direct product, p-group, metabelian, nilpotent (class 2), monomial
Aliases: C2×D4○C16, C8.24C24, C16.14C23, M5(2)⋊16C22, D4○(C2×C16), C16○(C2×D4), C16○(C2×Q8), Q8○(C2×C16), C4○(D4○C16), C8○(D4○C16), C4○D4.6C8, C8○D4.7C4, D4.8(C2×C8), Q8.9(C2×C8), C16○2(C4○D4), C16○2(C8○D4), C16○(D4○C16), (C2×D4).13C8, (C2×Q8).12C8, (C2×C16)⋊22C22, (C22×C16)⋊15C2, C16○2(C2×M4(2)), M4(2)○2(C2×C16), C16○2(C2×M5(2)), (C2×C16)○2M5(2), C23.24(C2×C8), C4.22(C22×C8), C2.11(C23×C8), C4.63(C23×C4), C8.50(C22×C4), (C2×M5(2))⋊22C2, (C2×C8).617C23, C8○D4.19C22, C22.4(C22×C8), (C2×M4(2)).38C4, M4(2).35(C2×C4), (C22×C8).587C22, C4○D4○(C2×C16), C16○(C2×C8○D4), C16○(C2×C4○D4), (C2×Q8)○(C2×C16), (C2×C16)○(C8○D4), (C2×C4).57(C2×C8), (C2×C8).198(C2×C4), (C2×C8○D4).24C2, (C2×C4○D4).35C4, C4○D4.38(C2×C4), (C2×C16)○(C2×M4(2)), (C2×C16)○(C2×M5(2)), (C2×C4).476(C22×C4), (C22×C4).421(C2×C4), (C2×C16)○(C2×C8○D4), (C2×C16)○(C2×C4○D4), SmallGroup(128,2138)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 196 in 184 conjugacy classes, 172 normal (14 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×2], C4 [×6], C22, C22 [×6], C22 [×6], C8 [×2], C8 [×6], C2×C4, C2×C4 [×15], D4 [×12], Q8 [×4], C23 [×3], C16 [×8], C2×C8, C2×C8 [×15], M4(2) [×12], C22×C4 [×3], C2×D4 [×3], C2×Q8, C4○D4 [×8], C2×C16, C2×C16 [×15], M5(2) [×12], C22×C8 [×3], C2×M4(2) [×3], C8○D4 [×8], C2×C4○D4, C22×C16 [×3], C2×M5(2) [×3], D4○C16 [×8], C2×C8○D4, C2×D4○C16
Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], C8 [×8], C2×C4 [×28], C23 [×15], C2×C8 [×28], C22×C4 [×14], C24, C22×C8 [×14], C23×C4, D4○C16 [×2], C23×C8, C2×D4○C16
Generators and relations
G = < a,b,c,d | a2=b4=c2=1, d8=b2, ab=ba, ac=ca, ad=da, cbc=b-1, bd=db, cd=dc >
►On 64 points
(1 24)(2 25)(3 26)(4 27)(5 28)(6 29)(7 30)(8 31)(9 32)(10 17)(11 18)(12 19)(13 20)(14 21)(15 22)(16 23)(33 55)(34 56)(35 57)(36 58)(37 59)(38 60)(39 61)(40 62)(41 63)(42 64)(43 49)(44 50)(45 51)(46 52)(47 53)(48 54)
(1 40 9 48)(2 41 10 33)(3 42 11 34)(4 43 12 35)(5 44 13 36)(6 45 14 37)(7 46 15 38)(8 47 16 39)(17 55 25 63)(18 56 26 64)(19 57 27 49)(20 58 28 50)(21 59 29 51)(22 60 30 52)(23 61 31 53)(24 62 32 54)
(1 48)(2 33)(3 34)(4 35)(5 36)(6 37)(7 38)(8 39)(9 40)(10 41)(11 42)(12 43)(13 44)(14 45)(15 46)(16 47)(17 63)(18 64)(19 49)(20 50)(21 51)(22 52)(23 53)(24 54)(25 55)(26 56)(27 57)(28 58)(29 59)(30 60)(31 61)(32 62)
(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)
G:=sub<Sym(64)| (1,24)(2,25)(3,26)(4,27)(5,28)(6,29)(7,30)(8,31)(9,32)(10,17)(11,18)(12,19)(13,20)(14,21)(15,22)(16,23)(33,55)(34,56)(35,57)(36,58)(37,59)(38,60)(39,61)(40,62)(41,63)(42,64)(43,49)(44,50)(45,51)(46,52)(47,53)(48,54), (1,40,9,48)(2,41,10,33)(3,42,11,34)(4,43,12,35)(5,44,13,36)(6,45,14,37)(7,46,15,38)(8,47,16,39)(17,55,25,63)(18,56,26,64)(19,57,27,49)(20,58,28,50)(21,59,29,51)(22,60,30,52)(23,61,31,53)(24,62,32,54), (1,48)(2,33)(3,34)(4,35)(5,36)(6,37)(7,38)(8,39)(9,40)(10,41)(11,42)(12,43)(13,44)(14,45)(15,46)(16,47)(17,63)(18,64)(19,49)(20,50)(21,51)(22,52)(23,53)(24,54)(25,55)(26,56)(27,57)(28,58)(29,59)(30,60)(31,61)(32,62), (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)>;
G:=Group( (1,24)(2,25)(3,26)(4,27)(5,28)(6,29)(7,30)(8,31)(9,32)(10,17)(11,18)(12,19)(13,20)(14,21)(15,22)(16,23)(33,55)(34,56)(35,57)(36,58)(37,59)(38,60)(39,61)(40,62)(41,63)(42,64)(43,49)(44,50)(45,51)(46,52)(47,53)(48,54), (1,40,9,48)(2,41,10,33)(3,42,11,34)(4,43,12,35)(5,44,13,36)(6,45,14,37)(7,46,15,38)(8,47,16,39)(17,55,25,63)(18,56,26,64)(19,57,27,49)(20,58,28,50)(21,59,29,51)(22,60,30,52)(23,61,31,53)(24,62,32,54), (1,48)(2,33)(3,34)(4,35)(5,36)(6,37)(7,38)(8,39)(9,40)(10,41)(11,42)(12,43)(13,44)(14,45)(15,46)(16,47)(17,63)(18,64)(19,49)(20,50)(21,51)(22,52)(23,53)(24,54)(25,55)(26,56)(27,57)(28,58)(29,59)(30,60)(31,61)(32,62), (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) );
G=PermutationGroup([(1,24),(2,25),(3,26),(4,27),(5,28),(6,29),(7,30),(8,31),(9,32),(10,17),(11,18),(12,19),(13,20),(14,21),(15,22),(16,23),(33,55),(34,56),(35,57),(36,58),(37,59),(38,60),(39,61),(40,62),(41,63),(42,64),(43,49),(44,50),(45,51),(46,52),(47,53),(48,54)], [(1,40,9,48),(2,41,10,33),(3,42,11,34),(4,43,12,35),(5,44,13,36),(6,45,14,37),(7,46,15,38),(8,47,16,39),(17,55,25,63),(18,56,26,64),(19,57,27,49),(20,58,28,50),(21,59,29,51),(22,60,30,52),(23,61,31,53),(24,62,32,54)], [(1,48),(2,33),(3,34),(4,35),(5,36),(6,37),(7,38),(8,39),(9,40),(10,41),(11,42),(12,43),(13,44),(14,45),(15,46),(16,47),(17,63),(18,64),(19,49),(20,50),(21,51),(22,52),(23,53),(24,54),(25,55),(26,56),(27,57),(28,58),(29,59),(30,60),(31,61),(32,62)], [(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)])
Matrix representation ►G ⊆ GL3(𝔽17) generated by
| 16 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 0 | 0 | 16 |
| 0 | 1 | 0 |
| 16 | 0 | 0 |
| 0 | 0 | 16 |
| 0 | 16 | 0 |
| 16 | 0 | 0 |
| 0 | 14 | 0 |
| 0 | 0 | 14 |
G:=sub<GL(3,GF(17))| [16,0,0,0,1,0,0,0,1],[1,0,0,0,0,1,0,16,0],[16,0,0,0,0,16,0,16,0],[16,0,0,0,14,0,0,0,14] >;
80 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | ··· | 2I | 4A | 4B | 4C | 4D | 4E | ··· | 4J | 8A | ··· | 8H | 8I | ··· | 8T | 16A | ··· | 16P | 16Q | ··· | 16AN |
| order | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 | 8 | ··· | 8 | 16 | ··· | 16 | 16 | ··· | 16 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 |
80 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 |
| type | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C8 | C8 | C8 | D4○C16 |
| kernel | C2×D4○C16 | C22×C16 | C2×M5(2) | D4○C16 | C2×C8○D4 | C2×M4(2) | C8○D4 | C2×C4○D4 | C2×D4 | C2×Q8 | C4○D4 | C2 |
| # reps | 1 | 3 | 3 | 8 | 1 | 6 | 8 | 2 | 12 | 4 | 16 | 16 |
In GAP, Magma, Sage, TeX
C_2\times D_4\circ C_{16} % in TeX
G:=Group("C2xD4oC16"); // GroupNames label
G:=SmallGroup(128,2138);
// by ID
G=gap.SmallGroup(128,2138);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,-2,-2,112,723,102,124]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^4=c^2=1,d^8=b^2,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,b*d=d*b,c*d=d*c>;
// generators/relations