direct product, p-group, metabelian, nilpotent (class 2), monomial
Aliases: C2×C42.12C4, C42.675C23, C42○2(C4⋊C8), C4⋊C8⋊93C22, (C4×C8)⋊67C22, (C22×C4)⋊10C8, C2.3(C23×C8), (C23×C4).41C4, C4.30(C22×C8), (C2×C42).55C4, C23.40(C2×C8), C42○2(C22⋊C8), C42.300(C2×C4), (C2×C8).472C23, C24.128(C2×C4), (C2×C4).636C24, (C2×C4).94M4(2), C4.70(C2×M4(2)), C4○(C42.12C4), C22.39(C23×C4), (C22×C42).32C2, C22.14(C22×C8), C2.5(C22×M4(2)), C4.80(C42⋊C2), C22⋊C8.243C22, (C23×C4).657C22, (C22×C8).506C22, C23.223(C22×C4), C22.62(C2×M4(2)), C42○(C42.12C4), (C22×C4).1270C23, (C2×C42).1104C22, C22.69(C42⋊C2), C4○2(C2×C4⋊C8), (C2×C4×C8)⋊15C2, C4⋊C8○(C2×C42), (C2×C4⋊C8)⋊54C2, (C2×C4)○3(C4⋊C8), (C2×C4)⋊12(C2×C8), C42○2(C2×C4⋊C8), C4○2(C2×C22⋊C8), C22⋊C8○(C2×C42), (C2×C4)○3(C22⋊C8), C4.287(C2×C4○D4), C42○2(C2×C22⋊C8), (C2×C22⋊C8).50C2, C2.5(C2×C42⋊C2), (C2×C4).952(C4○D4), (C22×C4).458(C2×C4), (C2×C4).626(C22×C4), (C2×C4)○(C42.12C4), (C2×C4)○2(C2×C4⋊C8), (C2×C42)○(C2×C4⋊C8), SmallGroup(128,1649)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C2×C42.12C4
G = < a,b,c,d | a2=b4=c4=1, d4=b2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1c2, cd=dc >
Subgroups: 332 in 264 conjugacy classes, 196 normal (24 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, C23, C23, C23, C42, C2×C8, C2×C8, C22×C4, C22×C4, C22×C4, C24, C4×C8, C22⋊C8, C4⋊C8, C2×C42, C2×C42, C22×C8, C23×C4, C2×C4×C8, C2×C22⋊C8, C2×C4⋊C8, C42.12C4, C22×C42, C2×C42.12C4
Quotients: C1, C2, C4, C22, C8, C2×C4, C23, C2×C8, M4(2), C22×C4, C4○D4, C24, C42⋊C2, C22×C8, C2×M4(2), C23×C4, C2×C4○D4, C42.12C4, C2×C42⋊C2, C23×C8, C22×M4(2), C2×C42.12C4
►On 64 points
(1 51)(2 52)(3 53)(4 54)(5 55)(6 56)(7 49)(8 50)(9 42)(10 43)(11 44)(12 45)(13 46)(14 47)(15 48)(16 41)(17 25)(18 26)(19 27)(20 28)(21 29)(22 30)(23 31)(24 32)(33 59)(34 60)(35 61)(36 62)(37 63)(38 64)(39 57)(40 58)
(1 3 5 7)(2 18 6 22)(4 20 8 24)(9 60 13 64)(10 12 14 16)(11 62 15 58)(17 19 21 23)(25 27 29 31)(26 56 30 52)(28 50 32 54)(33 35 37 39)(34 46 38 42)(36 48 40 44)(41 43 45 47)(49 51 53 55)(57 59 61 63)
(1 57 19 12)(2 58 20 13)(3 59 21 14)(4 60 22 15)(5 61 23 16)(6 62 24 9)(7 63 17 10)(8 64 18 11)(25 43 49 37)(26 44 50 38)(27 45 51 39)(28 46 52 40)(29 47 53 33)(30 48 54 34)(31 41 55 35)(32 42 56 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)
G:=sub<Sym(64)| (1,51)(2,52)(3,53)(4,54)(5,55)(6,56)(7,49)(8,50)(9,42)(10,43)(11,44)(12,45)(13,46)(14,47)(15,48)(16,41)(17,25)(18,26)(19,27)(20,28)(21,29)(22,30)(23,31)(24,32)(33,59)(34,60)(35,61)(36,62)(37,63)(38,64)(39,57)(40,58), (1,3,5,7)(2,18,6,22)(4,20,8,24)(9,60,13,64)(10,12,14,16)(11,62,15,58)(17,19,21,23)(25,27,29,31)(26,56,30,52)(28,50,32,54)(33,35,37,39)(34,46,38,42)(36,48,40,44)(41,43,45,47)(49,51,53,55)(57,59,61,63), (1,57,19,12)(2,58,20,13)(3,59,21,14)(4,60,22,15)(5,61,23,16)(6,62,24,9)(7,63,17,10)(8,64,18,11)(25,43,49,37)(26,44,50,38)(27,45,51,39)(28,46,52,40)(29,47,53,33)(30,48,54,34)(31,41,55,35)(32,42,56,36), (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)>;
G:=Group( (1,51)(2,52)(3,53)(4,54)(5,55)(6,56)(7,49)(8,50)(9,42)(10,43)(11,44)(12,45)(13,46)(14,47)(15,48)(16,41)(17,25)(18,26)(19,27)(20,28)(21,29)(22,30)(23,31)(24,32)(33,59)(34,60)(35,61)(36,62)(37,63)(38,64)(39,57)(40,58), (1,3,5,7)(2,18,6,22)(4,20,8,24)(9,60,13,64)(10,12,14,16)(11,62,15,58)(17,19,21,23)(25,27,29,31)(26,56,30,52)(28,50,32,54)(33,35,37,39)(34,46,38,42)(36,48,40,44)(41,43,45,47)(49,51,53,55)(57,59,61,63), (1,57,19,12)(2,58,20,13)(3,59,21,14)(4,60,22,15)(5,61,23,16)(6,62,24,9)(7,63,17,10)(8,64,18,11)(25,43,49,37)(26,44,50,38)(27,45,51,39)(28,46,52,40)(29,47,53,33)(30,48,54,34)(31,41,55,35)(32,42,56,36), (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) );
G=PermutationGroup([[(1,51),(2,52),(3,53),(4,54),(5,55),(6,56),(7,49),(8,50),(9,42),(10,43),(11,44),(12,45),(13,46),(14,47),(15,48),(16,41),(17,25),(18,26),(19,27),(20,28),(21,29),(22,30),(23,31),(24,32),(33,59),(34,60),(35,61),(36,62),(37,63),(38,64),(39,57),(40,58)], [(1,3,5,7),(2,18,6,22),(4,20,8,24),(9,60,13,64),(10,12,14,16),(11,62,15,58),(17,19,21,23),(25,27,29,31),(26,56,30,52),(28,50,32,54),(33,35,37,39),(34,46,38,42),(36,48,40,44),(41,43,45,47),(49,51,53,55),(57,59,61,63)], [(1,57,19,12),(2,58,20,13),(3,59,21,14),(4,60,22,15),(5,61,23,16),(6,62,24,9),(7,63,17,10),(8,64,18,11),(25,43,49,37),(26,44,50,38),(27,45,51,39),(28,46,52,40),(29,47,53,33),(30,48,54,34),(31,41,55,35),(32,42,56,36)], [(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)]])
80 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 4A | ··· | 4X | 4Y | ··· | 4AJ | 8A | ··· | 8AF |
| 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 | 2 | ··· | 2 |
80 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C8 | M4(2) | C4○D4 |
| kernel | C2×C42.12C4 | C2×C4×C8 | C2×C22⋊C8 | C2×C4⋊C8 | C42.12C4 | C22×C42 | C2×C42 | C23×C4 | C22×C4 | C2×C4 | C2×C4 |
| # reps | 1 | 2 | 2 | 2 | 8 | 1 | 12 | 4 | 32 | 8 | 8 |
Matrix representation of C2×C42.12C4 ►in GL4(𝔽17) generated by
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 4 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 1 |
| 13 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 13 | 0 |
| 0 | 0 | 0 | 13 |
| 8 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 16 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(17))| [16,0,0,0,0,16,0,0,0,0,1,0,0,0,0,1],[4,0,0,0,0,16,0,0,0,0,16,0,0,0,0,1],[13,0,0,0,0,16,0,0,0,0,13,0,0,0,0,13],[8,0,0,0,0,16,0,0,0,0,0,1,0,0,16,0] >;
C2×C42.12C4 in GAP, Magma, Sage, TeX
C_2\times C_4^2._{12}C_4 % in TeX
G:=Group("C2xC4^2.12C4"); // GroupNames label
G:=SmallGroup(128,1649);
// by ID
G=gap.SmallGroup(128,1649);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,100,124]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^4=c^4=1,d^4=b^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1*c^2,c*d=d*c>;
// generators/relations