(48,63,53,58)(49,64,54,59)(50,65,55,60), (1,76)(2,77)(3,78)(4,79)(5,80)(6,21)(7,22)(8,23)(9,24)(10,25)(11,26)(12,27)(13,28)(14,29)(15,30)(16,31)(17,32)(18,33)(19,34)(20,35)(36,56)(37,57)(38,58)(39,59)(40,60)(41,61)(42,62)(43,63)(44,64)(45,65)(46,66)(47,67)(48,68)(49,69)(50,70)(51,71)(52,72)(53,73)(54,74)(55,75) );
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)], [(1,46,26,36),(2,47,27,37),(3,48,28,38),(4,49,29,39),(5,50,30,40),(6,71,16,61),(7,72,17,62),(8,73,18,63),(9,74,19,64),(10,75,20,65),(11,56,76,66),(12,57,77,67),(13,58,78,68),(14,59,79,69),(15,60,80,70),(21,51,31,41),(22,52,32,42),(23,53,33,43),(24,54,34,44),(25,55,35,45)], [(1,36),(2,37),(3,38),(4,39),(5,40),(6,71),(7,72),(8,73),(9,74),(10,75),(11,56),(12,57),(13,58),(14,59),(15,60),(16,61),(17,62),(18,63),(19,64),(20,65),(21,41),(22,42),(23,43),(24,44),(25,45),(26,46),(27,47),(28,48),(29,49),(30,50),(31,51),(32,52),(33,53),(34,54),(35,55),(66,76),(67,77),(68,78),(69,79),(70,80)], [(1,16,21,11),(2,17,22,12),(3,18,23,13),(4,19,24,14),(5,20,25,15),(6,31,76,26),(7,32,77,27),(8,33,78,28),(9,34,79,29),(10,35,80,30),(36,71,41,66),(37,72,42,67),(38,73,43,68),(39,74,44,69),(40,75,45,70),(46,61,51,56),(47,62,52,57),(48,63,53,58),(49,64,54,59),(50,65,55,60)], [(1,76),(2,77),(3,78),(4,79),(5,80),(6,21),(7,22),(8,23),(9,24),(10,25),(11,26),(12,27),(13,28),(14,29),(15,30),(16,31),(17,32),(18,33),(19,34),(20,35),(36,56),(37,57),(38,58),(39,59),(40,60),(41,61),(42,62),(43,63),(44,64),(45,65),(46,66),(47,67),(48,68),(49,69),(50,70),(51,71),(52,72),(53,73),(54,74),(55,75)])
Matrix representation ►G ⊆ GL4(𝔽41) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 18 | 0 |
| 0 | 0 | 0 | 18 |
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 1 | 2 |
| 0 | 0 | 40 | 40 |
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 1 | 2 |
| 0 | 0 | 0 | 40 |
| 0 | 40 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 9 | 18 |
| 0 | 0 | 32 | 32 |
| 0 | 40 | 0 | 0 |
| 40 | 0 | 0 | 0 |
| 0 | 0 | 32 | 23 |
| 0 | 0 | 9 | 9 |
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,18,0,0,0,0,18],[40,0,0,0,0,40,0,0,0,0,1,40,0,0,2,40],[40,0,0,0,0,40,0,0,0,0,1,0,0,0,2,40],[0,1,0,0,40,0,0,0,0,0,9,32,0,0,18,32],[0,40,0,0,40,0,0,0,0,0,32,9,0,0,23,9] >;
125 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | ··· | 2I | 2J | 2K | 2L | 4A | ··· | 4F | 4G | ··· | 4L | 5A | 5B | 5C | 5D | 10A | ··· | 10L | 10M | ··· | 10AJ | 10AK | ··· | 10AV | 20A | ··· | 20X | 20Y | ··· | 20AV |
| order | 1 | 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 | 4 | 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 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | ||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C5 | C10 | C10 | C10 | C10 | C10 | C10 | C10 | C10 | C10 | D4 | C4○D4 | C5×D4 | C5×C4○D4 | 2+ (1+4) | C5×2+ (1+4) |
| kernel | C5×D4⋊5D4 | C10×C22⋊C4 | D4×C20 | C5×C22≀C2 | C5×C4⋊D4 | C5×C22⋊Q8 | C5×C22.D4 | C5×C4.4D4 | D4×C2×C10 | C10×C4○D4 | D4⋊5D4 | C2×C22⋊C4 | C4×D4 | C22≀C2 | C4⋊D4 | C22⋊Q8 | C22.D4 | C4.4D4 | C22×D4 | C2×C4○D4 | C5×D4 | C2×C10 | D4 | C22 | C10 | C2 |
| # reps | 1 | 2 | 2 | 2 | 3 | 1 | 2 | 1 | 1 | 1 | 4 | 8 | 8 | 8 | 12 | 4 | 8 | 4 | 4 | 4 | 4 | 4 | 16 | 16 | 1 | 4 |
In GAP, Magma, Sage, TeX
C_5\times D_4\rtimes_5D_4
% in TeX
G:=Group("C5xD4:5D4"); // GroupNames label
G:=SmallGroup(320,1548);
// by ID
G=gap.SmallGroup(320,1548);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-5,-2,-2,560,1149,3446,1242]);
// Polycyclic
G:=Group<a,b,c,d,e|a^5=b^4=c^2=d^4=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,d*c*d^-1=e*c*e=b^2*c,e*d*e=d^-1>;
// generators/relations