direct product, p-group, metabelian, nilpotent (class 3), monomial
Aliases: C2×C22⋊D8, C23⋊5D8, C24.173D4, D4⋊6(C2×D4), (C2×D4)⋊47D4, C4⋊C4⋊1C23, (C2×C8)⋊3C23, C22⋊3(C2×D8), (D4×C23)⋊7C2, (C2×D4)⋊1C23, (C22×D8)⋊5C2, C2.4(C22×D8), C4.33C22≀C2, (C2×D8)⋊35C22, (C22×C8)⋊7C22, C4.33(C22×D4), C4⋊D4⋊47C22, C22⋊C8⋊51C22, (C2×C4).215C24, (C22×C4).415D4, C23.846(C2×D4), D4⋊C4⋊61C22, (C22×D4)⋊55C22, C22.112C22≀C2, (C23×C4).535C22, (C22×C4).953C23, C22.475(C22×D4), C22.109(C8⋊C22), C2.7(C2×C8⋊C22), (C2×C4⋊D4)⋊43C2, (C2×C4⋊C4)⋊44C22, (C2×C22⋊C8)⋊20C2, (C2×D4⋊C4)⋊20C2, C2.33(C2×C22≀C2), (C2×C4).1089(C2×D4), SmallGroup(128,1728)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C2×C22⋊D8
G = < a,b,c,d,e | a2=b2=c2=d8=e2=1, ab=ba, ac=ca, ad=da, ae=ea, dbd-1=ebe=bc=cb, cd=dc, ce=ec, ede=d-1 >
Subgroups: 1276 in 524 conjugacy classes, 124 normal (18 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, D4, C23, C23, C23, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, D8, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C24, C24, C22⋊C8, D4⋊C4, C2×C22⋊C4, C2×C4⋊C4, C4⋊D4, C4⋊D4, C22×C8, C2×D8, C2×D8, C23×C4, C22×D4, C22×D4, C22×D4, C25, C2×C22⋊C8, C2×D4⋊C4, C22⋊D8, C2×C4⋊D4, C22×D8, D4×C23, C2×C22⋊D8
Quotients: C1, C2, C22, D4, C23, D8, C2×D4, C24, C22≀C2, C2×D8, C8⋊C22, C22×D4, C22⋊D8, C2×C22≀C2, C22×D8, C2×C8⋊C22, C2×C22⋊D8
►On 32 points
(1 16)(2 9)(3 10)(4 11)(5 12)(6 13)(7 14)(8 15)(17 28)(18 29)(19 30)(20 31)(21 32)(22 25)(23 26)(24 27)
(2 29)(4 31)(6 25)(8 27)(9 18)(11 20)(13 22)(15 24)
(1 28)(2 29)(3 30)(4 31)(5 32)(6 25)(7 26)(8 27)(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 29)(2 28)(3 27)(4 26)(5 25)(6 32)(7 31)(8 30)(9 17)(10 24)(11 23)(12 22)(13 21)(14 20)(15 19)(16 18)
G:=sub<Sym(32)| (1,16)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14)(8,15)(17,28)(18,29)(19,30)(20,31)(21,32)(22,25)(23,26)(24,27), (2,29)(4,31)(6,25)(8,27)(9,18)(11,20)(13,22)(15,24), (1,28)(2,29)(3,30)(4,31)(5,32)(6,25)(7,26)(8,27)(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), (1,29)(2,28)(3,27)(4,26)(5,25)(6,32)(7,31)(8,30)(9,17)(10,24)(11,23)(12,22)(13,21)(14,20)(15,19)(16,18)>;
G:=Group( (1,16)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14)(8,15)(17,28)(18,29)(19,30)(20,31)(21,32)(22,25)(23,26)(24,27), (2,29)(4,31)(6,25)(8,27)(9,18)(11,20)(13,22)(15,24), (1,28)(2,29)(3,30)(4,31)(5,32)(6,25)(7,26)(8,27)(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), (1,29)(2,28)(3,27)(4,26)(5,25)(6,32)(7,31)(8,30)(9,17)(10,24)(11,23)(12,22)(13,21)(14,20)(15,19)(16,18) );
G=PermutationGroup([[(1,16),(2,9),(3,10),(4,11),(5,12),(6,13),(7,14),(8,15),(17,28),(18,29),(19,30),(20,31),(21,32),(22,25),(23,26),(24,27)], [(2,29),(4,31),(6,25),(8,27),(9,18),(11,20),(13,22),(15,24)], [(1,28),(2,29),(3,30),(4,31),(5,32),(6,25),(7,26),(8,27),(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)], [(1,29),(2,28),(3,27),(4,26),(5,25),(6,32),(7,31),(8,30),(9,17),(10,24),(11,23),(12,22),(13,21),(14,20),(15,19),(16,18)]])
38 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 2L | ··· | 2S | 2T | 2U | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 8A | ··· | 8H |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | 8 | 2 | 2 | 2 | 2 | 4 | 4 | 8 | 8 | 4 | ··· | 4 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D8 | C8⋊C22 |
| kernel | C2×C22⋊D8 | C2×C22⋊C8 | C2×D4⋊C4 | C22⋊D8 | C2×C4⋊D4 | C22×D8 | D4×C23 | C22×C4 | C2×D4 | C24 | C23 | C22 |
| # reps | 1 | 1 | 2 | 8 | 1 | 2 | 1 | 3 | 8 | 1 | 8 | 2 |
Matrix representation of C2×C22⋊D8 ►in GL5(𝔽17)
| 16 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 16 |
| 16 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 | 16 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 16 |
| 16 | 0 | 0 | 0 | 0 |
| 0 | 3 | 14 | 0 | 0 |
| 0 | 3 | 3 | 0 | 0 |
| 0 | 0 | 0 | 1 | 15 |
| 0 | 0 | 0 | 1 | 16 |
| 16 | 0 | 0 | 0 | 0 |
| 0 | 3 | 3 | 0 | 0 |
| 0 | 3 | 14 | 0 | 0 |
| 0 | 0 | 0 | 1 | 15 |
| 0 | 0 | 0 | 0 | 16 |
G:=sub<GL(5,GF(17))| [16,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,16],[16,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,1,0,0,0,0,16],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,16],[16,0,0,0,0,0,3,3,0,0,0,14,3,0,0,0,0,0,1,1,0,0,0,15,16],[16,0,0,0,0,0,3,3,0,0,0,3,14,0,0,0,0,0,1,0,0,0,0,15,16] >;
C2×C22⋊D8 in GAP, Magma, Sage, TeX
C_2\times C_2^2\rtimes D_8
% in TeX
G:=Group("C2xC2^2:D8"); // GroupNames label
G:=SmallGroup(128,1728);
// by ID
G=gap.SmallGroup(128,1728);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,758,2804,1411,172]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^8=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,d*b*d^-1=e*b*e=b*c=c*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations