direct product, p-group, metabelian, nilpotent (class 2), monomial
Aliases: C4×C22⋊C8, C42.457D4, C23.33C42, (C22×C4)⋊6C8, C22⋊1(C4×C8), C4.155(C4×D4), (C2×C42).32C4, C23.30(C2×C8), (C2×C4).43C42, (C23×C4).27C4, C2.3(C4×M4(2)), C24.109(C2×C4), (C2×C4).90M4(2), (C22×C42).5C2, C22.27(C2×C42), C22.19(C22×C8), (C22×C8).469C22, (C23×C4).628C22, (C2×C42).987C22, C23.250(C22×C4), C22.36(C2×M4(2)), C2.2(C42.12C4), C4○3(C22.7C42), (C22×C4).1603C23, C22.7C42⋊42C2, C22.48(C42⋊C2), C42○2(C22.7C42), (C2×C4×C8)⋊5C2, C2.6(C2×C4×C8), (C2×C8)⋊27(C2×C4), (C2×C4).59(C2×C8), C2.2(C4×C22⋊C4), C2.2(C2×C22⋊C8), C42○(C2×C22⋊C8), (C2×C4).1493(C2×D4), (C2×C22⋊C8).48C2, (C2×C4).913(C4○D4), (C22×C4).433(C2×C4), (C2×C4).593(C22×C4), (C2×C4).395(C22⋊C4), C22.111(C2×C22⋊C4), (C2×C4)○2(C22.7C42), (C2×C4)○(C2×C22⋊C8), (C2×C42)○(C2×C22⋊C8), SmallGroup(128,480)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C4×C22⋊C8
G = < a,b,c,d | a4=b2=c2=d8=1, ab=ba, ac=ca, ad=da, dbd-1=bc=cb, cd=dc >
Subgroups: 316 in 218 conjugacy classes, 120 normal (16 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×12], C4 [×6], C22 [×3], C22 [×8], C22 [×12], C8 [×8], C2×C4 [×24], C2×C4 [×30], C23, C23 [×6], C23 [×4], C42 [×4], C42 [×6], C2×C8 [×8], C2×C8 [×8], C22×C4 [×2], C22×C4 [×16], C22×C4 [×12], C24, C4×C8 [×4], C22⋊C8 [×8], C2×C42 [×2], C2×C42 [×2], C2×C42 [×4], C22×C8 [×4], C23×C4, C23×C4 [×2], C22.7C42 [×2], C2×C4×C8 [×2], C2×C22⋊C8 [×2], C22×C42, C4×C22⋊C8
Quotients: C1, C2 [×7], C4 [×12], C22 [×7], C8 [×8], C2×C4 [×18], D4 [×4], C23, C42 [×4], C22⋊C4 [×4], C2×C8 [×12], M4(2) [×4], C22×C4 [×3], C2×D4 [×2], C4○D4 [×2], C4×C8 [×4], C22⋊C8 [×8], C2×C42, C2×C22⋊C4, C42⋊C2, C4×D4 [×4], C22×C8 [×2], C2×M4(2) [×2], C4×C22⋊C4, C2×C4×C8, C4×M4(2), C2×C22⋊C8 [×2], C42.12C4 [×2], C4×C22⋊C8
►On 64 points
(1 27 61 45)(2 28 62 46)(3 29 63 47)(4 30 64 48)(5 31 57 41)(6 32 58 42)(7 25 59 43)(8 26 60 44)(9 18 36 49)(10 19 37 50)(11 20 38 51)(12 21 39 52)(13 22 40 53)(14 23 33 54)(15 24 34 55)(16 17 35 56)
(2 21)(4 23)(6 17)(8 19)(10 44)(12 46)(14 48)(16 42)(26 37)(28 39)(30 33)(32 35)(50 60)(52 62)(54 64)(56 58)
(1 20)(2 21)(3 22)(4 23)(5 24)(6 17)(7 18)(8 19)(9 43)(10 44)(11 45)(12 46)(13 47)(14 48)(15 41)(16 42)(25 36)(26 37)(27 38)(28 39)(29 40)(30 33)(31 34)(32 35)(49 59)(50 60)(51 61)(52 62)(53 63)(54 64)(55 57)(56 58)
(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)| (1,27,61,45)(2,28,62,46)(3,29,63,47)(4,30,64,48)(5,31,57,41)(6,32,58,42)(7,25,59,43)(8,26,60,44)(9,18,36,49)(10,19,37,50)(11,20,38,51)(12,21,39,52)(13,22,40,53)(14,23,33,54)(15,24,34,55)(16,17,35,56), (2,21)(4,23)(6,17)(8,19)(10,44)(12,46)(14,48)(16,42)(26,37)(28,39)(30,33)(32,35)(50,60)(52,62)(54,64)(56,58), (1,20)(2,21)(3,22)(4,23)(5,24)(6,17)(7,18)(8,19)(9,43)(10,44)(11,45)(12,46)(13,47)(14,48)(15,41)(16,42)(25,36)(26,37)(27,38)(28,39)(29,40)(30,33)(31,34)(32,35)(49,59)(50,60)(51,61)(52,62)(53,63)(54,64)(55,57)(56,58), (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( (1,27,61,45)(2,28,62,46)(3,29,63,47)(4,30,64,48)(5,31,57,41)(6,32,58,42)(7,25,59,43)(8,26,60,44)(9,18,36,49)(10,19,37,50)(11,20,38,51)(12,21,39,52)(13,22,40,53)(14,23,33,54)(15,24,34,55)(16,17,35,56), (2,21)(4,23)(6,17)(8,19)(10,44)(12,46)(14,48)(16,42)(26,37)(28,39)(30,33)(32,35)(50,60)(52,62)(54,64)(56,58), (1,20)(2,21)(3,22)(4,23)(5,24)(6,17)(7,18)(8,19)(9,43)(10,44)(11,45)(12,46)(13,47)(14,48)(15,41)(16,42)(25,36)(26,37)(27,38)(28,39)(29,40)(30,33)(31,34)(32,35)(49,59)(50,60)(51,61)(52,62)(53,63)(54,64)(55,57)(56,58), (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([(1,27,61,45),(2,28,62,46),(3,29,63,47),(4,30,64,48),(5,31,57,41),(6,32,58,42),(7,25,59,43),(8,26,60,44),(9,18,36,49),(10,19,37,50),(11,20,38,51),(12,21,39,52),(13,22,40,53),(14,23,33,54),(15,24,34,55),(16,17,35,56)], [(2,21),(4,23),(6,17),(8,19),(10,44),(12,46),(14,48),(16,42),(26,37),(28,39),(30,33),(32,35),(50,60),(52,62),(54,64),(56,58)], [(1,20),(2,21),(3,22),(4,23),(5,24),(6,17),(7,18),(8,19),(9,43),(10,44),(11,45),(12,46),(13,47),(14,48),(15,41),(16,42),(25,36),(26,37),(27,38),(28,39),(29,40),(30,33),(31,34),(32,35),(49,59),(50,60),(51,61),(52,62),(53,63),(54,64),(55,57),(56,58)], [(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 | 2 |
| type | + | + | + | + | + | + | ||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C8 | D4 | M4(2) | C4○D4 |
| kernel | C4×C22⋊C8 | C22.7C42 | C2×C4×C8 | C2×C22⋊C8 | C22×C42 | C22⋊C8 | C2×C42 | C23×C4 | C22×C4 | C42 | C2×C4 | C2×C4 |
| # reps | 1 | 2 | 2 | 2 | 1 | 16 | 4 | 4 | 32 | 4 | 8 | 4 |
Matrix representation of C4×C22⋊C8 ►in GL4(𝔽17) generated by
| 1 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 |
| 0 | 0 | 13 | 0 |
| 0 | 0 | 0 | 13 |
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 16 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 9 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 13 |
| 0 | 0 | 13 | 0 |
G:=sub<GL(4,GF(17))| [1,0,0,0,0,13,0,0,0,0,13,0,0,0,0,13],[16,0,0,0,0,16,0,0,0,0,1,0,0,0,0,16],[1,0,0,0,0,1,0,0,0,0,16,0,0,0,0,16],[9,0,0,0,0,1,0,0,0,0,0,13,0,0,13,0] >;
C4×C22⋊C8 in GAP, Magma, Sage, TeX
C_4\times C_2^2\rtimes C_8
% in TeX
G:=Group("C4xC2^2:C8"); // GroupNames label
G:=SmallGroup(128,480);
// by ID
G=gap.SmallGroup(128,480);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,112,141,232,100,124]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=c^2=d^8=1,a*b=b*a,a*c=c*a,a*d=d*a,d*b*d^-1=b*c=c*b,c*d=d*c>;
// generators/relations