direct product, p-group, metabelian, nilpotent (class 3), monomial
Aliases: C4×C8.C4, C8.10C42, C42.460D4, (C4×C8).32C4, C4.173(C4×D4), C22.3(C4×Q8), C4.26(C2×C42), (C22×C4).76Q8, C23.77(C2×Q8), C42.318(C2×C4), C4○2(C4.C42), M4(2).21(C2×C4), (C4×M4(2)).21C2, C4.50(C42⋊C2), C4.C42.12C2, C42○(C4.C42), (C22×C8).548C22, (C2×C42).1053C22, (C22×C4).1318C23, (C2×M4(2)).312C22, (C2×C4×C8).43C2, C2.16(C4×C4⋊C4), (C2×C8).207(C2×C4), C2.5(C2×C8.C4), C22.61(C2×C4⋊C4), C42○(C2×C8.C4), (C2×C4).166(C4⋊C4), (C2×C4).1509(C2×D4), (C2×C8.C4).27C2, (C2×C4).549(C4○D4), (C2×C4).525(C22×C4), SmallGroup(128,509)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C4×C8.C4
G = < a,b,c | a4=b8=1, c4=b4, ab=ba, ac=ca, cbc-1=b-1 >
Subgroups: 148 in 114 conjugacy classes, 80 normal (18 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C8, C8, C2×C4, C2×C4, C2×C4, C23, C42, C42, C2×C8, C2×C8, C2×C8, M4(2), M4(2), C22×C4, C22×C4, C4×C8, C4×C8, C8⋊C4, C8.C4, C2×C42, C22×C8, C2×M4(2), C4.C42, C2×C4×C8, C4×M4(2), C2×C8.C4, C4×C8.C4
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C42, C4⋊C4, C22×C4, C2×D4, C2×Q8, C4○D4, C8.C4, C2×C42, C2×C4⋊C4, C42⋊C2, C4×D4, C4×Q8, C4×C4⋊C4, C2×C8.C4, C4×C8.C4
►On 64 points
(1 14 21 49)(2 15 22 50)(3 16 23 51)(4 9 24 52)(5 10 17 53)(6 11 18 54)(7 12 19 55)(8 13 20 56)(25 44 64 40)(26 45 57 33)(27 46 58 34)(28 47 59 35)(29 48 60 36)(30 41 61 37)(31 42 62 38)(32 43 63 39)
(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 32 19 57 5 28 23 61)(2 31 20 64 6 27 24 60)(3 30 21 63 7 26 17 59)(4 29 22 62 8 25 18 58)(9 48 50 38 13 44 54 34)(10 47 51 37 14 43 55 33)(11 46 52 36 15 42 56 40)(12 45 53 35 16 41 49 39)
G:=sub<Sym(64)| (1,14,21,49)(2,15,22,50)(3,16,23,51)(4,9,24,52)(5,10,17,53)(6,11,18,54)(7,12,19,55)(8,13,20,56)(25,44,64,40)(26,45,57,33)(27,46,58,34)(28,47,59,35)(29,48,60,36)(30,41,61,37)(31,42,62,38)(32,43,63,39), (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), (1,32,19,57,5,28,23,61)(2,31,20,64,6,27,24,60)(3,30,21,63,7,26,17,59)(4,29,22,62,8,25,18,58)(9,48,50,38,13,44,54,34)(10,47,51,37,14,43,55,33)(11,46,52,36,15,42,56,40)(12,45,53,35,16,41,49,39)>;
G:=Group( (1,14,21,49)(2,15,22,50)(3,16,23,51)(4,9,24,52)(5,10,17,53)(6,11,18,54)(7,12,19,55)(8,13,20,56)(25,44,64,40)(26,45,57,33)(27,46,58,34)(28,47,59,35)(29,48,60,36)(30,41,61,37)(31,42,62,38)(32,43,63,39), (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), (1,32,19,57,5,28,23,61)(2,31,20,64,6,27,24,60)(3,30,21,63,7,26,17,59)(4,29,22,62,8,25,18,58)(9,48,50,38,13,44,54,34)(10,47,51,37,14,43,55,33)(11,46,52,36,15,42,56,40)(12,45,53,35,16,41,49,39) );
G=PermutationGroup([[(1,14,21,49),(2,15,22,50),(3,16,23,51),(4,9,24,52),(5,10,17,53),(6,11,18,54),(7,12,19,55),(8,13,20,56),(25,44,64,40),(26,45,57,33),(27,46,58,34),(28,47,59,35),(29,48,60,36),(30,41,61,37),(31,42,62,38),(32,43,63,39)], [(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)], [(1,32,19,57,5,28,23,61),(2,31,20,64,6,27,24,60),(3,30,21,63,7,26,17,59),(4,29,22,62,8,25,18,58),(9,48,50,38,13,44,54,34),(10,47,51,37,14,43,55,33),(11,46,52,36,15,42,56,40),(12,45,53,35,16,41,49,39)]])
56 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | ··· | 4L | 4M | ··· | 4R | 8A | ··· | 8P | 8Q | ··· | 8AF |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | D4 | Q8 | C4○D4 | C8.C4 |
| kernel | C4×C8.C4 | C4.C42 | C2×C4×C8 | C4×M4(2) | C2×C8.C4 | C4×C8 | C8.C4 | C42 | C22×C4 | C2×C4 | C4 |
| # reps | 1 | 2 | 1 | 2 | 2 | 8 | 16 | 2 | 2 | 4 | 16 |
Matrix representation of C4×C8.C4 ►in GL3(𝔽17) generated by
| 13 | 0 | 0 |
| 0 | 16 | 0 |
| 0 | 0 | 16 |
| 16 | 0 | 0 |
| 0 | 9 | 0 |
| 0 | 0 | 2 |
| 1 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 4 | 0 |
G:=sub<GL(3,GF(17))| [13,0,0,0,16,0,0,0,16],[16,0,0,0,9,0,0,0,2],[1,0,0,0,0,4,0,1,0] >;
C4×C8.C4 in GAP, Magma, Sage, TeX
C_4\times C_8.C_4
% in TeX
G:=Group("C4xC8.C4"); // GroupNames label
G:=SmallGroup(128,509);
// by ID
G=gap.SmallGroup(128,509);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,112,141,232,142,1018,248,1411,172,124]);
// Polycyclic
G:=Group<a,b,c|a^4=b^8=1,c^4=b^4,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations