direct product, p-group, metabelian, nilpotent (class 2), monomial
Aliases: C2×C8⋊6D4, C42.680C23, C8⋊12(C2×D4), (C2×C8)⋊33D4, C4○(C8⋊6D4), C4⋊C8⋊86C22, (C4×C8)⋊80C22, (C4×D4).27C4, C4.184(C4×D4), C4⋊1(C2×M4(2)), (C2×C4)⋊8M4(2), C24.79(C2×C4), C22⋊C8⋊75C22, (C2×C8).477C23, C42.282(C2×C4), (C2×C4).647C24, (C22×D4).39C4, C4.193(C22×D4), C22.114(C4×D4), (C4×D4).285C22, C22.42(C8○D4), (C2×M4(2))⋊75C22, (C22×M4(2))⋊24C2, C22.174(C23×C4), C23.104(C22×C4), (C23×C4).525C22, (C22×C8).509C22, (C22×C4).915C23, C22.63(C2×M4(2)), C2.11(C22×M4(2)), (C2×C42).1109C22, (C2×C4×C8)⋊42C2, (C2×C4⋊C8)⋊48C2, C2.45(C2×C4×D4), (C2×C4×D4).71C2, (C2×C4⋊C4).71C4, (C2×C4)○(C8⋊6D4), C2.15(C2×C8○D4), C4⋊C4.222(C2×C4), (C2×C22⋊C8)⋊43C2, C4.298(C2×C4○D4), (C2×D4).231(C2×C4), (C2×C4).1572(C2×D4), (C2×C22⋊C4).48C4, C22⋊C4.72(C2×C4), (C2×C4).957(C4○D4), (C2×C4).463(C22×C4), (C22×C4).339(C2×C4), SmallGroup(128,1660)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 420 in 276 conjugacy classes, 156 normal (22 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×2], C4 [×6], C4 [×6], C22, C22 [×6], C22 [×20], C8 [×4], C8 [×6], C2×C4 [×2], C2×C4 [×16], C2×C4 [×22], D4 [×8], C23, C23 [×4], C23 [×12], C42 [×4], C22⋊C4 [×8], C4⋊C4 [×4], C2×C8 [×12], C2×C8 [×6], M4(2) [×16], C22×C4 [×3], C22×C4 [×10], C22×C4 [×8], C2×D4 [×4], C2×D4 [×4], C24 [×2], C4×C8 [×4], C22⋊C8 [×8], C4⋊C8 [×4], C2×C42, C2×C22⋊C4 [×2], C2×C4⋊C4, C4×D4 [×8], C22×C8 [×2], C22×C8 [×2], C2×M4(2) [×8], C2×M4(2) [×8], C23×C4 [×2], C22×D4, C2×C4×C8, C2×C22⋊C8 [×2], C2×C4⋊C8, C8⋊6D4 [×8], C2×C4×D4, C22×M4(2) [×2], C2×C8⋊6D4
Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], D4 [×4], C23 [×15], M4(2) [×4], C22×C4 [×14], C2×D4 [×6], C4○D4 [×2], C24, C4×D4 [×4], C2×M4(2) [×6], C8○D4 [×2], C23×C4, C22×D4, C2×C4○D4, C8⋊6D4 [×4], C2×C4×D4, C22×M4(2), C2×C8○D4, C2×C8⋊6D4
Generators and relations
G = < a,b,c,d | a2=b8=c4=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b5, dcd=c-1 >
►On 64 points
(1 19)(2 20)(3 21)(4 22)(5 23)(6 24)(7 17)(8 18)(9 41)(10 42)(11 43)(12 44)(13 45)(14 46)(15 47)(16 48)(25 39)(26 40)(27 33)(28 34)(29 35)(30 36)(31 37)(32 38)(49 60)(50 61)(51 62)(52 63)(53 64)(54 57)(55 58)(56 59)
(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 58 42 36)(2 59 43 37)(3 60 44 38)(4 61 45 39)(5 62 46 40)(6 63 47 33)(7 64 48 34)(8 57 41 35)(9 29 18 54)(10 30 19 55)(11 31 20 56)(12 32 21 49)(13 25 22 50)(14 26 23 51)(15 27 24 52)(16 28 17 53)
(1 36)(2 33)(3 38)(4 35)(5 40)(6 37)(7 34)(8 39)(9 50)(10 55)(11 52)(12 49)(13 54)(14 51)(15 56)(16 53)(17 28)(18 25)(19 30)(20 27)(21 32)(22 29)(23 26)(24 31)(41 61)(42 58)(43 63)(44 60)(45 57)(46 62)(47 59)(48 64)
G:=sub<Sym(64)| (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,17)(8,18)(9,41)(10,42)(11,43)(12,44)(13,45)(14,46)(15,47)(16,48)(25,39)(26,40)(27,33)(28,34)(29,35)(30,36)(31,37)(32,38)(49,60)(50,61)(51,62)(52,63)(53,64)(54,57)(55,58)(56,59), (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), (1,58,42,36)(2,59,43,37)(3,60,44,38)(4,61,45,39)(5,62,46,40)(6,63,47,33)(7,64,48,34)(8,57,41,35)(9,29,18,54)(10,30,19,55)(11,31,20,56)(12,32,21,49)(13,25,22,50)(14,26,23,51)(15,27,24,52)(16,28,17,53), (1,36)(2,33)(3,38)(4,35)(5,40)(6,37)(7,34)(8,39)(9,50)(10,55)(11,52)(12,49)(13,54)(14,51)(15,56)(16,53)(17,28)(18,25)(19,30)(20,27)(21,32)(22,29)(23,26)(24,31)(41,61)(42,58)(43,63)(44,60)(45,57)(46,62)(47,59)(48,64)>;
G:=Group( (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,17)(8,18)(9,41)(10,42)(11,43)(12,44)(13,45)(14,46)(15,47)(16,48)(25,39)(26,40)(27,33)(28,34)(29,35)(30,36)(31,37)(32,38)(49,60)(50,61)(51,62)(52,63)(53,64)(54,57)(55,58)(56,59), (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), (1,58,42,36)(2,59,43,37)(3,60,44,38)(4,61,45,39)(5,62,46,40)(6,63,47,33)(7,64,48,34)(8,57,41,35)(9,29,18,54)(10,30,19,55)(11,31,20,56)(12,32,21,49)(13,25,22,50)(14,26,23,51)(15,27,24,52)(16,28,17,53), (1,36)(2,33)(3,38)(4,35)(5,40)(6,37)(7,34)(8,39)(9,50)(10,55)(11,52)(12,49)(13,54)(14,51)(15,56)(16,53)(17,28)(18,25)(19,30)(20,27)(21,32)(22,29)(23,26)(24,31)(41,61)(42,58)(43,63)(44,60)(45,57)(46,62)(47,59)(48,64) );
G=PermutationGroup([(1,19),(2,20),(3,21),(4,22),(5,23),(6,24),(7,17),(8,18),(9,41),(10,42),(11,43),(12,44),(13,45),(14,46),(15,47),(16,48),(25,39),(26,40),(27,33),(28,34),(29,35),(30,36),(31,37),(32,38),(49,60),(50,61),(51,62),(52,63),(53,64),(54,57),(55,58),(56,59)], [(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)], [(1,58,42,36),(2,59,43,37),(3,60,44,38),(4,61,45,39),(5,62,46,40),(6,63,47,33),(7,64,48,34),(8,57,41,35),(9,29,18,54),(10,30,19,55),(11,31,20,56),(12,32,21,49),(13,25,22,50),(14,26,23,51),(15,27,24,52),(16,28,17,53)], [(1,36),(2,33),(3,38),(4,35),(5,40),(6,37),(7,34),(8,39),(9,50),(10,55),(11,52),(12,49),(13,54),(14,51),(15,56),(16,53),(17,28),(18,25),(19,30),(20,27),(21,32),(22,29),(23,26),(24,31),(41,61),(42,58),(43,63),(44,60),(45,57),(46,62),(47,59),(48,64)])
Matrix representation ►G ⊆ GL5(𝔽17)
| 16 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 16 |
| 16 | 0 | 0 | 0 | 0 |
| 0 | 4 | 7 | 0 | 0 |
| 0 | 8 | 13 | 0 | 0 |
| 0 | 0 | 0 | 13 | 0 |
| 0 | 0 | 0 | 0 | 13 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 8 | 14 | 0 | 0 |
| 0 | 16 | 9 | 0 | 0 |
| 0 | 0 | 0 | 0 | 13 |
| 0 | 0 | 0 | 13 | 0 |
| 16 | 0 | 0 | 0 | 0 |
| 0 | 9 | 5 | 0 | 0 |
| 0 | 1 | 8 | 0 | 0 |
| 0 | 0 | 0 | 0 | 13 |
| 0 | 0 | 0 | 4 | 0 |
G:=sub<GL(5,GF(17))| [16,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,16],[16,0,0,0,0,0,4,8,0,0,0,7,13,0,0,0,0,0,13,0,0,0,0,0,13],[1,0,0,0,0,0,8,16,0,0,0,14,9,0,0,0,0,0,0,13,0,0,0,13,0],[16,0,0,0,0,0,9,1,0,0,0,5,8,0,0,0,0,0,0,4,0,0,0,13,0] >;
56 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 4A | ··· | 4H | 4I | ··· | 4P | 4Q | 4R | 4S | 4T | 8A | ··· | 8P | 8Q | ··· | 8X |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | D4 | M4(2) | C4○D4 | C8○D4 |
| kernel | C2×C8⋊6D4 | C2×C4×C8 | C2×C22⋊C8 | C2×C4⋊C8 | C8⋊6D4 | C2×C4×D4 | C22×M4(2) | C2×C22⋊C4 | C2×C4⋊C4 | C4×D4 | C22×D4 | C2×C8 | C2×C4 | C2×C4 | C22 |
| # reps | 1 | 1 | 2 | 1 | 8 | 1 | 2 | 4 | 2 | 8 | 2 | 4 | 8 | 4 | 8 |
In GAP, Magma, Sage, TeX
C_2\times C_8\rtimes_6D_4
% in TeX
G:=Group("C2xC8:6D4"); // GroupNames label
G:=SmallGroup(128,1660);
// by ID
G=gap.SmallGroup(128,1660);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,1430,723,184,124]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^8=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=b^5,d*c*d=c^-1>;
// generators/relations