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×C8).556C23, (C2×C4).193C24, (C22×D8).17C2, C23.840(C2×D4), C22.115(C4×D4), (C22×C4).824D4, D4⋊C4⋊98C22, (C2×D4).364C23, (C2×D8).171C22, C22.83(C4○D8), (C22×C8).510C22, C22.137(C22×D4), (C22×C4).1509C23, (C2×C42).1111C22, (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
Generators and relations for C2×C4×D8
G = < a,b,c,d | a2=b4=c8=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Subgroups: 604 in 308 conjugacy classes, 156 normal (20 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C8, C8, C2×C4, C2×C4, C2×C4, D4, D4, C23, C23, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, D8, C22×C4, C22×C4, C2×D4, C2×D4, C24, C4×C8, D4⋊C4, C2.D8, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C4×D4, C4×D4, C22×C8, C2×D8, C23×C4, C22×D4, C2×C4×C8, C2×D4⋊C4, C2×C2.D8, C4×D8, C2×C4×D4, C22×D8, C2×C4×D8
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D8, C22×C4, C2×D4, C4○D4, C24, C4×D4, C2×D8, C4○D8, C23×C4, C22×D4, C2×C4○D4, C4×D8, C2×C4×D4, C22×D8, C2×C4○D8, C2×C4×D8
►On 64 points
(1 29)(2 30)(3 31)(4 32)(5 25)(6 26)(7 27)(8 28)(9 63)(10 64)(11 57)(12 58)(13 59)(14 60)(15 61)(16 62)(17 38)(18 39)(19 40)(20 33)(21 34)(22 35)(23 36)(24 37)(41 49)(42 50)(43 51)(44 52)(45 53)(46 54)(47 55)(48 56)
(1 34 53 15)(2 35 54 16)(3 36 55 9)(4 37 56 10)(5 38 49 11)(6 39 50 12)(7 40 51 13)(8 33 52 14)(17 41 57 25)(18 42 58 26)(19 43 59 27)(20 44 60 28)(21 45 61 29)(22 46 62 30)(23 47 63 31)(24 48 64 32)
(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 63)(10 62)(11 61)(12 60)(13 59)(14 58)(15 57)(16 64)(17 34)(18 33)(19 40)(20 39)(21 38)(22 37)(23 36)(24 35)(41 53)(42 52)(43 51)(44 50)(45 49)(46 56)(47 55)(48 54)
G:=sub<Sym(64)| (1,29)(2,30)(3,31)(4,32)(5,25)(6,26)(7,27)(8,28)(9,63)(10,64)(11,57)(12,58)(13,59)(14,60)(15,61)(16,62)(17,38)(18,39)(19,40)(20,33)(21,34)(22,35)(23,36)(24,37)(41,49)(42,50)(43,51)(44,52)(45,53)(46,54)(47,55)(48,56), (1,34,53,15)(2,35,54,16)(3,36,55,9)(4,37,56,10)(5,38,49,11)(6,39,50,12)(7,40,51,13)(8,33,52,14)(17,41,57,25)(18,42,58,26)(19,43,59,27)(20,44,60,28)(21,45,61,29)(22,46,62,30)(23,47,63,31)(24,48,64,32), (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)(9,63)(10,62)(11,61)(12,60)(13,59)(14,58)(15,57)(16,64)(17,34)(18,33)(19,40)(20,39)(21,38)(22,37)(23,36)(24,35)(41,53)(42,52)(43,51)(44,50)(45,49)(46,56)(47,55)(48,54)>;
G:=Group( (1,29)(2,30)(3,31)(4,32)(5,25)(6,26)(7,27)(8,28)(9,63)(10,64)(11,57)(12,58)(13,59)(14,60)(15,61)(16,62)(17,38)(18,39)(19,40)(20,33)(21,34)(22,35)(23,36)(24,37)(41,49)(42,50)(43,51)(44,52)(45,53)(46,54)(47,55)(48,56), (1,34,53,15)(2,35,54,16)(3,36,55,9)(4,37,56,10)(5,38,49,11)(6,39,50,12)(7,40,51,13)(8,33,52,14)(17,41,57,25)(18,42,58,26)(19,43,59,27)(20,44,60,28)(21,45,61,29)(22,46,62,30)(23,47,63,31)(24,48,64,32), (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)(9,63)(10,62)(11,61)(12,60)(13,59)(14,58)(15,57)(16,64)(17,34)(18,33)(19,40)(20,39)(21,38)(22,37)(23,36)(24,35)(41,53)(42,52)(43,51)(44,50)(45,49)(46,56)(47,55)(48,54) );
G=PermutationGroup([[(1,29),(2,30),(3,31),(4,32),(5,25),(6,26),(7,27),(8,28),(9,63),(10,64),(11,57),(12,58),(13,59),(14,60),(15,61),(16,62),(17,38),(18,39),(19,40),(20,33),(21,34),(22,35),(23,36),(24,37),(41,49),(42,50),(43,51),(44,52),(45,53),(46,54),(47,55),(48,56)], [(1,34,53,15),(2,35,54,16),(3,36,55,9),(4,37,56,10),(5,38,49,11),(6,39,50,12),(7,40,51,13),(8,33,52,14),(17,41,57,25),(18,42,58,26),(19,43,59,27),(20,44,60,28),(21,45,61,29),(22,46,62,30),(23,47,63,31),(24,48,64,32)], [(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),(9,63),(10,62),(11,61),(12,60),(13,59),(14,58),(15,57),(16,64),(17,34),(18,33),(19,40),(20,39),(21,38),(22,37),(23,36),(24,35),(41,53),(42,52),(43,51),(44,50),(45,49),(46,56),(47,55),(48,54)]])
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 |
Matrix representation of C2×C4×D8 ►in 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] >;
C2×C4×D8 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