direct product, p-group, metabelian, nilpotent (class 2), monomial
Aliases: C22×C22⋊C8, C24⋊4C8, C25.7C4, C23.38M4(2), (C23×C8)⋊3C2, C23⋊6(C2×C8), (C2×C8)⋊12C23, (C24×C4).7C2, C2.1(C23×C8), (C23×C4).34C4, C22⋊2(C22×C8), C24.124(C2×C4), (C2×C4).626C24, (C22×C8)⋊60C22, (C22×C4).818D4, C4.175(C22×D4), C22.37(C23×C4), C2.3(C22×M4(2)), C23.218(C22×C4), (C23×C4).656C22, C22.59(C2×M4(2)), C23.229(C22⋊C4), (C22×C4).1269C23, C4○(C2×C22⋊C8), (C2×C4)○2(C22⋊C8), (C2×C4).1561(C2×D4), C4.119(C2×C22⋊C4), (C22×C4)○(C22⋊C8), C2.3(C22×C22⋊C4), (C22×C4).456(C2×C4), (C2×C4).624(C22×C4), (C2×C4).403(C22⋊C4), C22.134(C2×C22⋊C4), (C2×C4)○2(C2×C22⋊C8), (C22×C4)○(C2×C22⋊C8), SmallGroup(128,1608)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 812 in 536 conjugacy classes, 260 normal (12 characteristic)
C1, C2 [×3], C2 [×12], C2 [×8], C4 [×8], C4 [×4], C22, C22 [×42], C22 [×56], C8 [×8], C2×C4, C2×C4 [×31], C2×C4 [×44], C23 [×43], C23 [×56], C2×C8 [×8], C2×C8 [×24], C22×C4 [×36], C22×C4 [×52], C24, C24 [×14], C24 [×8], C22⋊C8 [×16], C22×C8 [×12], C22×C8 [×8], C23×C4 [×2], C23×C4 [×12], C23×C4 [×8], C25, C2×C22⋊C8 [×12], C23×C8 [×2], C24×C4, C22×C22⋊C8
Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], C8 [×8], C2×C4 [×28], D4 [×8], C23 [×15], C22⋊C4 [×16], C2×C8 [×28], M4(2) [×4], C22×C4 [×14], C2×D4 [×12], C24, C22⋊C8 [×16], C2×C22⋊C4 [×12], C22×C8 [×14], C2×M4(2) [×6], C23×C4, C22×D4 [×2], C2×C22⋊C8 [×12], C22×C22⋊C4, C23×C8, C22×M4(2), C22×C22⋊C8
Generators and relations
G = < a,b,c,d,e | a2=b2=c2=d2=e8=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, ece-1=cd=dc, de=ed >
►On 64 points
(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 9)(8 10)(17 48)(18 41)(19 42)(20 43)(21 44)(22 45)(23 46)(24 47)(25 62)(26 63)(27 64)(28 57)(29 58)(30 59)(31 60)(32 61)(33 51)(34 52)(35 53)(36 54)(37 55)(38 56)(39 49)(40 50)
(1 20)(2 21)(3 22)(4 23)(5 24)(6 17)(7 18)(8 19)(9 41)(10 42)(11 43)(12 44)(13 45)(14 46)(15 47)(16 48)(25 40)(26 33)(27 34)(28 35)(29 36)(30 37)(31 38)(32 39)(49 61)(50 62)(51 63)(52 64)(53 57)(54 58)(55 59)(56 60)
(1 24)(2 56)(3 18)(4 50)(5 20)(6 52)(7 22)(8 54)(9 45)(10 36)(11 47)(12 38)(13 41)(14 40)(15 43)(16 34)(17 64)(19 58)(21 60)(23 62)(25 46)(26 37)(27 48)(28 39)(29 42)(30 33)(31 44)(32 35)(49 57)(51 59)(53 61)(55 63)
(1 63)(2 64)(3 57)(4 58)(5 59)(6 60)(7 61)(8 62)(9 32)(10 25)(11 26)(12 27)(13 28)(14 29)(15 30)(16 31)(17 56)(18 49)(19 50)(20 51)(21 52)(22 53)(23 54)(24 55)(33 43)(34 44)(35 45)(36 46)(37 47)(38 48)(39 41)(40 42)
(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)| (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,9)(8,10)(17,48)(18,41)(19,42)(20,43)(21,44)(22,45)(23,46)(24,47)(25,62)(26,63)(27,64)(28,57)(29,58)(30,59)(31,60)(32,61)(33,51)(34,52)(35,53)(36,54)(37,55)(38,56)(39,49)(40,50), (1,20)(2,21)(3,22)(4,23)(5,24)(6,17)(7,18)(8,19)(9,41)(10,42)(11,43)(12,44)(13,45)(14,46)(15,47)(16,48)(25,40)(26,33)(27,34)(28,35)(29,36)(30,37)(31,38)(32,39)(49,61)(50,62)(51,63)(52,64)(53,57)(54,58)(55,59)(56,60), (1,24)(2,56)(3,18)(4,50)(5,20)(6,52)(7,22)(8,54)(9,45)(10,36)(11,47)(12,38)(13,41)(14,40)(15,43)(16,34)(17,64)(19,58)(21,60)(23,62)(25,46)(26,37)(27,48)(28,39)(29,42)(30,33)(31,44)(32,35)(49,57)(51,59)(53,61)(55,63), (1,63)(2,64)(3,57)(4,58)(5,59)(6,60)(7,61)(8,62)(9,32)(10,25)(11,26)(12,27)(13,28)(14,29)(15,30)(16,31)(17,56)(18,49)(19,50)(20,51)(21,52)(22,53)(23,54)(24,55)(33,43)(34,44)(35,45)(36,46)(37,47)(38,48)(39,41)(40,42), (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( (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,9)(8,10)(17,48)(18,41)(19,42)(20,43)(21,44)(22,45)(23,46)(24,47)(25,62)(26,63)(27,64)(28,57)(29,58)(30,59)(31,60)(32,61)(33,51)(34,52)(35,53)(36,54)(37,55)(38,56)(39,49)(40,50), (1,20)(2,21)(3,22)(4,23)(5,24)(6,17)(7,18)(8,19)(9,41)(10,42)(11,43)(12,44)(13,45)(14,46)(15,47)(16,48)(25,40)(26,33)(27,34)(28,35)(29,36)(30,37)(31,38)(32,39)(49,61)(50,62)(51,63)(52,64)(53,57)(54,58)(55,59)(56,60), (1,24)(2,56)(3,18)(4,50)(5,20)(6,52)(7,22)(8,54)(9,45)(10,36)(11,47)(12,38)(13,41)(14,40)(15,43)(16,34)(17,64)(19,58)(21,60)(23,62)(25,46)(26,37)(27,48)(28,39)(29,42)(30,33)(31,44)(32,35)(49,57)(51,59)(53,61)(55,63), (1,63)(2,64)(3,57)(4,58)(5,59)(6,60)(7,61)(8,62)(9,32)(10,25)(11,26)(12,27)(13,28)(14,29)(15,30)(16,31)(17,56)(18,49)(19,50)(20,51)(21,52)(22,53)(23,54)(24,55)(33,43)(34,44)(35,45)(36,46)(37,47)(38,48)(39,41)(40,42), (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([(1,11),(2,12),(3,13),(4,14),(5,15),(6,16),(7,9),(8,10),(17,48),(18,41),(19,42),(20,43),(21,44),(22,45),(23,46),(24,47),(25,62),(26,63),(27,64),(28,57),(29,58),(30,59),(31,60),(32,61),(33,51),(34,52),(35,53),(36,54),(37,55),(38,56),(39,49),(40,50)], [(1,20),(2,21),(3,22),(4,23),(5,24),(6,17),(7,18),(8,19),(9,41),(10,42),(11,43),(12,44),(13,45),(14,46),(15,47),(16,48),(25,40),(26,33),(27,34),(28,35),(29,36),(30,37),(31,38),(32,39),(49,61),(50,62),(51,63),(52,64),(53,57),(54,58),(55,59),(56,60)], [(1,24),(2,56),(3,18),(4,50),(5,20),(6,52),(7,22),(8,54),(9,45),(10,36),(11,47),(12,38),(13,41),(14,40),(15,43),(16,34),(17,64),(19,58),(21,60),(23,62),(25,46),(26,37),(27,48),(28,39),(29,42),(30,33),(31,44),(32,35),(49,57),(51,59),(53,61),(55,63)], [(1,63),(2,64),(3,57),(4,58),(5,59),(6,60),(7,61),(8,62),(9,32),(10,25),(11,26),(12,27),(13,28),(14,29),(15,30),(16,31),(17,56),(18,49),(19,50),(20,51),(21,52),(22,53),(23,54),(24,55),(33,43),(34,44),(35,45),(36,46),(37,47),(38,48),(39,41),(40,42)], [(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 ⊆ GL5(𝔽17)
| 16 | 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 | 16 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 16 |
| 1 | 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 | 8 | 0 | 0 | 0 |
| 0 | 0 | 8 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 |
| 0 | 0 | 0 | 16 | 0 |
G:=sub<GL(5,GF(17))| [16,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,1,0,0,0,0,0,1],[16,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,1,0,0,0,0,0,16],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,16],[16,0,0,0,0,0,8,0,0,0,0,0,8,0,0,0,0,0,0,16,0,0,0,16,0] >;
80 conjugacy classes
| class | 1 | 2A | ··· | 2O | 2P | ··· | 2W | 4A | ··· | 4P | 4Q | ··· | 4X | 8A | ··· | 8AF |
| order | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
80 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C8 | D4 | M4(2) |
| kernel | C22×C22⋊C8 | C2×C22⋊C8 | C23×C8 | C24×C4 | C23×C4 | C25 | C24 | C22×C4 | C23 |
| # reps | 1 | 12 | 2 | 1 | 14 | 2 | 32 | 8 | 8 |
In GAP, Magma, Sage, TeX
C_2^2\times C_2^2\rtimes C_8
% in TeX
G:=Group("C2^2xC2^2:C8"); // GroupNames label
G:=SmallGroup(128,1608);
// by ID
G=gap.SmallGroup(128,1608);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^2=e^8=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,e*c*e^-1=c*d=d*c,d*e=e*d>;
// generators/relations