p-group, metabelian, nilpotent (class 3), monomial
Aliases: C4.125(C4×D4), (C2×C8).200D4, D4⋊C4⋊11C4, C2.3(C8⋊2D4), C2.5(C8⋊D4), C23.787(C2×D4), C22.170(C4×D4), (C22×C4).131D4, C2.10(D8⋊C4), C22.4Q16⋊49C2, C4.4(C42⋊2C2), C4.10(C42⋊C2), C22.88(C8⋊C22), (C22×C8).398C22, (C2×C42).303C22, (C22×D4).40C22, C2.17(SD16⋊C4), C22.133(C4⋊D4), (C22×C4).1387C23, C23.65C23⋊4C2, C4.95(C22.D4), C22.61(C4.4D4), C22.77(C8.C22), C24.3C22.9C2, C2.2(C42.29C22), C2.3(C42.28C22), C2.21(C24.C22), C4⋊C4.84(C2×C4), (C2×C8⋊C4)⋊24C2, (C2×C8).147(C2×C4), (C2×D4).100(C2×C4), (C2×C4).1344(C2×D4), (C2×C4⋊C4).73C22, (C2×D4⋊C4).34C2, (C2×C4).582(C4○D4), (C2×C4).405(C22×C4), SmallGroup(128,668)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C2.(C8⋊2D4)
G = < a,b,c,d | a2=b8=c4=1, d2=a, ab=ba, ac=ca, ad=da, cbc-1=ab3, dbd-1=ab-1, dcd-1=b6c-1 >
Subgroups: 340 in 142 conjugacy classes, 56 normal (44 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, D4, C23, C23, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C22×C4, C2×D4, C2×D4, C24, C2.C42, C8⋊C4, D4⋊C4, D4⋊C4, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C22×C8, C22×D4, C22.4Q16, C23.65C23, C24.3C22, C2×C8⋊C4, C2×D4⋊C4, C2.(C8⋊2D4)
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22×C4, C2×D4, C4○D4, C42⋊C2, C4×D4, C4⋊D4, C22.D4, C4.4D4, C42⋊2C2, C8⋊C22, C8.C22, C24.C22, SD16⋊C4, D8⋊C4, C8⋊D4, C8⋊2D4, C42.28C22, C42.29C22, C2.(C8⋊2D4)
►On 64 points
(1 31)(2 32)(3 25)(4 26)(5 27)(6 28)(7 29)(8 30)(9 47)(10 48)(11 41)(12 42)(13 43)(14 44)(15 45)(16 46)(17 50)(18 51)(19 52)(20 53)(21 54)(22 55)(23 56)(24 49)(33 57)(34 58)(35 59)(36 60)(37 61)(38 62)(39 63)(40 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 20 39 11)(2 56 40 44)(3 18 33 9)(4 54 34 42)(5 24 35 15)(6 52 36 48)(7 22 37 13)(8 50 38 46)(10 28 19 60)(12 26 21 58)(14 32 23 64)(16 30 17 62)(25 51 57 47)(27 49 59 45)(29 55 61 43)(31 53 63 41)
(1 8 31 30)(2 29 32 7)(3 6 25 28)(4 27 26 5)(9 54 47 21)(10 20 48 53)(11 52 41 19)(12 18 42 51)(13 50 43 17)(14 24 44 49)(15 56 45 23)(16 22 46 55)(33 36 57 60)(34 59 58 35)(37 40 61 64)(38 63 62 39)
G:=sub<Sym(64)| (1,31)(2,32)(3,25)(4,26)(5,27)(6,28)(7,29)(8,30)(9,47)(10,48)(11,41)(12,42)(13,43)(14,44)(15,45)(16,46)(17,50)(18,51)(19,52)(20,53)(21,54)(22,55)(23,56)(24,49)(33,57)(34,58)(35,59)(36,60)(37,61)(38,62)(39,63)(40,64), (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), (1,20,39,11)(2,56,40,44)(3,18,33,9)(4,54,34,42)(5,24,35,15)(6,52,36,48)(7,22,37,13)(8,50,38,46)(10,28,19,60)(12,26,21,58)(14,32,23,64)(16,30,17,62)(25,51,57,47)(27,49,59,45)(29,55,61,43)(31,53,63,41), (1,8,31,30)(2,29,32,7)(3,6,25,28)(4,27,26,5)(9,54,47,21)(10,20,48,53)(11,52,41,19)(12,18,42,51)(13,50,43,17)(14,24,44,49)(15,56,45,23)(16,22,46,55)(33,36,57,60)(34,59,58,35)(37,40,61,64)(38,63,62,39)>;
G:=Group( (1,31)(2,32)(3,25)(4,26)(5,27)(6,28)(7,29)(8,30)(9,47)(10,48)(11,41)(12,42)(13,43)(14,44)(15,45)(16,46)(17,50)(18,51)(19,52)(20,53)(21,54)(22,55)(23,56)(24,49)(33,57)(34,58)(35,59)(36,60)(37,61)(38,62)(39,63)(40,64), (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), (1,20,39,11)(2,56,40,44)(3,18,33,9)(4,54,34,42)(5,24,35,15)(6,52,36,48)(7,22,37,13)(8,50,38,46)(10,28,19,60)(12,26,21,58)(14,32,23,64)(16,30,17,62)(25,51,57,47)(27,49,59,45)(29,55,61,43)(31,53,63,41), (1,8,31,30)(2,29,32,7)(3,6,25,28)(4,27,26,5)(9,54,47,21)(10,20,48,53)(11,52,41,19)(12,18,42,51)(13,50,43,17)(14,24,44,49)(15,56,45,23)(16,22,46,55)(33,36,57,60)(34,59,58,35)(37,40,61,64)(38,63,62,39) );
G=PermutationGroup([[(1,31),(2,32),(3,25),(4,26),(5,27),(6,28),(7,29),(8,30),(9,47),(10,48),(11,41),(12,42),(13,43),(14,44),(15,45),(16,46),(17,50),(18,51),(19,52),(20,53),(21,54),(22,55),(23,56),(24,49),(33,57),(34,58),(35,59),(36,60),(37,61),(38,62),(39,63),(40,64)], [(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)], [(1,20,39,11),(2,56,40,44),(3,18,33,9),(4,54,34,42),(5,24,35,15),(6,52,36,48),(7,22,37,13),(8,50,38,46),(10,28,19,60),(12,26,21,58),(14,32,23,64),(16,30,17,62),(25,51,57,47),(27,49,59,45),(29,55,61,43),(31,53,63,41)], [(1,8,31,30),(2,29,32,7),(3,6,25,28),(4,27,26,5),(9,54,47,21),(10,20,48,53),(11,52,41,19),(12,18,42,51),(13,50,43,17),(14,24,44,49),(15,56,45,23),(16,22,46,55),(33,36,57,60),(34,59,58,35),(37,40,61,64),(38,63,62,39)]])
32 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4N | 8A | ··· | 8H |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 8 | 8 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | D4 | C4○D4 | C8⋊C22 | C8.C22 |
| kernel | C2.(C8⋊2D4) | C22.4Q16 | C23.65C23 | C24.3C22 | C2×C8⋊C4 | C2×D4⋊C4 | D4⋊C4 | C2×C8 | C22×C4 | C2×C4 | C22 | C22 |
| # reps | 1 | 2 | 1 | 1 | 1 | 2 | 8 | 2 | 2 | 8 | 3 | 1 |
Matrix representation of C2.(C8⋊2D4) ►in GL8(𝔽17)
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 16 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 13 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 | 8 | 14 | 0 |
| 0 | 0 | 0 | 0 | 9 | 8 | 0 | 14 |
| 0 | 0 | 0 | 0 | 0 | 14 | 9 | 9 |
| 0 | 0 | 0 | 0 | 3 | 0 | 8 | 9 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 13 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 14 | 0 | 9 | 8 |
| 0 | 0 | 0 | 0 | 0 | 3 | 8 | 8 |
| 0 | 0 | 0 | 0 | 9 | 8 | 0 | 14 |
| 0 | 0 | 0 | 0 | 8 | 8 | 14 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 | 8 | 14 | 0 |
| 0 | 0 | 0 | 0 | 8 | 9 | 0 | 3 |
| 0 | 0 | 0 | 0 | 14 | 0 | 9 | 8 |
| 0 | 0 | 0 | 0 | 0 | 3 | 8 | 8 |
G:=sub<GL(8,GF(17))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,13,4,0,0,0,0,0,0,9,4,0,0,0,0,0,0,0,0,8,9,0,3,0,0,0,0,8,8,14,0,0,0,0,0,14,0,9,8,0,0,0,0,0,14,9,9],[0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,13,4,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,14,0,9,8,0,0,0,0,0,3,8,8,0,0,0,0,9,8,0,14,0,0,0,0,8,8,14,0],[16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,8,13,0,0,0,0,0,0,0,0,8,8,14,0,0,0,0,0,8,9,0,3,0,0,0,0,14,0,9,8,0,0,0,0,0,3,8,8] >;
C2.(C8⋊2D4) in GAP, Magma, Sage, TeX
C_2.(C_8\rtimes_2D_4)
% in TeX
G:=Group("C2.(C8:2D4)"); // GroupNames label
G:=SmallGroup(128,668);
// by ID
G=gap.SmallGroup(128,668);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,736,422,723,58,2019,248,2804,172]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^8=c^4=1,d^2=a,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=a*b^3,d*b*d^-1=a*b^-1,d*c*d^-1=b^6*c^-1>;
// generators/relations