direct product, p-group, metabelian, nilpotent (class 2), monomial
Aliases: C2×C4⋊M4(2), C42.673C23, C4⋊C8⋊79C22, C4⋊3(C2×M4(2)), (C2×C4)⋊11M4(2), (C23×C4).40C4, (C2×C42).54C4, C4.58(C22×Q8), (C22×C4).99Q8, C23.73(C4⋊C4), C4○(C4⋊M4(2)), C24.127(C2×C4), (C2×C4).632C24, C42.333(C2×C4), (C2×C8).393C23, (C22×C4).602D4, C4.184(C22×D4), (C22×C42).31C2, C2.7(C22×M4(2)), C22.161(C23×C4), C23.222(C22×C4), (C23×C4).692C22, (C22×C8).426C22, C22.61(C2×M4(2)), (C2×C42).1103C22, (C22×C4).1500C23, (C22×M4(2)).27C2, (C2×M4(2)).335C22, (C2×C4⋊C8)⋊41C2, C4.60(C2×C4⋊C4), C22.32(C2×C4⋊C4), C2.18(C22×C4⋊C4), (C2×C4).356(C2×Q8), (C2×C4).147(C4⋊C4), (C2×C4).1566(C2×D4), (C2×C4)○(C4⋊M4(2)), (C22×C4).457(C2×C4), (C2×C4).570(C22×C4), SmallGroup(128,1635)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 380 in 288 conjugacy classes, 196 normal (16 characteristic)
C1, C2, C2 [×6], C2 [×4], C4 [×2], C4 [×14], C4 [×4], C22, C22 [×10], C22 [×12], C8 [×8], C2×C4 [×2], C2×C4 [×42], C2×C4 [×20], C23, C23 [×6], C23 [×4], C42 [×16], C2×C8 [×8], C2×C8 [×8], M4(2) [×16], C22×C4 [×2], C22×C4 [×24], C22×C4 [×8], C24, C4⋊C8 [×16], C2×C42 [×2], C2×C42 [×10], C22×C8 [×4], C2×M4(2) [×8], C2×M4(2) [×8], C23×C4, C23×C4 [×2], C2×C4⋊C8 [×4], C4⋊M4(2) [×8], C22×C42, C22×M4(2) [×2], C2×C4⋊M4(2)
Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], D4 [×4], Q8 [×4], C23 [×15], C4⋊C4 [×16], M4(2) [×8], C22×C4 [×14], C2×D4 [×6], C2×Q8 [×6], C24, C2×C4⋊C4 [×12], C2×M4(2) [×12], C23×C4, C22×D4, C22×Q8, C4⋊M4(2) [×4], C22×C4⋊C4, C22×M4(2) [×2], C2×C4⋊M4(2)
Generators and relations
G = < a,b,c,d | a2=b4=c8=d2=1, ab=ba, ac=ca, ad=da, cbc-1=b-1, bd=db, dcd=c5 >
►On 64 points
(1 55)(2 56)(3 49)(4 50)(5 51)(6 52)(7 53)(8 54)(9 61)(10 62)(11 63)(12 64)(13 57)(14 58)(15 59)(16 60)(17 39)(18 40)(19 33)(20 34)(21 35)(22 36)(23 37)(24 38)(25 47)(26 48)(27 41)(28 42)(29 43)(30 44)(31 45)(32 46)
(1 61 23 26)(2 27 24 62)(3 63 17 28)(4 29 18 64)(5 57 19 30)(6 31 20 58)(7 59 21 32)(8 25 22 60)(9 37 48 55)(10 56 41 38)(11 39 42 49)(12 50 43 40)(13 33 44 51)(14 52 45 34)(15 35 46 53)(16 54 47 36)
(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)
(2 6)(4 8)(10 14)(12 16)(18 22)(20 24)(25 29)(27 31)(34 38)(36 40)(41 45)(43 47)(50 54)(52 56)(58 62)(60 64)
G:=sub<Sym(64)| (1,55)(2,56)(3,49)(4,50)(5,51)(6,52)(7,53)(8,54)(9,61)(10,62)(11,63)(12,64)(13,57)(14,58)(15,59)(16,60)(17,39)(18,40)(19,33)(20,34)(21,35)(22,36)(23,37)(24,38)(25,47)(26,48)(27,41)(28,42)(29,43)(30,44)(31,45)(32,46), (1,61,23,26)(2,27,24,62)(3,63,17,28)(4,29,18,64)(5,57,19,30)(6,31,20,58)(7,59,21,32)(8,25,22,60)(9,37,48,55)(10,56,41,38)(11,39,42,49)(12,50,43,40)(13,33,44,51)(14,52,45,34)(15,35,46,53)(16,54,47,36), (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), (2,6)(4,8)(10,14)(12,16)(18,22)(20,24)(25,29)(27,31)(34,38)(36,40)(41,45)(43,47)(50,54)(52,56)(58,62)(60,64)>;
G:=Group( (1,55)(2,56)(3,49)(4,50)(5,51)(6,52)(7,53)(8,54)(9,61)(10,62)(11,63)(12,64)(13,57)(14,58)(15,59)(16,60)(17,39)(18,40)(19,33)(20,34)(21,35)(22,36)(23,37)(24,38)(25,47)(26,48)(27,41)(28,42)(29,43)(30,44)(31,45)(32,46), (1,61,23,26)(2,27,24,62)(3,63,17,28)(4,29,18,64)(5,57,19,30)(6,31,20,58)(7,59,21,32)(8,25,22,60)(9,37,48,55)(10,56,41,38)(11,39,42,49)(12,50,43,40)(13,33,44,51)(14,52,45,34)(15,35,46,53)(16,54,47,36), (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), (2,6)(4,8)(10,14)(12,16)(18,22)(20,24)(25,29)(27,31)(34,38)(36,40)(41,45)(43,47)(50,54)(52,56)(58,62)(60,64) );
G=PermutationGroup([(1,55),(2,56),(3,49),(4,50),(5,51),(6,52),(7,53),(8,54),(9,61),(10,62),(11,63),(12,64),(13,57),(14,58),(15,59),(16,60),(17,39),(18,40),(19,33),(20,34),(21,35),(22,36),(23,37),(24,38),(25,47),(26,48),(27,41),(28,42),(29,43),(30,44),(31,45),(32,46)], [(1,61,23,26),(2,27,24,62),(3,63,17,28),(4,29,18,64),(5,57,19,30),(6,31,20,58),(7,59,21,32),(8,25,22,60),(9,37,48,55),(10,56,41,38),(11,39,42,49),(12,50,43,40),(13,33,44,51),(14,52,45,34),(15,35,46,53),(16,54,47,36)], [(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)], [(2,6),(4,8),(10,14),(12,16),(18,22),(20,24),(25,29),(27,31),(34,38),(36,40),(41,45),(43,47),(50,54),(52,56),(58,62),(60,64)])
Matrix representation ►G ⊆ GL5(𝔽17)
| 16 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 16 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 15 | 0 | 0 |
| 0 | 1 | 16 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 13 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 16 | 2 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 4 | 0 |
| 16 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 16 |
G:=sub<GL(5,GF(17))| [16,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,16],[1,0,0,0,0,0,1,1,0,0,0,15,16,0,0,0,0,0,4,0,0,0,0,0,13],[1,0,0,0,0,0,16,0,0,0,0,2,1,0,0,0,0,0,0,4,0,0,0,1,0],[16,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,16] >;
56 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 4A | ··· | 4H | 4I | ··· | 4AB | 8A | ··· | 8P |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | D4 | Q8 | M4(2) |
| kernel | C2×C4⋊M4(2) | C2×C4⋊C8 | C4⋊M4(2) | C22×C42 | C22×M4(2) | C2×C42 | C23×C4 | C22×C4 | C22×C4 | C2×C4 |
| # reps | 1 | 4 | 8 | 1 | 2 | 12 | 4 | 4 | 4 | 16 |
In GAP, Magma, Sage, TeX
C_2\times C_4\rtimes M_{4(2)} % in TeX
G:=Group("C2xC4:M4(2)"); // GroupNames label
G:=SmallGroup(128,1635);
// by ID
G=gap.SmallGroup(128,1635);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,120,1430,124]);
// 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,c*b*c^-1=b^-1,b*d=d*b,d*c*d=c^5>;
// generators/relations