direct product, p-group, metabelian, nilpotent (class 3), monomial
Aliases: C2×C23.19D4, C24.113D4, C2.D8⋊52C22, C4.Q8⋊61C22, C4⋊C4.388C23, C22⋊C8⋊59C22, (C2×C4).285C24, (C2×C8).141C23, (C2×D4).76C23, C23.239(C2×D4), (C22×C4).436D4, D4⋊C4⋊66C22, C22.95(C4○D8), C42⋊C2⋊78C22, C4⋊D4.152C22, (C23×C4).555C22, (C22×C8).146C22, C22.545(C22×D4), C22.120(C8⋊C22), (C22×C4).1545C23, C4.57(C22.D4), (C22×D4).357C22, C22.108(C22.D4), (C2×C2.D8)⋊24C2, (C2×C4.Q8)⋊32C2, C2.20(C2×C4○D8), C4.95(C2×C4○D4), (C2×C22⋊C8)⋊26C2, C2.25(C2×C8⋊C22), (C2×D4⋊C4)⋊24C2, (C2×C4).1216(C2×D4), (C2×C4⋊D4).56C2, (C2×C42⋊C2)⋊44C2, (C2×C4).843(C4○D4), (C2×C4⋊C4).609C22, C2.50(C2×C22.D4), SmallGroup(128,1819)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 492 in 234 conjugacy classes, 100 normal (28 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×2], C4 [×2], C4 [×8], C22, C22 [×6], C22 [×20], C8 [×4], C2×C4 [×2], C2×C4 [×6], C2×C4 [×22], D4 [×14], C23, C23 [×2], C23 [×14], C42 [×4], C22⋊C4 [×8], C4⋊C4 [×6], C4⋊C4 [×3], C2×C8 [×4], C2×C8 [×4], C22×C4 [×2], C22×C4 [×4], C22×C4 [×7], C2×D4 [×2], C2×D4 [×13], C24, C24, C22⋊C8 [×4], D4⋊C4 [×8], C4.Q8 [×4], C2.D8 [×4], C2×C42, C2×C22⋊C4 [×2], C2×C4⋊C4 [×3], C42⋊C2 [×4], C42⋊C2 [×2], C4⋊D4 [×4], C4⋊D4 [×2], C22×C8 [×2], C23×C4, C22×D4, C22×D4, C2×C22⋊C8, C2×D4⋊C4 [×2], C2×C4.Q8, C2×C2.D8, C23.19D4 [×8], C2×C42⋊C2, C2×C4⋊D4, C2×C23.19D4
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×4], C24, C22.D4 [×4], C4○D8 [×2], C8⋊C22 [×2], C22×D4, C2×C4○D4 [×2], C23.19D4 [×4], C2×C22.D4, C2×C4○D8, C2×C8⋊C22, C2×C23.19D4
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=bc=cb, bd=db, fbf=bcd, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef=cde3 >
►On 64 points
(1 35)(2 36)(3 37)(4 38)(5 39)(6 40)(7 33)(8 34)(9 26)(10 27)(11 28)(12 29)(13 30)(14 31)(15 32)(16 25)(17 44)(18 45)(19 46)(20 47)(21 48)(22 41)(23 42)(24 43)(49 58)(50 59)(51 60)(52 61)(53 62)(54 63)(55 64)(56 57)
(1 50)(2 21)(3 52)(4 23)(5 54)(6 17)(7 56)(8 19)(9 64)(10 45)(11 58)(12 47)(13 60)(14 41)(15 62)(16 43)(18 27)(20 29)(22 31)(24 25)(26 55)(28 49)(30 51)(32 53)(33 57)(34 46)(35 59)(36 48)(37 61)(38 42)(39 63)(40 44)
(1 29)(2 30)(3 31)(4 32)(5 25)(6 26)(7 27)(8 28)(9 40)(10 33)(11 34)(12 35)(13 36)(14 37)(15 38)(16 39)(17 55)(18 56)(19 49)(20 50)(21 51)(22 52)(23 53)(24 54)(41 61)(42 62)(43 63)(44 64)(45 57)(46 58)(47 59)(48 60)
(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 16)(2 38)(3 14)(4 36)(5 12)(6 34)(7 10)(8 40)(9 28)(11 26)(13 32)(15 30)(17 62)(18 41)(19 60)(20 47)(21 58)(22 45)(23 64)(24 43)(25 35)(27 33)(29 39)(31 37)(42 55)(44 53)(46 51)(48 49)(50 59)(52 57)(54 63)(56 61)
G:=sub<Sym(64)| (1,35)(2,36)(3,37)(4,38)(5,39)(6,40)(7,33)(8,34)(9,26)(10,27)(11,28)(12,29)(13,30)(14,31)(15,32)(16,25)(17,44)(18,45)(19,46)(20,47)(21,48)(22,41)(23,42)(24,43)(49,58)(50,59)(51,60)(52,61)(53,62)(54,63)(55,64)(56,57), (1,50)(2,21)(3,52)(4,23)(5,54)(6,17)(7,56)(8,19)(9,64)(10,45)(11,58)(12,47)(13,60)(14,41)(15,62)(16,43)(18,27)(20,29)(22,31)(24,25)(26,55)(28,49)(30,51)(32,53)(33,57)(34,46)(35,59)(36,48)(37,61)(38,42)(39,63)(40,44), (1,29)(2,30)(3,31)(4,32)(5,25)(6,26)(7,27)(8,28)(9,40)(10,33)(11,34)(12,35)(13,36)(14,37)(15,38)(16,39)(17,55)(18,56)(19,49)(20,50)(21,51)(22,52)(23,53)(24,54)(41,61)(42,62)(43,63)(44,64)(45,57)(46,58)(47,59)(48,60), (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), (1,16)(2,38)(3,14)(4,36)(5,12)(6,34)(7,10)(8,40)(9,28)(11,26)(13,32)(15,30)(17,62)(18,41)(19,60)(20,47)(21,58)(22,45)(23,64)(24,43)(25,35)(27,33)(29,39)(31,37)(42,55)(44,53)(46,51)(48,49)(50,59)(52,57)(54,63)(56,61)>;
G:=Group( (1,35)(2,36)(3,37)(4,38)(5,39)(6,40)(7,33)(8,34)(9,26)(10,27)(11,28)(12,29)(13,30)(14,31)(15,32)(16,25)(17,44)(18,45)(19,46)(20,47)(21,48)(22,41)(23,42)(24,43)(49,58)(50,59)(51,60)(52,61)(53,62)(54,63)(55,64)(56,57), (1,50)(2,21)(3,52)(4,23)(5,54)(6,17)(7,56)(8,19)(9,64)(10,45)(11,58)(12,47)(13,60)(14,41)(15,62)(16,43)(18,27)(20,29)(22,31)(24,25)(26,55)(28,49)(30,51)(32,53)(33,57)(34,46)(35,59)(36,48)(37,61)(38,42)(39,63)(40,44), (1,29)(2,30)(3,31)(4,32)(5,25)(6,26)(7,27)(8,28)(9,40)(10,33)(11,34)(12,35)(13,36)(14,37)(15,38)(16,39)(17,55)(18,56)(19,49)(20,50)(21,51)(22,52)(23,53)(24,54)(41,61)(42,62)(43,63)(44,64)(45,57)(46,58)(47,59)(48,60), (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), (1,16)(2,38)(3,14)(4,36)(5,12)(6,34)(7,10)(8,40)(9,28)(11,26)(13,32)(15,30)(17,62)(18,41)(19,60)(20,47)(21,58)(22,45)(23,64)(24,43)(25,35)(27,33)(29,39)(31,37)(42,55)(44,53)(46,51)(48,49)(50,59)(52,57)(54,63)(56,61) );
G=PermutationGroup([(1,35),(2,36),(3,37),(4,38),(5,39),(6,40),(7,33),(8,34),(9,26),(10,27),(11,28),(12,29),(13,30),(14,31),(15,32),(16,25),(17,44),(18,45),(19,46),(20,47),(21,48),(22,41),(23,42),(24,43),(49,58),(50,59),(51,60),(52,61),(53,62),(54,63),(55,64),(56,57)], [(1,50),(2,21),(3,52),(4,23),(5,54),(6,17),(7,56),(8,19),(9,64),(10,45),(11,58),(12,47),(13,60),(14,41),(15,62),(16,43),(18,27),(20,29),(22,31),(24,25),(26,55),(28,49),(30,51),(32,53),(33,57),(34,46),(35,59),(36,48),(37,61),(38,42),(39,63),(40,44)], [(1,29),(2,30),(3,31),(4,32),(5,25),(6,26),(7,27),(8,28),(9,40),(10,33),(11,34),(12,35),(13,36),(14,37),(15,38),(16,39),(17,55),(18,56),(19,49),(20,50),(21,51),(22,52),(23,53),(24,54),(41,61),(42,62),(43,63),(44,64),(45,57),(46,58),(47,59),(48,60)], [(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)], [(1,16),(2,38),(3,14),(4,36),(5,12),(6,34),(7,10),(8,40),(9,28),(11,26),(13,32),(15,30),(17,62),(18,41),(19,60),(20,47),(21,58),(22,45),(23,64),(24,43),(25,35),(27,33),(29,39),(31,37),(42,55),(44,53),(46,51),(48,49),(50,59),(52,57),(54,63),(56,61)])
Matrix representation ►G ⊆ GL6(𝔽17)
| 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 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 13 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 0 | 0 | 0 | 0 | 13 | 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 | 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 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 14 | 3 |
| 0 | 0 | 0 | 0 | 14 | 14 |
| 16 | 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 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(17))| [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],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,4,0,0,0,0,13,0,0,0,0,0,0,0,0,13,0,0,0,0,4,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[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],[0,1,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,16,0,0,0,0,0,0,0,14,14,0,0,0,0,3,14],[16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1] >;
38 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 4A | ··· | 4H | 4I | ··· | 4P | 4Q | 4R | 8A | ··· | 8H |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | 4 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 4 | 4 | 8 | 8 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 | 4 | ··· | 4 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | C4○D4 | C4○D8 | C8⋊C22 |
| kernel | C2×C23.19D4 | C2×C22⋊C8 | C2×D4⋊C4 | C2×C4.Q8 | C2×C2.D8 | C23.19D4 | C2×C42⋊C2 | C2×C4⋊D4 | C22×C4 | C24 | C2×C4 | C22 | C22 |
| # reps | 1 | 1 | 2 | 1 | 1 | 8 | 1 | 1 | 3 | 1 | 8 | 8 | 2 |
In GAP, Magma, Sage, TeX
C_2\times C_2^3._{19}D_4 % in TeX
G:=Group("C2xC2^3.19D4"); // GroupNames label
G:=SmallGroup(128,1819);
// by ID
G=gap.SmallGroup(128,1819);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,456,758,100,4037,1027,124]);
// 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=b*c=c*b,b*d=d*b,f*b*f=b*c*d,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f=c*d*e^3>;
// generators/relations