direct product, p-group, metabelian, nilpotent (class 3), monomial
Aliases: C23×D8, C8⋊2C24, D4⋊1C24, C4.1C25, C24.195D4, (C23×C8)⋊9C2, (C2×C8)⋊14C23, (D4×C23)⋊17C2, (C2×D4)⋊20C23, C2.36(D4×C23), C4.27(C22×D4), (C2×C4).607C24, (C22×C8)⋊66C22, (C22×C4).627D4, C23.893(C2×D4), (C22×D4)⋊64C22, (C23×C4).711C22, C22.164(C22×D4), (C22×C4).1589C23, (C2×C4).880(C2×D4), SmallGroup(128,2306)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 2012 in 988 conjugacy classes, 476 normal (7 characteristic)
C1, C2, C2 [×14], C2 [×16], C4, C4 [×7], C22 [×35], C22 [×128], C8 [×8], C2×C4 [×28], D4 [×16], D4 [×56], C23 [×15], C23 [×168], C2×C8 [×28], D8 [×64], C22×C4 [×14], C2×D4 [×56], C2×D4 [×84], C24, C24 [×44], C22×C8 [×14], C2×D8 [×112], C23×C4, C22×D4 [×28], C22×D4 [×14], C25 [×2], C23×C8, C22×D8 [×28], D4×C23 [×2], C23×D8
Quotients:
C1, C2 [×31], C22 [×155], D4 [×8], C23 [×155], D8 [×8], C2×D4 [×28], C24 [×31], C2×D8 [×28], C22×D4 [×14], C25, C22×D8 [×14], D4×C23, C23×D8
Generators and relations
G = < a,b,c,d,e | a2=b2=c2=d8=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >
►On 64 points
Generators in S64
(1 24)(2 17)(3 18)(4 19)(5 20)(6 21)(7 22)(8 23)(9 26)(10 27)(11 28)(12 29)(13 30)(14 31)(15 32)(16 25)(33 56)(34 49)(35 50)(36 51)(37 52)(38 53)(39 54)(40 55)(41 60)(42 61)(43 62)(44 63)(45 64)(46 57)(47 58)(48 59)
(1 53)(2 54)(3 55)(4 56)(5 49)(6 50)(7 51)(8 52)(9 47)(10 48)(11 41)(12 42)(13 43)(14 44)(15 45)(16 46)(17 39)(18 40)(19 33)(20 34)(21 35)(22 36)(23 37)(24 38)(25 57)(26 58)(27 59)(28 60)(29 61)(30 62)(31 63)(32 64)
(1 60)(2 61)(3 62)(4 63)(5 64)(6 57)(7 58)(8 59)(9 36)(10 37)(11 38)(12 39)(13 40)(14 33)(15 34)(16 35)(17 42)(18 43)(19 44)(20 45)(21 46)(22 47)(23 48)(24 41)(25 50)(26 51)(27 52)(28 53)(29 54)(30 55)(31 56)(32 49)
(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 28)(2 27)(3 26)(4 25)(5 32)(6 31)(7 30)(8 29)(9 18)(10 17)(11 24)(12 23)(13 22)(14 21)(15 20)(16 19)(33 46)(34 45)(35 44)(36 43)(37 42)(38 41)(39 48)(40 47)(49 64)(50 63)(51 62)(52 61)(53 60)(54 59)(55 58)(56 57)
G:=sub<Sym(64)| (1,24)(2,17)(3,18)(4,19)(5,20)(6,21)(7,22)(8,23)(9,26)(10,27)(11,28)(12,29)(13,30)(14,31)(15,32)(16,25)(33,56)(34,49)(35,50)(36,51)(37,52)(38,53)(39,54)(40,55)(41,60)(42,61)(43,62)(44,63)(45,64)(46,57)(47,58)(48,59), (1,53)(2,54)(3,55)(4,56)(5,49)(6,50)(7,51)(8,52)(9,47)(10,48)(11,41)(12,42)(13,43)(14,44)(15,45)(16,46)(17,39)(18,40)(19,33)(20,34)(21,35)(22,36)(23,37)(24,38)(25,57)(26,58)(27,59)(28,60)(29,61)(30,62)(31,63)(32,64), (1,60)(2,61)(3,62)(4,63)(5,64)(6,57)(7,58)(8,59)(9,36)(10,37)(11,38)(12,39)(13,40)(14,33)(15,34)(16,35)(17,42)(18,43)(19,44)(20,45)(21,46)(22,47)(23,48)(24,41)(25,50)(26,51)(27,52)(28,53)(29,54)(30,55)(31,56)(32,49), (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), (1,28)(2,27)(3,26)(4,25)(5,32)(6,31)(7,30)(8,29)(9,18)(10,17)(11,24)(12,23)(13,22)(14,21)(15,20)(16,19)(33,46)(34,45)(35,44)(36,43)(37,42)(38,41)(39,48)(40,47)(49,64)(50,63)(51,62)(52,61)(53,60)(54,59)(55,58)(56,57)>;
G:=Group( (1,24)(2,17)(3,18)(4,19)(5,20)(6,21)(7,22)(8,23)(9,26)(10,27)(11,28)(12,29)(13,30)(14,31)(15,32)(16,25)(33,56)(34,49)(35,50)(36,51)(37,52)(38,53)(39,54)(40,55)(41,60)(42,61)(43,62)(44,63)(45,64)(46,57)(47,58)(48,59), (1,53)(2,54)(3,55)(4,56)(5,49)(6,50)(7,51)(8,52)(9,47)(10,48)(11,41)(12,42)(13,43)(14,44)(15,45)(16,46)(17,39)(18,40)(19,33)(20,34)(21,35)(22,36)(23,37)(24,38)(25,57)(26,58)(27,59)(28,60)(29,61)(30,62)(31,63)(32,64), (1,60)(2,61)(3,62)(4,63)(5,64)(6,57)(7,58)(8,59)(9,36)(10,37)(11,38)(12,39)(13,40)(14,33)(15,34)(16,35)(17,42)(18,43)(19,44)(20,45)(21,46)(22,47)(23,48)(24,41)(25,50)(26,51)(27,52)(28,53)(29,54)(30,55)(31,56)(32,49), (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), (1,28)(2,27)(3,26)(4,25)(5,32)(6,31)(7,30)(8,29)(9,18)(10,17)(11,24)(12,23)(13,22)(14,21)(15,20)(16,19)(33,46)(34,45)(35,44)(36,43)(37,42)(38,41)(39,48)(40,47)(49,64)(50,63)(51,62)(52,61)(53,60)(54,59)(55,58)(56,57) );
G=PermutationGroup([(1,24),(2,17),(3,18),(4,19),(5,20),(6,21),(7,22),(8,23),(9,26),(10,27),(11,28),(12,29),(13,30),(14,31),(15,32),(16,25),(33,56),(34,49),(35,50),(36,51),(37,52),(38,53),(39,54),(40,55),(41,60),(42,61),(43,62),(44,63),(45,64),(46,57),(47,58),(48,59)], [(1,53),(2,54),(3,55),(4,56),(5,49),(6,50),(7,51),(8,52),(9,47),(10,48),(11,41),(12,42),(13,43),(14,44),(15,45),(16,46),(17,39),(18,40),(19,33),(20,34),(21,35),(22,36),(23,37),(24,38),(25,57),(26,58),(27,59),(28,60),(29,61),(30,62),(31,63),(32,64)], [(1,60),(2,61),(3,62),(4,63),(5,64),(6,57),(7,58),(8,59),(9,36),(10,37),(11,38),(12,39),(13,40),(14,33),(15,34),(16,35),(17,42),(18,43),(19,44),(20,45),(21,46),(22,47),(23,48),(24,41),(25,50),(26,51),(27,52),(28,53),(29,54),(30,55),(31,56),(32,49)], [(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)], [(1,28),(2,27),(3,26),(4,25),(5,32),(6,31),(7,30),(8,29),(9,18),(10,17),(11,24),(12,23),(13,22),(14,21),(15,20),(16,19),(33,46),(34,45),(35,44),(36,43),(37,42),(38,41),(39,48),(40,47),(49,64),(50,63),(51,62),(52,61),(53,60),(54,59),(55,58),(56,57)])
Matrix representation ►G ⊆ GL5(𝔽17)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 16 |
| 16 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 16 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 16 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 |
| 0 | 0 | 0 | 3 | 11 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 | 1 |
G:=sub<GL(5,GF(17))| [1,0,0,0,0,0,16,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,16],[16,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,16],[1,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,1,0,0,0,0,0,1],[16,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,3,0,0,0,11,11],[1,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,16,16,0,0,0,0,1] >;
56 conjugacy classes
| class | 1 | 2A | ··· | 2O | 2P | ··· | 2AE | 4A | ··· | 4H | 8A | ··· | 8P |
| order | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 4 | ··· | 4 | 2 | ··· | 2 | 2 | ··· | 2 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | D4 | D4 | D8 |
| kernel | C23×D8 | C23×C8 | C22×D8 | D4×C23 | C22×C4 | C24 | C23 |
| # reps | 1 | 1 | 28 | 2 | 7 | 1 | 16 |
In GAP, Magma, Sage, TeX
C_2^3\times D_8
% in TeX
G:=Group("C2^3xD8"); // GroupNames label
G:=SmallGroup(128,2306);
// by ID
G=gap.SmallGroup(128,2306);
# by ID
G:=PCGroup([7,-2,2,2,2,2,-2,-2,477,4037,2028,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^8=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀