direct product, p-group, metabelian, nilpotent (class 3), monomial
Aliases: C2×C23.46D4, C24.181D4, C23.44SD16, C4⋊C4.50C23, C4.Q8⋊63C22, C22⋊C8⋊67C22, (C2×C8).311C23, (C2×C4).287C24, (C2×D4).77C23, (C22×C4).438D4, C23.663(C2×D4), D4⋊C4⋊77C22, C2.13(C22×SD16), C22.35(C2×SD16), C4⋊D4.153C22, (C23×C4).557C22, (C22×C8).348C22, C22.547(C22×D4), C22.121(C8⋊C22), (C22×C4).1004C23, C4.59(C22.D4), (C22×D4).358C22, C22.110(C22.D4), (C2×C4.Q8)⋊34C2, C4.97(C2×C4○D4), (C2×C22⋊C8)⋊35C2, (C22×C4⋊C4)⋊34C2, (C2×C4).847(C2×D4), C2.26(C2×C8⋊C22), (C2×D4⋊C4)⋊39C2, (C2×C4⋊C4)⋊117C22, (C2×C4⋊D4).57C2, (C2×C4).845(C4○D4), C2.52(C2×C22.D4), SmallGroup(128,1821)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 540 in 256 conjugacy classes, 108 normal (18 characteristic)
C1, C2 [×3], C2 [×4], C2 [×6], C4 [×4], C4 [×8], C22, C22 [×10], C22 [×22], C8 [×4], C2×C4 [×2], C2×C4 [×6], C2×C4 [×30], D4 [×14], C23, C23 [×6], C23 [×12], C22⋊C4 [×4], C4⋊C4 [×6], C4⋊C4 [×7], C2×C8 [×4], C2×C8 [×4], C22×C4 [×2], C22×C4 [×4], C22×C4 [×15], C2×D4 [×2], C2×D4 [×13], C24, C24, C22⋊C8 [×4], D4⋊C4 [×8], C4.Q8 [×8], C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4 [×6], C2×C4⋊C4 [×3], C4⋊D4 [×4], C4⋊D4 [×2], C22×C8 [×2], C23×C4, C23×C4, C22×D4, C22×D4, C2×C22⋊C8, C2×D4⋊C4 [×2], C2×C4.Q8 [×2], C23.46D4 [×8], C22×C4⋊C4, C2×C4⋊D4, C2×C23.46D4
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], SD16 [×4], C2×D4 [×6], C4○D4 [×4], C24, C22.D4 [×4], C2×SD16 [×6], C8⋊C22 [×2], C22×D4, C2×C4○D4 [×2], C23.46D4 [×4], C2×C22.D4, C22×SD16, C2×C8⋊C22, C2×C23.46D4
Generators and relations
G = < a,b,c,d,e,f | a2=b2=c2=d2=f2=1, e4=d, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ebe-1=fbf=bc=cb, bd=db, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef=ce3 >
►On 64 points
(1 35)(2 36)(3 37)(4 38)(5 39)(6 40)(7 33)(8 34)(9 30)(10 31)(11 32)(12 25)(13 26)(14 27)(15 28)(16 29)(17 52)(18 53)(19 54)(20 55)(21 56)(22 49)(23 50)(24 51)(41 64)(42 57)(43 58)(44 59)(45 60)(46 61)(47 62)(48 63)
(1 19)(2 43)(3 21)(4 45)(5 23)(6 47)(7 17)(8 41)(9 55)(10 59)(11 49)(12 61)(13 51)(14 63)(15 53)(16 57)(18 28)(20 30)(22 32)(24 26)(25 46)(27 48)(29 42)(31 44)(33 52)(34 64)(35 54)(36 58)(37 56)(38 60)(39 50)(40 62)
(1 29)(2 30)(3 31)(4 32)(5 25)(6 26)(7 27)(8 28)(9 36)(10 37)(11 38)(12 39)(13 40)(14 33)(15 34)(16 35)(17 48)(18 41)(19 42)(20 43)(21 44)(22 45)(23 46)(24 47)(49 60)(50 61)(51 62)(52 63)(53 64)(54 57)(55 58)(56 59)
(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)(25 29)(26 30)(27 31)(28 32)(33 37)(34 38)(35 39)(36 40)(41 45)(42 46)(43 47)(44 48)(49 53)(50 54)(51 55)(52 56)(57 61)(58 62)(59 63)(60 64)
(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)
(1 12)(2 34)(3 10)(4 40)(5 16)(6 38)(7 14)(8 36)(9 28)(11 26)(13 32)(15 30)(17 52)(18 58)(19 50)(20 64)(21 56)(22 62)(23 54)(24 60)(25 35)(27 33)(29 39)(31 37)(41 55)(42 61)(43 53)(44 59)(45 51)(46 57)(47 49)(48 63)
G:=sub<Sym(64)| (1,35)(2,36)(3,37)(4,38)(5,39)(6,40)(7,33)(8,34)(9,30)(10,31)(11,32)(12,25)(13,26)(14,27)(15,28)(16,29)(17,52)(18,53)(19,54)(20,55)(21,56)(22,49)(23,50)(24,51)(41,64)(42,57)(43,58)(44,59)(45,60)(46,61)(47,62)(48,63), (1,19)(2,43)(3,21)(4,45)(5,23)(6,47)(7,17)(8,41)(9,55)(10,59)(11,49)(12,61)(13,51)(14,63)(15,53)(16,57)(18,28)(20,30)(22,32)(24,26)(25,46)(27,48)(29,42)(31,44)(33,52)(34,64)(35,54)(36,58)(37,56)(38,60)(39,50)(40,62), (1,29)(2,30)(3,31)(4,32)(5,25)(6,26)(7,27)(8,28)(9,36)(10,37)(11,38)(12,39)(13,40)(14,33)(15,34)(16,35)(17,48)(18,41)(19,42)(20,43)(21,44)(22,45)(23,46)(24,47)(49,60)(50,61)(51,62)(52,63)(53,64)(54,57)(55,58)(56,59), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32)(33,37)(34,38)(35,39)(36,40)(41,45)(42,46)(43,47)(44,48)(49,53)(50,54)(51,55)(52,56)(57,61)(58,62)(59,63)(60,64), (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), (1,12)(2,34)(3,10)(4,40)(5,16)(6,38)(7,14)(8,36)(9,28)(11,26)(13,32)(15,30)(17,52)(18,58)(19,50)(20,64)(21,56)(22,62)(23,54)(24,60)(25,35)(27,33)(29,39)(31,37)(41,55)(42,61)(43,53)(44,59)(45,51)(46,57)(47,49)(48,63)>;
G:=Group( (1,35)(2,36)(3,37)(4,38)(5,39)(6,40)(7,33)(8,34)(9,30)(10,31)(11,32)(12,25)(13,26)(14,27)(15,28)(16,29)(17,52)(18,53)(19,54)(20,55)(21,56)(22,49)(23,50)(24,51)(41,64)(42,57)(43,58)(44,59)(45,60)(46,61)(47,62)(48,63), (1,19)(2,43)(3,21)(4,45)(5,23)(6,47)(7,17)(8,41)(9,55)(10,59)(11,49)(12,61)(13,51)(14,63)(15,53)(16,57)(18,28)(20,30)(22,32)(24,26)(25,46)(27,48)(29,42)(31,44)(33,52)(34,64)(35,54)(36,58)(37,56)(38,60)(39,50)(40,62), (1,29)(2,30)(3,31)(4,32)(5,25)(6,26)(7,27)(8,28)(9,36)(10,37)(11,38)(12,39)(13,40)(14,33)(15,34)(16,35)(17,48)(18,41)(19,42)(20,43)(21,44)(22,45)(23,46)(24,47)(49,60)(50,61)(51,62)(52,63)(53,64)(54,57)(55,58)(56,59), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32)(33,37)(34,38)(35,39)(36,40)(41,45)(42,46)(43,47)(44,48)(49,53)(50,54)(51,55)(52,56)(57,61)(58,62)(59,63)(60,64), (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), (1,12)(2,34)(3,10)(4,40)(5,16)(6,38)(7,14)(8,36)(9,28)(11,26)(13,32)(15,30)(17,52)(18,58)(19,50)(20,64)(21,56)(22,62)(23,54)(24,60)(25,35)(27,33)(29,39)(31,37)(41,55)(42,61)(43,53)(44,59)(45,51)(46,57)(47,49)(48,63) );
G=PermutationGroup([(1,35),(2,36),(3,37),(4,38),(5,39),(6,40),(7,33),(8,34),(9,30),(10,31),(11,32),(12,25),(13,26),(14,27),(15,28),(16,29),(17,52),(18,53),(19,54),(20,55),(21,56),(22,49),(23,50),(24,51),(41,64),(42,57),(43,58),(44,59),(45,60),(46,61),(47,62),(48,63)], [(1,19),(2,43),(3,21),(4,45),(5,23),(6,47),(7,17),(8,41),(9,55),(10,59),(11,49),(12,61),(13,51),(14,63),(15,53),(16,57),(18,28),(20,30),(22,32),(24,26),(25,46),(27,48),(29,42),(31,44),(33,52),(34,64),(35,54),(36,58),(37,56),(38,60),(39,50),(40,62)], [(1,29),(2,30),(3,31),(4,32),(5,25),(6,26),(7,27),(8,28),(9,36),(10,37),(11,38),(12,39),(13,40),(14,33),(15,34),(16,35),(17,48),(18,41),(19,42),(20,43),(21,44),(22,45),(23,46),(24,47),(49,60),(50,61),(51,62),(52,63),(53,64),(54,57),(55,58),(56,59)], [(1,5),(2,6),(3,7),(4,8),(9,13),(10,14),(11,15),(12,16),(17,21),(18,22),(19,23),(20,24),(25,29),(26,30),(27,31),(28,32),(33,37),(34,38),(35,39),(36,40),(41,45),(42,46),(43,47),(44,48),(49,53),(50,54),(51,55),(52,56),(57,61),(58,62),(59,63),(60,64)], [(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)], [(1,12),(2,34),(3,10),(4,40),(5,16),(6,38),(7,14),(8,36),(9,28),(11,26),(13,32),(15,30),(17,52),(18,58),(19,50),(20,64),(21,56),(22,62),(23,54),(24,60),(25,35),(27,33),(29,39),(31,37),(41,55),(42,61),(43,53),(44,59),(45,51),(46,57),(47,49),(48,63)])
Matrix representation ►G ⊆ GL6(𝔽17)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 8 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 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 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 13 | 2 | 0 | 0 | 0 | 0 |
| 1 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 13 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 5 |
| 0 | 0 | 0 | 0 | 12 | 12 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 13 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,8,16,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[13,1,0,0,0,0,2,4,0,0,0,0,0,0,4,0,0,0,0,0,0,13,0,0,0,0,0,0,12,12,0,0,0,0,5,12],[16,13,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1] >;
38 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 2L | 2M | 4A | 4B | 4C | 4D | 4E | ··· | 4N | 4O | 4P | 8A | ··· | 8H |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 8 | 8 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | 8 | 4 | ··· | 4 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | C4○D4 | SD16 | C8⋊C22 |
| kernel | C2×C23.46D4 | C2×C22⋊C8 | C2×D4⋊C4 | C2×C4.Q8 | C23.46D4 | C22×C4⋊C4 | C2×C4⋊D4 | C22×C4 | C24 | C2×C4 | C23 | C22 |
| # reps | 1 | 1 | 2 | 2 | 8 | 1 | 1 | 3 | 1 | 8 | 8 | 2 |
In GAP, Magma, Sage, TeX
C_2\times C_2^3._{46}D_4 % in TeX
G:=Group("C2xC2^3.46D4"); // GroupNames label
G:=SmallGroup(128,1821);
// by ID
G=gap.SmallGroup(128,1821);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,758,436,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=f^2=1,e^4=d,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,e*b*e^-1=f*b*f=b*c=c*b,b*d=d*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f=c*e^3>;
// generators/relations