direct product, p-group, metabelian, nilpotent (class 2), monomial
Aliases: C8×C4○D4, C42.690C23, D4○2(C4×C8), C4○2(C8×D4), C8○2(C8×D4), C8○3(C4×D4), C4○2(C8×Q8), C8○2(C8×Q8), C8○3(C4×Q8), Q8○2(C4×C8), D4⋊6(C2×C8), Q8⋊6(C2×C8), C42○(C8×D4), C42○(C8×Q8), (C8×D4)⋊54C2, (C8×Q8)⋊39C2, (C4×D4).43C4, C2.6(C23×C8), (C4×Q8).40C4, C4.63(C8○D4), C4.18(C22×C8), C8○3(C42⋊C2), C4⋊C8.375C22, (C2×C8).480C23, (C4×C8).381C22, C42.248(C2×C4), (C2×C4).661C24, C22.1(C22×C8), C42⋊C2.40C4, (C4×D4).361C22, C22.42(C23×C4), (C4×Q8).331C22, C8○3(C42.12C4), C42.12C4⋊61C2, C22⋊C8.244C22, (C22×C8).515C22, C23.145(C22×C4), (C2×C42).1118C22, (C22×C4).1277C23, C4⋊C4○(C4×C8), (C2×C4×C8)⋊19C2, (C2×D4)○(C4×C8), (C4×C8)○(C8×D4), (C4×C8)○(C8×Q8), (C2×C4)⋊9(C2×C8), (C4×C8)○2(C4×D4), C22⋊C4○(C4×C8), (C4×C8)○2(C4×Q8), (C4×C8)○2(C4⋊C8), C2.4(C4×C4○D4), C2.5(C2×C8○D4), C4⋊C4.249(C2×C4), (C4×C8)○2(C22⋊C8), (C4×C4○D4).33C2, (C2×C4○D4).34C4, C4.312(C2×C4○D4), (C2×D4).250(C2×C4), C22⋊C4.91(C2×C4), (C2×Q8).227(C2×C4), (C4×C8)○2(C42⋊C2), (C22×C4).387(C2×C4), (C2×C4).296(C22×C4), (C4×C8)○(C42.12C4), (C2×C8)○3(C42.12C4), (C4×C8)○(C4×C4○D4), (C2×C8)○2(C4×C4○D4), (C4×C8)○2(C2×C4○D4), SmallGroup(128,1696)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C8×C4○D4
G = < a,b,c,d | a8=b4=d2=1, c2=b2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b2c >
Subgroups: 276 in 228 conjugacy classes, 180 normal (16 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C8, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C42, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C2×D4, C2×Q8, C4○D4, C4×C8, C4×C8, C22⋊C8, C4⋊C8, C2×C42, C42⋊C2, C4×D4, C4×Q8, C22×C8, C2×C4○D4, C2×C4×C8, C42.12C4, C8×D4, C8×Q8, C4×C4○D4, C8×C4○D4
Quotients: C1, C2, C4, C22, C8, C2×C4, C23, C2×C8, C22×C4, C4○D4, C24, C22×C8, C8○D4, C23×C4, C2×C4○D4, C4×C4○D4, C23×C8, C2×C8○D4, C8×C4○D4
►On 64 points
(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 29 11 39)(2 30 12 40)(3 31 13 33)(4 32 14 34)(5 25 15 35)(6 26 16 36)(7 27 9 37)(8 28 10 38)(17 61 44 52)(18 62 45 53)(19 63 46 54)(20 64 47 55)(21 57 48 56)(22 58 41 49)(23 59 42 50)(24 60 43 51)
(1 35 11 25)(2 36 12 26)(3 37 13 27)(4 38 14 28)(5 39 15 29)(6 40 16 30)(7 33 9 31)(8 34 10 32)(17 57 44 56)(18 58 45 49)(19 59 46 50)(20 60 47 51)(21 61 48 52)(22 62 41 53)(23 63 42 54)(24 64 43 55)
(1 19)(2 20)(3 21)(4 22)(5 23)(6 24)(7 17)(8 18)(9 44)(10 45)(11 46)(12 47)(13 48)(14 41)(15 42)(16 43)(25 59)(26 60)(27 61)(28 62)(29 63)(30 64)(31 57)(32 58)(33 56)(34 49)(35 50)(36 51)(37 52)(38 53)(39 54)(40 55)
G:=sub<Sym(64)| (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), (1,29,11,39)(2,30,12,40)(3,31,13,33)(4,32,14,34)(5,25,15,35)(6,26,16,36)(7,27,9,37)(8,28,10,38)(17,61,44,52)(18,62,45,53)(19,63,46,54)(20,64,47,55)(21,57,48,56)(22,58,41,49)(23,59,42,50)(24,60,43,51), (1,35,11,25)(2,36,12,26)(3,37,13,27)(4,38,14,28)(5,39,15,29)(6,40,16,30)(7,33,9,31)(8,34,10,32)(17,57,44,56)(18,58,45,49)(19,59,46,50)(20,60,47,51)(21,61,48,52)(22,62,41,53)(23,63,42,54)(24,64,43,55), (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,17)(8,18)(9,44)(10,45)(11,46)(12,47)(13,48)(14,41)(15,42)(16,43)(25,59)(26,60)(27,61)(28,62)(29,63)(30,64)(31,57)(32,58)(33,56)(34,49)(35,50)(36,51)(37,52)(38,53)(39,54)(40,55)>;
G:=Group( (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), (1,29,11,39)(2,30,12,40)(3,31,13,33)(4,32,14,34)(5,25,15,35)(6,26,16,36)(7,27,9,37)(8,28,10,38)(17,61,44,52)(18,62,45,53)(19,63,46,54)(20,64,47,55)(21,57,48,56)(22,58,41,49)(23,59,42,50)(24,60,43,51), (1,35,11,25)(2,36,12,26)(3,37,13,27)(4,38,14,28)(5,39,15,29)(6,40,16,30)(7,33,9,31)(8,34,10,32)(17,57,44,56)(18,58,45,49)(19,59,46,50)(20,60,47,51)(21,61,48,52)(22,62,41,53)(23,63,42,54)(24,64,43,55), (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,17)(8,18)(9,44)(10,45)(11,46)(12,47)(13,48)(14,41)(15,42)(16,43)(25,59)(26,60)(27,61)(28,62)(29,63)(30,64)(31,57)(32,58)(33,56)(34,49)(35,50)(36,51)(37,52)(38,53)(39,54)(40,55) );
G=PermutationGroup([[(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)], [(1,29,11,39),(2,30,12,40),(3,31,13,33),(4,32,14,34),(5,25,15,35),(6,26,16,36),(7,27,9,37),(8,28,10,38),(17,61,44,52),(18,62,45,53),(19,63,46,54),(20,64,47,55),(21,57,48,56),(22,58,41,49),(23,59,42,50),(24,60,43,51)], [(1,35,11,25),(2,36,12,26),(3,37,13,27),(4,38,14,28),(5,39,15,29),(6,40,16,30),(7,33,9,31),(8,34,10,32),(17,57,44,56),(18,58,45,49),(19,59,46,50),(20,60,47,51),(21,61,48,52),(22,62,41,53),(23,63,42,54),(24,64,43,55)], [(1,19),(2,20),(3,21),(4,22),(5,23),(6,24),(7,17),(8,18),(9,44),(10,45),(11,46),(12,47),(13,48),(14,41),(15,42),(16,43),(25,59),(26,60),(27,61),(28,62),(29,63),(30,64),(31,57),(32,58),(33,56),(34,49),(35,50),(36,51),(37,52),(38,53),(39,54),(40,55)]])
80 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | ··· | 2I | 4A | ··· | 4L | 4M | ··· | 4AD | 8A | ··· | 8P | 8Q | ··· | 8AN |
| 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 | 1 | ··· | 1 | 2 | ··· | 2 |
80 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | C8 | C4○D4 | C8○D4 |
| kernel | C8×C4○D4 | C2×C4×C8 | C42.12C4 | C8×D4 | C8×Q8 | C4×C4○D4 | C42⋊C2 | C4×D4 | C4×Q8 | C2×C4○D4 | C4○D4 | C8 | C4 |
| # reps | 1 | 3 | 3 | 6 | 2 | 1 | 6 | 6 | 2 | 2 | 32 | 8 | 8 |
Matrix representation of C8×C4○D4 ►in GL3(𝔽17) generated by
| 8 | 0 | 0 |
| 0 | 8 | 0 |
| 0 | 0 | 8 |
| 1 | 0 | 0 |
| 0 | 13 | 0 |
| 0 | 0 | 13 |
| 1 | 0 | 0 |
| 0 | 13 | 0 |
| 0 | 0 | 4 |
| 16 | 0 | 0 |
| 0 | 0 | 13 |
| 0 | 4 | 0 |
G:=sub<GL(3,GF(17))| [8,0,0,0,8,0,0,0,8],[1,0,0,0,13,0,0,0,13],[1,0,0,0,13,0,0,0,4],[16,0,0,0,0,4,0,13,0] >;
C8×C4○D4 in GAP, Magma, Sage, TeX
C_8\times C_4\circ D_4
% in TeX
G:=Group("C8xC4oD4"); // GroupNames label
G:=SmallGroup(128,1696);
// by ID
G=gap.SmallGroup(128,1696);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,184,80,124]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^4=d^2=1,c^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=b^2*c>;
// generators/relations