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), (C2×C4).636C24, C24.128(C2×C4), (C2×C8).472C23, C42.300(C2×C4), (C2×C4).94M4(2), C4.70(C2×M4(2)), C4○(C42.12C4), (C22×C42).32C2, C22.14(C22×C8), C22.39(C23×C4), 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), (C2×C42).1104C22, (C22×C4).1270C23, 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
Subgroups: 332 in 264 conjugacy classes, 196 normal (24 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×16], 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], C22×C4 [×6], C22×C4 [×20], C22×C4 [×8], C24, C4×C8 [×8], C22⋊C8 [×8], C4⋊C8 [×8], C2×C42 [×4], C2×C42 [×8], C22×C8 [×4], C23×C4 [×3], C2×C4×C8 [×2], C2×C22⋊C8 [×2], C2×C4⋊C8 [×2], C42.12C4 [×8], C22×C42, C2×C42.12C4
Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], C8 [×8], C2×C4 [×28], C23 [×15], C2×C8 [×28], M4(2) [×4], C22×C4 [×14], C4○D4 [×4], C24, C42⋊C2 [×4], C22×C8 [×14], C2×M4(2) [×6], C23×C4, C2×C4○D4 [×2], C42.12C4 [×4], C2×C42⋊C2, C23×C8, C22×M4(2), C2×C42.12C4
Generators and relations
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 >
►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 29)(18 30)(19 31)(20 32)(21 25)(22 26)(23 27)(24 28)(33 41)(34 42)(35 43)(36 44)(37 45)(38 46)(39 47)(40 48)
(1 3 5 7)(2 18 6 22)(4 20 8 24)(9 11 13 15)(10 44 14 48)(12 46 16 42)(17 19 21 23)(25 27 29 31)(26 52 30 56)(28 54 32 50)(33 35 37 39)(34 60 38 64)(36 62 40 58)(41 43 45 47)(49 51 53 55)(57 59 61 63)
(1 57 19 37)(2 58 20 38)(3 59 21 39)(4 60 22 40)(5 61 23 33)(6 62 24 34)(7 63 17 35)(8 64 18 36)(9 31 45 51)(10 32 46 52)(11 25 47 53)(12 26 48 54)(13 27 41 55)(14 28 42 56)(15 29 43 49)(16 30 44 50)
(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)(9,57)(10,58)(11,59)(12,60)(13,61)(14,62)(15,63)(16,64)(17,29)(18,30)(19,31)(20,32)(21,25)(22,26)(23,27)(24,28)(33,41)(34,42)(35,43)(36,44)(37,45)(38,46)(39,47)(40,48), (1,3,5,7)(2,18,6,22)(4,20,8,24)(9,11,13,15)(10,44,14,48)(12,46,16,42)(17,19,21,23)(25,27,29,31)(26,52,30,56)(28,54,32,50)(33,35,37,39)(34,60,38,64)(36,62,40,58)(41,43,45,47)(49,51,53,55)(57,59,61,63), (1,57,19,37)(2,58,20,38)(3,59,21,39)(4,60,22,40)(5,61,23,33)(6,62,24,34)(7,63,17,35)(8,64,18,36)(9,31,45,51)(10,32,46,52)(11,25,47,53)(12,26,48,54)(13,27,41,55)(14,28,42,56)(15,29,43,49)(16,30,44,50), (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)(9,57)(10,58)(11,59)(12,60)(13,61)(14,62)(15,63)(16,64)(17,29)(18,30)(19,31)(20,32)(21,25)(22,26)(23,27)(24,28)(33,41)(34,42)(35,43)(36,44)(37,45)(38,46)(39,47)(40,48), (1,3,5,7)(2,18,6,22)(4,20,8,24)(9,11,13,15)(10,44,14,48)(12,46,16,42)(17,19,21,23)(25,27,29,31)(26,52,30,56)(28,54,32,50)(33,35,37,39)(34,60,38,64)(36,62,40,58)(41,43,45,47)(49,51,53,55)(57,59,61,63), (1,57,19,37)(2,58,20,38)(3,59,21,39)(4,60,22,40)(5,61,23,33)(6,62,24,34)(7,63,17,35)(8,64,18,36)(9,31,45,51)(10,32,46,52)(11,25,47,53)(12,26,48,54)(13,27,41,55)(14,28,42,56)(15,29,43,49)(16,30,44,50), (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),(9,57),(10,58),(11,59),(12,60),(13,61),(14,62),(15,63),(16,64),(17,29),(18,30),(19,31),(20,32),(21,25),(22,26),(23,27),(24,28),(33,41),(34,42),(35,43),(36,44),(37,45),(38,46),(39,47),(40,48)], [(1,3,5,7),(2,18,6,22),(4,20,8,24),(9,11,13,15),(10,44,14,48),(12,46,16,42),(17,19,21,23),(25,27,29,31),(26,52,30,56),(28,54,32,50),(33,35,37,39),(34,60,38,64),(36,62,40,58),(41,43,45,47),(49,51,53,55),(57,59,61,63)], [(1,57,19,37),(2,58,20,38),(3,59,21,39),(4,60,22,40),(5,61,23,33),(6,62,24,34),(7,63,17,35),(8,64,18,36),(9,31,45,51),(10,32,46,52),(11,25,47,53),(12,26,48,54),(13,27,41,55),(14,28,42,56),(15,29,43,49),(16,30,44,50)], [(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 ⊆ 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] >;
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 |
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