direct product, p-group, metabelian, nilpotent (class 3), monomial
Aliases: C2×C4×D8, C42.349D4, C42.683C23, C4○(C4×D8), C8⋊5(C22×C4), C4.42(C4×D4), (C4×C8)⋊72C22, D4⋊1(C22×C4), C2.3(C22×D8), (C4×D4)⋊78C22, C4.13(C23×C4), C22.69(C2×D8), C2.D8⋊74C22, C4⋊C4.353C23, (C2×C4).193C24, (C2×C8).556C23, (C22×D8).17C2, C22.115(C4×D4), (C22×C4).824D4, C23.840(C2×D4), D4⋊C4⋊98C22, (C2×D8).171C22, (C2×D4).364C23, C22.83(C4○D8), (C22×C8).510C22, C22.137(C22×D4), (C2×C42).1111C22, (C22×C4).1509C23, (C22×D4).555C22, (C2×C4×C8)⋊24C2, (C2×C4)○(C4×D8), (C2×C4×D4)⋊56C2, C4○2(C2×C2.D8), C2.53(C2×C4×D4), (C2×C8)⋊29(C2×C4), C4.1(C2×C4○D4), C2.3(C2×C4○D8), C4○2(C2×D4⋊C4), (C2×D4)⋊32(C2×C4), (C2×C2.D8)⋊45C2, (C2×C4)○3(C2.D8), (C2×C4)○3(D4⋊C4), (C2×D4⋊C4)⋊59C2, (C2×C4).1573(C2×D4), (C2×C4).685(C4○D4), (C2×C4⋊C4).906C22, (C2×C4).464(C22×C4), (C2×C4)○2(C2×C2.D8), (C2×C4)○2(C2×D4⋊C4), SmallGroup(128,1668)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 604 in 308 conjugacy classes, 156 normal (20 characteristic)
C1, C2 [×3], C2 [×4], C2 [×8], C4 [×2], C4 [×6], C4 [×6], C22, C22 [×6], C22 [×32], C8 [×4], C8 [×2], C2×C4 [×2], C2×C4 [×12], C2×C4 [×26], D4 [×8], D4 [×12], C23, C23 [×20], C42 [×4], C22⋊C4 [×8], C4⋊C4 [×4], C4⋊C4 [×2], C2×C8 [×8], C2×C8 [×2], D8 [×16], C22×C4 [×3], C22×C4 [×18], C2×D4 [×12], C2×D4 [×6], C24 [×2], C4×C8 [×4], D4⋊C4 [×8], C2.D8 [×4], C2×C42, C2×C22⋊C4 [×2], C2×C4⋊C4 [×2], C4×D4 [×8], C4×D4 [×4], C22×C8 [×2], C2×D8 [×12], C23×C4 [×2], C22×D4 [×2], C2×C4×C8, C2×D4⋊C4 [×2], C2×C2.D8, C4×D8 [×8], C2×C4×D4 [×2], C22×D8, C2×C4×D8
Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], D4 [×4], C23 [×15], D8 [×4], C22×C4 [×14], C2×D4 [×6], C4○D4 [×2], C24, C4×D4 [×4], C2×D8 [×6], C4○D8 [×2], C23×C4, C22×D4, C2×C4○D4, C4×D8 [×4], C2×C4×D4, C22×D8, C2×C4○D8, C2×C4×D8
Generators and relations
G = < a,b,c,d | a2=b4=c8=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
►On 64 points
(1 29)(2 30)(3 31)(4 32)(5 25)(6 26)(7 27)(8 28)(9 57)(10 58)(11 59)(12 60)(13 61)(14 62)(15 63)(16 64)(17 34)(18 35)(19 36)(20 37)(21 38)(22 39)(23 40)(24 33)(41 53)(42 54)(43 55)(44 56)(45 49)(46 50)(47 51)(48 52)
(1 34 53 13)(2 35 54 14)(3 36 55 15)(4 37 56 16)(5 38 49 9)(6 39 50 10)(7 40 51 11)(8 33 52 12)(17 41 61 29)(18 42 62 30)(19 43 63 31)(20 44 64 32)(21 45 57 25)(22 46 58 26)(23 47 59 27)(24 48 60 28)
(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 25)(2 32)(3 31)(4 30)(5 29)(6 28)(7 27)(8 26)(9 61)(10 60)(11 59)(12 58)(13 57)(14 64)(15 63)(16 62)(17 38)(18 37)(19 36)(20 35)(21 34)(22 33)(23 40)(24 39)(41 49)(42 56)(43 55)(44 54)(45 53)(46 52)(47 51)(48 50)
G:=sub<Sym(64)| (1,29)(2,30)(3,31)(4,32)(5,25)(6,26)(7,27)(8,28)(9,57)(10,58)(11,59)(12,60)(13,61)(14,62)(15,63)(16,64)(17,34)(18,35)(19,36)(20,37)(21,38)(22,39)(23,40)(24,33)(41,53)(42,54)(43,55)(44,56)(45,49)(46,50)(47,51)(48,52), (1,34,53,13)(2,35,54,14)(3,36,55,15)(4,37,56,16)(5,38,49,9)(6,39,50,10)(7,40,51,11)(8,33,52,12)(17,41,61,29)(18,42,62,30)(19,43,63,31)(20,44,64,32)(21,45,57,25)(22,46,58,26)(23,47,59,27)(24,48,60,28), (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), (1,25)(2,32)(3,31)(4,30)(5,29)(6,28)(7,27)(8,26)(9,61)(10,60)(11,59)(12,58)(13,57)(14,64)(15,63)(16,62)(17,38)(18,37)(19,36)(20,35)(21,34)(22,33)(23,40)(24,39)(41,49)(42,56)(43,55)(44,54)(45,53)(46,52)(47,51)(48,50)>;
G:=Group( (1,29)(2,30)(3,31)(4,32)(5,25)(6,26)(7,27)(8,28)(9,57)(10,58)(11,59)(12,60)(13,61)(14,62)(15,63)(16,64)(17,34)(18,35)(19,36)(20,37)(21,38)(22,39)(23,40)(24,33)(41,53)(42,54)(43,55)(44,56)(45,49)(46,50)(47,51)(48,52), (1,34,53,13)(2,35,54,14)(3,36,55,15)(4,37,56,16)(5,38,49,9)(6,39,50,10)(7,40,51,11)(8,33,52,12)(17,41,61,29)(18,42,62,30)(19,43,63,31)(20,44,64,32)(21,45,57,25)(22,46,58,26)(23,47,59,27)(24,48,60,28), (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), (1,25)(2,32)(3,31)(4,30)(5,29)(6,28)(7,27)(8,26)(9,61)(10,60)(11,59)(12,58)(13,57)(14,64)(15,63)(16,62)(17,38)(18,37)(19,36)(20,35)(21,34)(22,33)(23,40)(24,39)(41,49)(42,56)(43,55)(44,54)(45,53)(46,52)(47,51)(48,50) );
G=PermutationGroup([(1,29),(2,30),(3,31),(4,32),(5,25),(6,26),(7,27),(8,28),(9,57),(10,58),(11,59),(12,60),(13,61),(14,62),(15,63),(16,64),(17,34),(18,35),(19,36),(20,37),(21,38),(22,39),(23,40),(24,33),(41,53),(42,54),(43,55),(44,56),(45,49),(46,50),(47,51),(48,52)], [(1,34,53,13),(2,35,54,14),(3,36,55,15),(4,37,56,16),(5,38,49,9),(6,39,50,10),(7,40,51,11),(8,33,52,12),(17,41,61,29),(18,42,62,30),(19,43,63,31),(20,44,64,32),(21,45,57,25),(22,46,58,26),(23,47,59,27),(24,48,60,28)], [(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)], [(1,25),(2,32),(3,31),(4,30),(5,29),(6,28),(7,27),(8,26),(9,61),(10,60),(11,59),(12,58),(13,57),(14,64),(15,63),(16,62),(17,38),(18,37),(19,36),(20,35),(21,34),(22,33),(23,40),(24,39),(41,49),(42,56),(43,55),(44,54),(45,53),(46,52),(47,51),(48,50)])
Matrix representation ►G ⊆ GL4(𝔽17) generated by
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 16 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 11 | 14 |
| 0 | 0 | 6 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 1 |
| 0 | 0 | 0 | 16 |
G:=sub<GL(4,GF(17))| [16,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[16,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[1,0,0,0,0,1,0,0,0,0,11,6,0,0,14,0],[1,0,0,0,0,1,0,0,0,0,1,0,0,0,1,16] >;
56 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | ··· | 2O | 4A | ··· | 4H | 4I | ··· | 4P | 4Q | ··· | 4X | 8A | ··· | 8P |
| order | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 4 | ··· | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | D4 | D8 | C4○D4 | C4○D8 |
| kernel | C2×C4×D8 | C2×C4×C8 | C2×D4⋊C4 | C2×C2.D8 | C4×D8 | C2×C4×D4 | C22×D8 | C2×D8 | C42 | C22×C4 | C2×C4 | C2×C4 | C22 |
| # reps | 1 | 1 | 2 | 1 | 8 | 2 | 1 | 16 | 2 | 2 | 8 | 4 | 8 |
In GAP, Magma, Sage, TeX
C_2\times C_4\times D_8
% in TeX
G:=Group("C2xC4xD8"); // GroupNames label
G:=SmallGroup(128,1668);
// by ID
G=gap.SmallGroup(128,1668);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,184,2804,1411,172]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^4=c^8=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations