direct product, p-group, metabelian, nilpotent (class 2), monomial
Aliases: C2×C8○2M4(2), C23.38C42, C42.587C23, (C22×C8)⋊21C4, (C4×C8)⋊79C22, C8○3(C2×M4(2)), (C2×C8)○3M4(2), (C2×C4).77C42, C8.46(C22×C4), C4.58(C23×C4), C24.97(C2×C4), C4.31(C2×C42), (C23×C8).25C2, M4(2)○(C22×C8), C8○(C22×M4(2)), C8○2(C42⋊C2), C8⋊C4⋊68C22, C8○(C8○2M4(2)), C4○(C8○2M4(2)), M4(2)⋊22(C2×C4), (C2×M4(2))⋊21C4, (C2×C8).637C23, (C2×C4).622C24, C42.246(C2×C4), C42⋊C2.39C4, C22.36(C8○D4), C2.14(C22×C42), C22.33(C23×C4), C22.17(C2×C42), (C23×C4).686C22, C23.288(C22×C4), (C22×C8).583C22, (C2×C42).1022C22, (C22×C4).1648C23, (C22×M4(2)).33C2, C42⋊C2.345C22, (C2×M4(2)).379C22, C8○(C2×C4⋊C4), C4⋊C4○2(C2×C8), (C2×C4×C8)⋊40C2, C8○(C2×C22⋊C4), C8○2(C2×C8⋊C4), (C2×C8)⋊40(C2×C4), C22⋊C4○2(C2×C8), C2.1(C2×C8○D4), (C2×C4⋊C4).83C4, C8○(C2×C42⋊C2), (C2×C8⋊C4)⋊41C2, (C2×C8)○3(C8⋊C4), C4⋊C4.242(C2×C4), C8⋊C4○2(C22×C8), (C2×C8)○2(C2×M4(2)), C22⋊C4.85(C2×C4), (C2×C22⋊C4).54C4, C42⋊C2○(C22×C8), (C2×C8)○(C22×M4(2)), (C22×C8)○(C2×M4(2)), (C2×C8)○2(C42⋊C2), (C2×C8)○(C8○2M4(2)), (C2×C4)○(C8○2M4(2)), (C22×C4).381(C2×C4), (C2×C4).291(C22×C4), (C2×C42⋊C2).67C2, (C2×C8)○(C2×C4⋊C4), (C2×C8)○2(C2×C8⋊C4), (C22×C8)○(C2×C8⋊C4), (C2×C8)○(C2×C42⋊C2), (C22×C8)○(C2×C42⋊C2), SmallGroup(128,1604)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C2×C8○2M4(2)
G = < a,b,c,d | a2=b8=d2=1, c4=b4, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b4c >
Subgroups: 332 in 296 conjugacy classes, 260 normal (18 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, C23, C23, C23, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, C22×C4, C24, C4×C8, C8⋊C4, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C22×C8, C22×C8, C2×M4(2), C23×C4, C2×C4×C8, C2×C8⋊C4, C8○2M4(2), C2×C42⋊C2, C23×C8, C22×M4(2), C2×C8○2M4(2)
Quotients: C1, C2, C4, C22, C2×C4, C23, C42, C22×C4, C24, C2×C42, C8○D4, C23×C4, C8○2M4(2), C22×C42, C2×C8○D4, C2×C8○2M4(2)
►On 64 points
(1 55)(2 56)(3 49)(4 50)(5 51)(6 52)(7 53)(8 54)(9 20)(10 21)(11 22)(12 23)(13 24)(14 17)(15 18)(16 19)(25 60)(26 61)(27 62)(28 63)(29 64)(30 57)(31 58)(32 59)(33 44)(34 45)(35 46)(36 47)(37 48)(38 41)(39 42)(40 43)
(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 24 38 5 62 20 34)(2 59 17 39 6 63 21 35)(3 60 18 40 7 64 22 36)(4 61 19 33 8 57 23 37)(9 45 55 31 13 41 51 27)(10 46 56 32 14 42 52 28)(11 47 49 25 15 43 53 29)(12 48 50 26 16 44 54 30)
(25 29)(26 30)(27 31)(28 32)(33 37)(34 38)(35 39)(36 40)(41 45)(42 46)(43 47)(44 48)(57 61)(58 62)(59 63)(60 64)
G:=sub<Sym(64)| (1,55)(2,56)(3,49)(4,50)(5,51)(6,52)(7,53)(8,54)(9,20)(10,21)(11,22)(12,23)(13,24)(14,17)(15,18)(16,19)(25,60)(26,61)(27,62)(28,63)(29,64)(30,57)(31,58)(32,59)(33,44)(34,45)(35,46)(36,47)(37,48)(38,41)(39,42)(40,43), (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), (1,58,24,38,5,62,20,34)(2,59,17,39,6,63,21,35)(3,60,18,40,7,64,22,36)(4,61,19,33,8,57,23,37)(9,45,55,31,13,41,51,27)(10,46,56,32,14,42,52,28)(11,47,49,25,15,43,53,29)(12,48,50,26,16,44,54,30), (25,29)(26,30)(27,31)(28,32)(33,37)(34,38)(35,39)(36,40)(41,45)(42,46)(43,47)(44,48)(57,61)(58,62)(59,63)(60,64)>;
G:=Group( (1,55)(2,56)(3,49)(4,50)(5,51)(6,52)(7,53)(8,54)(9,20)(10,21)(11,22)(12,23)(13,24)(14,17)(15,18)(16,19)(25,60)(26,61)(27,62)(28,63)(29,64)(30,57)(31,58)(32,59)(33,44)(34,45)(35,46)(36,47)(37,48)(38,41)(39,42)(40,43), (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), (1,58,24,38,5,62,20,34)(2,59,17,39,6,63,21,35)(3,60,18,40,7,64,22,36)(4,61,19,33,8,57,23,37)(9,45,55,31,13,41,51,27)(10,46,56,32,14,42,52,28)(11,47,49,25,15,43,53,29)(12,48,50,26,16,44,54,30), (25,29)(26,30)(27,31)(28,32)(33,37)(34,38)(35,39)(36,40)(41,45)(42,46)(43,47)(44,48)(57,61)(58,62)(59,63)(60,64) );
G=PermutationGroup([[(1,55),(2,56),(3,49),(4,50),(5,51),(6,52),(7,53),(8,54),(9,20),(10,21),(11,22),(12,23),(13,24),(14,17),(15,18),(16,19),(25,60),(26,61),(27,62),(28,63),(29,64),(30,57),(31,58),(32,59),(33,44),(34,45),(35,46),(36,47),(37,48),(38,41),(39,42),(40,43)], [(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)], [(1,58,24,38,5,62,20,34),(2,59,17,39,6,63,21,35),(3,60,18,40,7,64,22,36),(4,61,19,33,8,57,23,37),(9,45,55,31,13,41,51,27),(10,46,56,32,14,42,52,28),(11,47,49,25,15,43,53,29),(12,48,50,26,16,44,54,30)], [(25,29),(26,30),(27,31),(28,32),(33,37),(34,38),(35,39),(36,40),(41,45),(42,46),(43,47),(44,48),(57,61),(58,62),(59,63),(60,64)]])
80 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 4A | ··· | 4H | 4I | ··· | 4AB | 8A | ··· | 8P | 8Q | ··· | 8AN |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 |
80 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 |
| type | + | + | + | + | + | + | + | ||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | C4 | C8○D4 |
| kernel | C2×C8○2M4(2) | C2×C4×C8 | C2×C8⋊C4 | C8○2M4(2) | C2×C42⋊C2 | C23×C8 | C22×M4(2) | C2×C22⋊C4 | C2×C4⋊C4 | C42⋊C2 | C22×C8 | C2×M4(2) | C22 |
| # reps | 1 | 2 | 2 | 8 | 1 | 1 | 1 | 4 | 4 | 8 | 16 | 16 | 16 |
Matrix representation of C2×C8○2M4(2) ►in GL4(𝔽17) generated by
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 13 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 8 |
| 1 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 13 | 8 |
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 1 | 13 |
| 0 | 0 | 0 | 16 |
G:=sub<GL(4,GF(17))| [16,0,0,0,0,16,0,0,0,0,1,0,0,0,0,1],[13,0,0,0,0,16,0,0,0,0,8,0,0,0,0,8],[1,0,0,0,0,16,0,0,0,0,9,13,0,0,0,8],[16,0,0,0,0,16,0,0,0,0,1,0,0,0,13,16] >;
C2×C8○2M4(2) in GAP, Magma, Sage, TeX
C_2\times C_8\circ_2M_4(2)
% in TeX
G:=Group("C2xC8o2M4(2)"); // GroupNames label
G:=SmallGroup(128,1604);
// by ID
G=gap.SmallGroup(128,1604);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,112,232,723,172]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^8=d^2=1,c^4=b^4,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=b^4*c>;
// generators/relations