direct product, p-group, metabelian, nilpotent (class 2), monomial
Aliases: C4xC22:C8, C42.457D4, C23.33C42, (C22xC4):6C8, C22:1(C4xC8), C4.155(C4xD4), (C2xC42).32C4, C23.30(C2xC8), (C2xC4).43C42, (C23xC4).27C4, C2.3(C4xM4(2)), C24.109(C2xC4), (C2xC4).90M4(2), (C22xC42).5C2, C22.27(C2xC42), C22.19(C22xC8), (C22xC8).469C22, (C23xC4).628C22, (C2xC42).987C22, C23.250(C22xC4), C22.36(C2xM4(2)), C2.2(C42.12C4), C4o3(C22.7C42), (C22xC4).1603C23, C22.7C42:42C2, C22.48(C42:C2), C42o2(C22.7C42), (C2xC4xC8):5C2, C2.6(C2xC4xC8), (C2xC8):27(C2xC4), (C2xC4).59(C2xC8), C2.2(C4xC22:C4), C2.2(C2xC22:C8), C42o(C2xC22:C8), (C2xC4).1493(C2xD4), (C2xC22:C8).48C2, (C2xC4).913(C4oD4), (C22xC4).433(C2xC4), (C2xC4).593(C22xC4), (C2xC4).395(C22:C4), C22.111(C2xC22:C4), (C2xC4)o2(C22.7C42), (C2xC4)o(C2xC22:C8), (C2xC42)o(C2xC22:C8), SmallGroup(128,480)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C4xC22: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, C2, C2, C4, C4, C22, C22, C22, C8, C2xC4, C2xC4, C23, C23, C23, C42, C42, C2xC8, C2xC8, C22xC4, C22xC4, C22xC4, C24, C4xC8, C22:C8, C2xC42, C2xC42, C2xC42, C22xC8, C23xC4, C23xC4, C22.7C42, C2xC4xC8, C2xC22:C8, C22xC42, C4xC22:C8
Quotients: C1, C2, C4, C22, C8, C2xC4, D4, C23, C42, C22:C4, C2xC8, M4(2), C22xC4, C2xD4, C4oD4, C4xC8, C22:C8, C2xC42, C2xC22:C4, C42:C2, C4xD4, C22xC8, C2xM4(2), C4xC22:C4, C2xC4xC8, C4xM4(2), C2xC22:C8, C42.12C4, C4xC22:C8
►On 64 points
(1 29 61 45)(2 30 62 46)(3 31 63 47)(4 32 64 48)(5 25 57 41)(6 26 58 42)(7 27 59 43)(8 28 60 44)(9 18 38 49)(10 19 39 50)(11 20 40 51)(12 21 33 52)(13 22 34 53)(14 23 35 54)(15 24 36 55)(16 17 37 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)| (1,29,61,45)(2,30,62,46)(3,31,63,47)(4,32,64,48)(5,25,57,41)(6,26,58,42)(7,27,59,43)(8,28,60,44)(9,18,38,49)(10,19,39,50)(11,20,40,51)(12,21,33,52)(13,22,34,53)(14,23,35,54)(15,24,36,55)(16,17,37,56), (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( (1,29,61,45)(2,30,62,46)(3,31,63,47)(4,32,64,48)(5,25,57,41)(6,26,58,42)(7,27,59,43)(8,28,60,44)(9,18,38,49)(10,19,39,50)(11,20,40,51)(12,21,33,52)(13,22,34,53)(14,23,35,54)(15,24,36,55)(16,17,37,56), (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([[(1,29,61,45),(2,30,62,46),(3,31,63,47),(4,32,64,48),(5,25,57,41),(6,26,58,42),(7,27,59,43),(8,28,60,44),(9,18,38,49),(10,19,39,50),(11,20,40,51),(12,21,33,52),(13,22,34,53),(14,23,35,54),(15,24,36,55),(16,17,37,56)], [(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) | C4oD4 |
| kernel | C4xC22:C8 | C22.7C42 | C2xC4xC8 | C2xC22:C8 | C22xC42 | C22:C8 | C2xC42 | C23xC4 | C22xC4 | C42 | C2xC4 | C2xC4 |
| # reps | 1 | 2 | 2 | 2 | 1 | 16 | 4 | 4 | 32 | 4 | 8 | 4 |
Matrix representation of C4xC22: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] >;
C4xC22: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