direct product, p-group, metabelian, nilpotent (class 2), monomial
Aliases: C2×C42.6C22, C42.256C23, C4⋊C8⋊80C22, (C23×C8).12C2, C24.98(C2×C4), C4.59(C22×Q8), C23.74(C4⋊C4), (C2×C4).633C24, C42.200(C2×C4), (C2×C8).394C23, (C22×C4).783D4, C4.185(C22×D4), (C22×C4).100Q8, C42⋊C2.26C4, C22.38(C8○D4), C4○(C42.6C22), C22.162(C23×C4), C23.291(C22×C4), (C22×C8).505C22, (C2×C42).751C22, (C23×C4).693C22, (C22×C4).1501C23, (C22×M4(2)).28C2, C42⋊C2.282C22, (C2×M4(2)).336C22, (C2×C4⋊C8)⋊42C2, C4.61(C2×C4⋊C4), C2.8(C2×C8○D4), (C2×C4⋊C4).65C4, C4⋊C4.212(C2×C4), C2.19(C22×C4⋊C4), C22.33(C2×C4⋊C4), (C2×C4).235(C2×Q8), (C2×C4).148(C4⋊C4), (C2×C4).1408(C2×D4), (C2×C22⋊C4).42C4, C22⋊C4.63(C2×C4), (C22×C4).327(C2×C4), (C2×C4).247(C22×C4), (C2×C42⋊C2).51C2, (C2×C4)○(C42.6C22), SmallGroup(128,1636)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 332 in 256 conjugacy classes, 180 normal (16 characteristic)
C1, C2, C2 [×6], C2 [×4], C4 [×2], C4 [×6], C4 [×8], C22, C22 [×10], C22 [×12], C8 [×8], C2×C4 [×2], C2×C4 [×34], C2×C4 [×8], C23, C23 [×6], C23 [×4], C42 [×8], C22⋊C4 [×8], C4⋊C4 [×8], C2×C8 [×8], C2×C8 [×16], M4(2) [×8], C22×C4 [×2], C22×C4 [×16], C24, C4⋊C8 [×16], C2×C42 [×2], C2×C22⋊C4 [×2], C2×C4⋊C4 [×2], C42⋊C2 [×8], C22×C8 [×8], C22×C8 [×4], C2×M4(2) [×4], C2×M4(2) [×4], C23×C4, C2×C4⋊C8 [×4], C42.6C22 [×8], C2×C42⋊C2, C23×C8, C22×M4(2), C2×C42.6C22
Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], D4 [×4], Q8 [×4], C23 [×15], C4⋊C4 [×16], C22×C4 [×14], C2×D4 [×6], C2×Q8 [×6], C24, C2×C4⋊C4 [×12], C8○D4 [×4], C23×C4, C22×D4, C22×Q8, C42.6C22 [×4], C22×C4⋊C4, C2×C8○D4 [×2], C2×C42.6C22
Generators and relations
G = < a,b,c,d,e | a2=b4=c4=1, d2=c, e2=b2c2, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=b-1c2, ebe-1=bc2, cd=dc, ce=ec, ede-1=b2c2d >
►On 64 points
(1 51)(2 52)(3 53)(4 54)(5 55)(6 56)(7 49)(8 50)(9 57)(10 58)(11 59)(12 60)(13 61)(14 62)(15 63)(16 64)(17 31)(18 32)(19 25)(20 26)(21 27)(22 28)(23 29)(24 30)(33 47)(34 48)(35 41)(36 42)(37 43)(38 44)(39 45)(40 46)
(1 59 19 33)(2 38 20 64)(3 61 21 35)(4 40 22 58)(5 63 23 37)(6 34 24 60)(7 57 17 39)(8 36 18 62)(9 31 45 49)(10 54 46 28)(11 25 47 51)(12 56 48 30)(13 27 41 53)(14 50 42 32)(15 29 43 55)(16 52 44 26)
(1 3 5 7)(2 4 6 8)(9 11 13 15)(10 12 14 16)(17 19 21 23)(18 20 22 24)(25 27 29 31)(26 28 30 32)(33 35 37 39)(34 36 38 40)(41 43 45 47)(42 44 46 48)(49 51 53 55)(50 52 54 56)(57 59 61 63)(58 60 62 64)
(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 63 23 33)(2 34 24 64)(3 57 17 35)(4 36 18 58)(5 59 19 37)(6 38 20 60)(7 61 21 39)(8 40 22 62)(9 31 41 53)(10 54 42 32)(11 25 43 55)(12 56 44 26)(13 27 45 49)(14 50 46 28)(15 29 47 51)(16 52 48 30)
G:=sub<Sym(64)| (1,51)(2,52)(3,53)(4,54)(5,55)(6,56)(7,49)(8,50)(9,57)(10,58)(11,59)(12,60)(13,61)(14,62)(15,63)(16,64)(17,31)(18,32)(19,25)(20,26)(21,27)(22,28)(23,29)(24,30)(33,47)(34,48)(35,41)(36,42)(37,43)(38,44)(39,45)(40,46), (1,59,19,33)(2,38,20,64)(3,61,21,35)(4,40,22,58)(5,63,23,37)(6,34,24,60)(7,57,17,39)(8,36,18,62)(9,31,45,49)(10,54,46,28)(11,25,47,51)(12,56,48,30)(13,27,41,53)(14,50,42,32)(15,29,43,55)(16,52,44,26), (1,3,5,7)(2,4,6,8)(9,11,13,15)(10,12,14,16)(17,19,21,23)(18,20,22,24)(25,27,29,31)(26,28,30,32)(33,35,37,39)(34,36,38,40)(41,43,45,47)(42,44,46,48)(49,51,53,55)(50,52,54,56)(57,59,61,63)(58,60,62,64), (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), (1,63,23,33)(2,34,24,64)(3,57,17,35)(4,36,18,58)(5,59,19,37)(6,38,20,60)(7,61,21,39)(8,40,22,62)(9,31,41,53)(10,54,42,32)(11,25,43,55)(12,56,44,26)(13,27,45,49)(14,50,46,28)(15,29,47,51)(16,52,48,30)>;
G:=Group( (1,51)(2,52)(3,53)(4,54)(5,55)(6,56)(7,49)(8,50)(9,57)(10,58)(11,59)(12,60)(13,61)(14,62)(15,63)(16,64)(17,31)(18,32)(19,25)(20,26)(21,27)(22,28)(23,29)(24,30)(33,47)(34,48)(35,41)(36,42)(37,43)(38,44)(39,45)(40,46), (1,59,19,33)(2,38,20,64)(3,61,21,35)(4,40,22,58)(5,63,23,37)(6,34,24,60)(7,57,17,39)(8,36,18,62)(9,31,45,49)(10,54,46,28)(11,25,47,51)(12,56,48,30)(13,27,41,53)(14,50,42,32)(15,29,43,55)(16,52,44,26), (1,3,5,7)(2,4,6,8)(9,11,13,15)(10,12,14,16)(17,19,21,23)(18,20,22,24)(25,27,29,31)(26,28,30,32)(33,35,37,39)(34,36,38,40)(41,43,45,47)(42,44,46,48)(49,51,53,55)(50,52,54,56)(57,59,61,63)(58,60,62,64), (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), (1,63,23,33)(2,34,24,64)(3,57,17,35)(4,36,18,58)(5,59,19,37)(6,38,20,60)(7,61,21,39)(8,40,22,62)(9,31,41,53)(10,54,42,32)(11,25,43,55)(12,56,44,26)(13,27,45,49)(14,50,46,28)(15,29,47,51)(16,52,48,30) );
G=PermutationGroup([(1,51),(2,52),(3,53),(4,54),(5,55),(6,56),(7,49),(8,50),(9,57),(10,58),(11,59),(12,60),(13,61),(14,62),(15,63),(16,64),(17,31),(18,32),(19,25),(20,26),(21,27),(22,28),(23,29),(24,30),(33,47),(34,48),(35,41),(36,42),(37,43),(38,44),(39,45),(40,46)], [(1,59,19,33),(2,38,20,64),(3,61,21,35),(4,40,22,58),(5,63,23,37),(6,34,24,60),(7,57,17,39),(8,36,18,62),(9,31,45,49),(10,54,46,28),(11,25,47,51),(12,56,48,30),(13,27,41,53),(14,50,42,32),(15,29,43,55),(16,52,44,26)], [(1,3,5,7),(2,4,6,8),(9,11,13,15),(10,12,14,16),(17,19,21,23),(18,20,22,24),(25,27,29,31),(26,28,30,32),(33,35,37,39),(34,36,38,40),(41,43,45,47),(42,44,46,48),(49,51,53,55),(50,52,54,56),(57,59,61,63),(58,60,62,64)], [(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)], [(1,63,23,33),(2,34,24,64),(3,57,17,35),(4,36,18,58),(5,59,19,37),(6,38,20,60),(7,61,21,39),(8,40,22,62),(9,31,41,53),(10,54,42,32),(11,25,43,55),(12,56,44,26),(13,27,45,49),(14,50,46,28),(15,29,47,51),(16,52,48,30)])
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 | 0 | 1 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 16 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 4 |
| 16 | 0 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 | 0 |
| 0 | 0 | 15 | 0 | 0 |
| 0 | 0 | 0 | 15 | 0 |
| 0 | 0 | 0 | 0 | 15 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 1 | 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],[16,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,16,0,0,0,1,0],[1,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,4],[16,0,0,0,0,0,2,0,0,0,0,0,15,0,0,0,0,0,15,0,0,0,0,0,15],[1,0,0,0,0,0,0,16,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,1,0] >;
56 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 4A | ··· | 4H | 4I | 4J | 4K | 4L | 4M | ··· | 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 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | D4 | Q8 | C8○D4 |
| kernel | C2×C42.6C22 | C2×C4⋊C8 | C42.6C22 | C2×C42⋊C2 | C23×C8 | C22×M4(2) | C2×C22⋊C4 | C2×C4⋊C4 | C42⋊C2 | C22×C4 | C22×C4 | C22 |
| # reps | 1 | 4 | 8 | 1 | 1 | 1 | 4 | 4 | 8 | 4 | 4 | 16 |
In GAP, Magma, Sage, TeX
C_2\times C_4^2._6C_2^2
% in TeX
G:=Group("C2xC4^2.6C2^2"); // GroupNames label
G:=SmallGroup(128,1636);
// by ID
G=gap.SmallGroup(128,1636);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,120,723,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^4=c^4=1,d^2=c,e^2=b^2*c^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,d*b*d^-1=b^-1*c^2,e*b*e^-1=b*c^2,c*d=d*c,c*e=e*c,e*d*e^-1=b^2*c^2*d>;
// generators/relations