direct product, p-group, metabelian, nilpotent (class 2), monomial
Aliases: C2×C8○D4, C8.14C23, C4.16C24, M4(2)⋊11C22, D4○(C2×C8), C8○(C2×D4), C8○(C2×Q8), Q8○(C2×C8), C4○(C8○D4), C8○(C8○D4), C8○2(C4○D4), C4○D4.4C4, D4.8(C2×C4), Q8.9(C2×C4), (C2×C8)⋊16C22, (C22×C8)⋊13C2, (C2×D4).12C4, C8○2(C2×M4(2)), (C2×C8)○2M4(2), (C2×Q8).10C4, C2.11(C23×C4), C23.19(C2×C4), C4.22(C22×C4), (C2×M4(2))⋊17C2, (C2×C4).162C23, C4○D4.14C22, C22.4(C22×C4), (C22×C4).128C22, C8○(C2×C4○D4), (C2×C8)○(C2×Q8), (C2×C8)○(C4○D4), (C2×C4).50(C2×C4), (C2×C8)○(C2×M4(2)), (C2×C4○D4).14C2, (C2×C8)○(C2×C4○D4), SmallGroup(64,248)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C2×C8○D4
G = < a,b,c,d | a2=b8=d2=1, c2=b4, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b4c >
Subgroups: 145 in 133 conjugacy classes, 121 normal (10 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, D4, Q8, C23, C2×C8, C2×C8, M4(2), C22×C4, C2×D4, C2×Q8, C4○D4, C22×C8, C2×M4(2), C8○D4, C2×C4○D4, C2×C8○D4
Quotients: C1, C2, C4, C22, C2×C4, C23, C22×C4, C24, C8○D4, C23×C4, C2×C8○D4
►On 32 points
(1 31)(2 32)(3 25)(4 26)(5 27)(6 28)(7 29)(8 30)(9 18)(10 19)(11 20)(12 21)(13 22)(14 23)(15 24)(16 17)
(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)
(1 14 5 10)(2 15 6 11)(3 16 7 12)(4 9 8 13)(17 29 21 25)(18 30 22 26)(19 31 23 27)(20 32 24 28)
(1 10)(2 11)(3 12)(4 13)(5 14)(6 15)(7 16)(8 9)(17 29)(18 30)(19 31)(20 32)(21 25)(22 26)(23 27)(24 28)
G:=sub<Sym(32)| (1,31)(2,32)(3,25)(4,26)(5,27)(6,28)(7,29)(8,30)(9,18)(10,19)(11,20)(12,21)(13,22)(14,23)(15,24)(16,17), (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), (1,14,5,10)(2,15,6,11)(3,16,7,12)(4,9,8,13)(17,29,21,25)(18,30,22,26)(19,31,23,27)(20,32,24,28), (1,10)(2,11)(3,12)(4,13)(5,14)(6,15)(7,16)(8,9)(17,29)(18,30)(19,31)(20,32)(21,25)(22,26)(23,27)(24,28)>;
G:=Group( (1,31)(2,32)(3,25)(4,26)(5,27)(6,28)(7,29)(8,30)(9,18)(10,19)(11,20)(12,21)(13,22)(14,23)(15,24)(16,17), (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), (1,14,5,10)(2,15,6,11)(3,16,7,12)(4,9,8,13)(17,29,21,25)(18,30,22,26)(19,31,23,27)(20,32,24,28), (1,10)(2,11)(3,12)(4,13)(5,14)(6,15)(7,16)(8,9)(17,29)(18,30)(19,31)(20,32)(21,25)(22,26)(23,27)(24,28) );
G=PermutationGroup([[(1,31),(2,32),(3,25),(4,26),(5,27),(6,28),(7,29),(8,30),(9,18),(10,19),(11,20),(12,21),(13,22),(14,23),(15,24),(16,17)], [(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)], [(1,14,5,10),(2,15,6,11),(3,16,7,12),(4,9,8,13),(17,29,21,25),(18,30,22,26),(19,31,23,27),(20,32,24,28)], [(1,10),(2,11),(3,12),(4,13),(5,14),(6,15),(7,16),(8,9),(17,29),(18,30),(19,31),(20,32),(21,25),(22,26),(23,27),(24,28)]])
C2×C8○D4 is a maximal subgroup of
C23.5C42 Q8.C42 D4.3C42 (C2×D4).24Q8 (C2×C8).103D4 C8○D4⋊C4 C4○D4.4Q8 C4○D4.5Q8 M4(2).42D4 M4(2).43D4 M4(2).48D4 M4(2).49D4 (C2×D4).5C8 M5(2).19C22 M5(2)⋊12C22 D4.5C42 D4○(C22⋊C8) 2+ 1+4⋊5C4 2- 1+4⋊4C4 C42.674C23 C4○D4.7Q8 C4○D4.8Q8 M4(2).29C23 C42.264C23 C42.265C23 C42.681C23 C42.266C23 M4(2)⋊22D4 M4(2)⋊23D4 C42.283C23 (C2×C8)⋊11D4 (C2×C8)⋊12D4 C8.D4⋊C2 (C2×C8)⋊13D4 (C2×C8)⋊14D4 M4(2)⋊16D4 M4(2)⋊17D4 M4(2).10C23 Q8○M5(2) C4.22C25 C8.C24 Dic5.21C24
C2×C8○D4 is a maximal quotient of
D4○(C22⋊C8) C42.674C23 C42.260C23 C42.262C23 C42.678C23 D4×C2×C8 C42.264C23 C42.681C23 M4(2)⋊22D4 M4(2)⋊23D4 Q8×C2×C8 C42.286C23 M4(2)⋊9Q8 C8×C4○D4 C42.290C23 C42.291C23 C42.293C23 C42.294C23 D4⋊7M4(2) C42.297C23 C42.298C23 C42.694C23 C42.301C23 Q8.4M4(2) C42.696C23 C42.304C23 D4⋊8M4(2) Q8⋊7M4(2) C42.307C23 C42.308C23 C42.309C23 Dic5.21C24
40 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | ··· | 2I | 4A | 4B | 4C | 4D | 4E | ··· | 4J | 8A | ··· | 8H | 8I | ··· | 8T |
| order | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 |
40 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 |
| type | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C8○D4 |
| kernel | C2×C8○D4 | C22×C8 | C2×M4(2) | C8○D4 | C2×C4○D4 | C2×D4 | C2×Q8 | C4○D4 | C2 |
| # reps | 1 | 3 | 3 | 8 | 1 | 6 | 2 | 8 | 8 |
Matrix representation of C2×C8○D4 ►in GL3(𝔽17) generated by
| 16 | 0 | 0 |
| 0 | 16 | 0 |
| 0 | 0 | 16 |
| 4 | 0 | 0 |
| 0 | 8 | 0 |
| 0 | 0 | 8 |
| 16 | 0 | 0 |
| 0 | 16 | 15 |
| 0 | 1 | 1 |
| 1 | 0 | 0 |
| 0 | 16 | 15 |
| 0 | 0 | 1 |
G:=sub<GL(3,GF(17))| [16,0,0,0,16,0,0,0,16],[4,0,0,0,8,0,0,0,8],[16,0,0,0,16,1,0,15,1],[1,0,0,0,16,0,0,15,1] >;
C2×C8○D4 in GAP, Magma, Sage, TeX
C_2\times C_8\circ D_4
% in TeX
G:=Group("C2xC8oD4"); // GroupNames label
G:=SmallGroup(64,248);
// by ID
G=gap.SmallGroup(64,248);
# by ID
G:=PCGroup([6,-2,2,2,2,-2,-2,96,332,88]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^8=d^2=1,c^2=b^4,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=b^4*c>;
// generators/relations