direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C5×C23⋊3D4, C10.1512+ (1+4), C23⋊3(C5×D4), C4⋊D4⋊5C10, C24⋊3(C2×C10), (C22×C10)⋊6D4, C22≀C2⋊2C10, (C22×D4)⋊6C10, C22.2(D4×C10), (D4×C10)⋊35C22, (C23×C10)⋊3C22, (C2×C10).354C24, (C2×C20).663C23, (C22×C20)⋊47C22, C22.D4⋊2C10, C10.189(C22×D4), C2.3(C5×2+ (1+4)), C23.36(C22×C10), (C22×C10).90C23, C22.28(C23×C10), C4⋊C4⋊3(C2×C10), (D4×C2×C10)⋊21C2, C2.13(D4×C2×C10), (C2×D4)⋊3(C2×C10), (C5×C4⋊D4)⋊32C2, (C5×C4⋊C4)⋊37C22, C22⋊C4⋊3(C2×C10), (C22×C4)⋊7(C2×C10), (C2×C10).90(C2×D4), (C5×C22≀C2)⋊12C2, (C2×C22⋊C4)⋊12C10, (C10×C22⋊C4)⋊32C2, (C5×C22⋊C4)⋊38C22, (C2×C4).21(C22×C10), (C5×C22.D4)⋊21C2, SmallGroup(320,1536)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 642 in 346 conjugacy classes, 162 normal (14 characteristic)
C1, C2, C2 [×2], C2 [×10], C4 [×8], C22, C22 [×6], C22 [×30], C5, C2×C4 [×8], C2×C4 [×6], D4 [×20], C23, C23 [×10], C23 [×10], C10, C10 [×2], C10 [×10], C22⋊C4 [×12], C4⋊C4 [×4], C22×C4 [×4], C2×D4 [×12], C2×D4 [×8], C24, C24 [×2], C20 [×8], C2×C10, C2×C10 [×6], C2×C10 [×30], C2×C22⋊C4, C22≀C2 [×4], C4⋊D4 [×4], C22.D4 [×4], C22×D4 [×2], C2×C20 [×8], C2×C20 [×6], C5×D4 [×20], C22×C10, C22×C10 [×10], C22×C10 [×10], C23⋊3D4, C5×C22⋊C4 [×12], C5×C4⋊C4 [×4], C22×C20 [×4], D4×C10 [×12], D4×C10 [×8], C23×C10, C23×C10 [×2], C10×C22⋊C4, C5×C22≀C2 [×4], C5×C4⋊D4 [×4], C5×C22.D4 [×4], D4×C2×C10 [×2], C5×C23⋊3D4
Quotients:
C1, C2 [×15], C22 [×35], C5, D4 [×4], C23 [×15], C10 [×15], C2×D4 [×6], C24, C2×C10 [×35], C22×D4, 2+ (1+4) [×2], C5×D4 [×4], C22×C10 [×15], C23⋊3D4, D4×C10 [×6], C23×C10, D4×C2×C10, C5×2+ (1+4) [×2], C5×C23⋊3D4
Generators and relations
G = < a,b,c,d,e,f | a5=b2=c2=d2=e4=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, fbf=bd=db, be=eb, ece-1=fcf=cd=dc, de=ed, df=fd, fef=e-1 >
►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 36)(2 37)(3 38)(4 39)(5 40)(6 63)(7 64)(8 65)(9 61)(10 62)(11 56)(12 57)(13 58)(14 59)(15 60)(16 70)(17 66)(18 67)(19 68)(20 69)(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)(71 77)(72 78)(73 79)(74 80)(75 76)
(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 50)(37 46)(38 47)(39 48)(40 49)(41 53)(42 54)(43 55)(44 51)(45 52)(56 63)(57 64)(58 65)(59 61)(60 62)(66 71)(67 72)(68 73)(69 74)(70 75)
(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 70 30 56)(2 66 26 57)(3 67 27 58)(4 68 28 59)(5 69 29 60)(6 43 76 55)(7 44 77 51)(8 45 78 52)(9 41 79 53)(10 42 80 54)(11 36 16 50)(12 37 17 46)(13 38 18 47)(14 39 19 48)(15 40 20 49)(21 73 33 61)(22 74 34 62)(23 75 35 63)(24 71 31 64)(25 72 32 65)
(1 56)(2 57)(3 58)(4 59)(5 60)(6 50)(7 46)(8 47)(9 48)(10 49)(11 55)(12 51)(13 52)(14 53)(15 54)(16 43)(17 44)(18 45)(19 41)(20 42)(21 61)(22 62)(23 63)(24 64)(25 65)(26 66)(27 67)(28 68)(29 69)(30 70)(31 71)(32 72)(33 73)(34 74)(35 75)(36 76)(37 77)(38 78)(39 79)(40 80)
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), (1,36)(2,37)(3,38)(4,39)(5,40)(6,63)(7,64)(8,65)(9,61)(10,62)(11,56)(12,57)(13,58)(14,59)(15,60)(16,70)(17,66)(18,67)(19,68)(20,69)(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)(71,77)(72,78)(73,79)(74,80)(75,76), (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)(36,50)(37,46)(38,47)(39,48)(40,49)(41,53)(42,54)(43,55)(44,51)(45,52)(56,63)(57,64)(58,65)(59,61)(60,62)(66,71)(67,72)(68,73)(69,74)(70,75), (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), (1,70,30,56)(2,66,26,57)(3,67,27,58)(4,68,28,59)(5,69,29,60)(6,43,76,55)(7,44,77,51)(8,45,78,52)(9,41,79,53)(10,42,80,54)(11,36,16,50)(12,37,17,46)(13,38,18,47)(14,39,19,48)(15,40,20,49)(21,73,33,61)(22,74,34,62)(23,75,35,63)(24,71,31,64)(25,72,32,65), (1,56)(2,57)(3,58)(4,59)(5,60)(6,50)(7,46)(8,47)(9,48)(10,49)(11,55)(12,51)(13,52)(14,53)(15,54)(16,43)(17,44)(18,45)(19,41)(20,42)(21,61)(22,62)(23,63)(24,64)(25,65)(26,66)(27,67)(28,68)(29,69)(30,70)(31,71)(32,72)(33,73)(34,74)(35,75)(36,76)(37,77)(38,78)(39,79)(40,80)>;
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), (1,36)(2,37)(3,38)(4,39)(5,40)(6,63)(7,64)(8,65)(9,61)(10,62)(11,56)(12,57)(13,58)(14,59)(15,60)(16,70)(17,66)(18,67)(19,68)(20,69)(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)(71,77)(72,78)(73,79)(74,80)(75,76), (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)(36,50)(37,46)(38,47)(39,48)(40,49)(41,53)(42,54)(43,55)(44,51)(45,52)(56,63)(57,64)(58,65)(59,61)(60,62)(66,71)(67,72)(68,73)(69,74)(70,75), (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), (1,70,30,56)(2,66,26,57)(3,67,27,58)(4,68,28,59)(5,69,29,60)(6,43,76,55)(7,44,77,51)(8,45,78,52)(9,41,79,53)(10,42,80,54)(11,36,16,50)(12,37,17,46)(13,38,18,47)(14,39,19,48)(15,40,20,49)(21,73,33,61)(22,74,34,62)(23,75,35,63)(24,71,31,64)(25,72,32,65), (1,56)(2,57)(3,58)(4,59)(5,60)(6,50)(7,46)(8,47)(9,48)(10,49)(11,55)(12,51)(13,52)(14,53)(15,54)(16,43)(17,44)(18,45)(19,41)(20,42)(21,61)(22,62)(23,63)(24,64)(25,65)(26,66)(27,67)(28,68)(29,69)(30,70)(31,71)(32,72)(33,73)(34,74)(35,75)(36,76)(37,77)(38,78)(39,79)(40,80) );
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)], [(1,36),(2,37),(3,38),(4,39),(5,40),(6,63),(7,64),(8,65),(9,61),(10,62),(11,56),(12,57),(13,58),(14,59),(15,60),(16,70),(17,66),(18,67),(19,68),(20,69),(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),(71,77),(72,78),(73,79),(74,80),(75,76)], [(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),(36,50),(37,46),(38,47),(39,48),(40,49),(41,53),(42,54),(43,55),(44,51),(45,52),(56,63),(57,64),(58,65),(59,61),(60,62),(66,71),(67,72),(68,73),(69,74),(70,75)], [(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)], [(1,70,30,56),(2,66,26,57),(3,67,27,58),(4,68,28,59),(5,69,29,60),(6,43,76,55),(7,44,77,51),(8,45,78,52),(9,41,79,53),(10,42,80,54),(11,36,16,50),(12,37,17,46),(13,38,18,47),(14,39,19,48),(15,40,20,49),(21,73,33,61),(22,74,34,62),(23,75,35,63),(24,71,31,64),(25,72,32,65)], [(1,56),(2,57),(3,58),(4,59),(5,60),(6,50),(7,46),(8,47),(9,48),(10,49),(11,55),(12,51),(13,52),(14,53),(15,54),(16,43),(17,44),(18,45),(19,41),(20,42),(21,61),(22,62),(23,63),(24,64),(25,65),(26,66),(27,67),(28,68),(29,69),(30,70),(31,71),(32,72),(33,73),(34,74),(35,75),(36,76),(37,77),(38,78),(39,79),(40,80)])
Matrix representation ►G ⊆ GL6(𝔽41)
| 18 | 0 | 0 | 0 | 0 | 0 |
| 0 | 18 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 40 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 2 | 40 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 39 | 1 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
G:=sub<GL(6,GF(41))| [18,0,0,0,0,0,0,18,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0,0,40,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[0,1,0,0,0,0,40,0,0,0,0,0,0,0,0,0,40,39,0,0,0,0,0,1,0,0,1,2,0,0,0,0,0,40,0,0],[0,40,0,0,0,0,40,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0] >;
110 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | ··· | 2I | 2J | 2K | 2L | 2M | 4A | ··· | 4H | 5A | 5B | 5C | 5D | 10A | ··· | 10L | 10M | ··· | 10AJ | 10AK | ··· | 10AZ | 20A | ··· | 20AF |
| order | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 5 | 5 | 5 | 5 | 10 | ··· | 10 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
110 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | ||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C5 | C10 | C10 | C10 | C10 | C10 | D4 | C5×D4 | 2+ (1+4) | C5×2+ (1+4) |
| kernel | C5×C23⋊3D4 | C10×C22⋊C4 | C5×C22≀C2 | C5×C4⋊D4 | C5×C22.D4 | D4×C2×C10 | C23⋊3D4 | C2×C22⋊C4 | C22≀C2 | C4⋊D4 | C22.D4 | C22×D4 | C22×C10 | C23 | C10 | C2 |
| # reps | 1 | 1 | 4 | 4 | 4 | 2 | 4 | 4 | 16 | 16 | 16 | 8 | 4 | 16 | 2 | 8 |
In GAP, Magma, Sage, TeX
C_5\times C_2^3\rtimes_3D_4
% in TeX
G:=Group("C5xC2^3:3D4"); // GroupNames label
G:=SmallGroup(320,1536);
// by ID
G=gap.SmallGroup(320,1536);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-5,-2,-2,1149,3446,891,2467]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^5=b^2=c^2=d^2=e^4=f^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,f*b*f=b*d=d*b,b*e=e*b,e*c*e^-1=f*c*f=c*d=d*c,d*e=e*d,d*f=f*d,f*e*f=e^-1>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀