p-group, metabelian, nilpotent (class 3), monomial
Aliases: (C2×C4)⋊9D8, (C2×D8)⋊6C4, C2.7(C4×D8), C4.83(C4×D4), C4⋊C4.312D4, D4⋊1(C22⋊C4), (C2×D4).205D4, C2.2(C4⋊D8), C4.3(C4⋊D4), (C22×D8).1C2, C22.39(C2×D8), C2.7(D8⋊C4), C2.5(C22⋊D8), C2.6(D4⋊D4), C23.770(C2×D4), (C22×C4).689D4, C22.153(C4×D4), C22.4Q16⋊35C2, C22.95C22≀C2, C2.3(D4.2D4), C22.57(C4○D8), (C22×C8).37C22, C22.77(C8⋊C22), (C2×C42).279C22, C24.3C22⋊2C2, C22.116(C4⋊D4), (C22×C4).1366C23, C22.7C42⋊16C2, C4.65(C22.D4), (C22×D4).461C22, C2.18(C23.23D4), (C2×C4×D4)⋊1C2, (C2×C8)⋊6(C2×C4), (C2×D4)⋊6(C2×C4), (C2×D4⋊C4)⋊5C2, (C2×C4).998(C2×D4), C4.13(C2×C22⋊C4), (C2×C4).563(C4○D4), (C2×C4⋊C4).767C22, (C2×C4).384(C22×C4), SmallGroup(128,611)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for (C2×C4)⋊9D8
G = < a,b,c,d | a2=b4=c8=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=ab-1, dcd=c-1 >
Subgroups: 532 in 219 conjugacy classes, 66 normal (44 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, D4, C23, C23, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, D8, C22×C4, C22×C4, C2×D4, C2×D4, C24, D4⋊C4, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C4×D4, C22×C8, C2×D8, C2×D8, C23×C4, C22×D4, C22.7C42, C22.4Q16, C24.3C22, C2×D4⋊C4, C2×C4×D4, C22×D8, (C2×C4)⋊9D8
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, D8, C22×C4, C2×D4, C4○D4, C2×C22⋊C4, C4×D4, C22≀C2, C4⋊D4, C22.D4, C2×D8, C4○D8, C8⋊C22, C23.23D4, C4×D8, D8⋊C4, C22⋊D8, D4⋊D4, C4⋊D8, D4.2D4, (C2×C4)⋊9D8
►On 64 points
(1 27)(2 28)(3 29)(4 30)(5 31)(6 32)(7 25)(8 26)(9 44)(10 45)(11 46)(12 47)(13 48)(14 41)(15 42)(16 43)(17 56)(18 49)(19 50)(20 51)(21 52)(22 53)(23 54)(24 55)(33 61)(34 62)(35 63)(36 64)(37 57)(38 58)(39 59)(40 60)
(1 24 44 40)(2 61 45 56)(3 18 46 34)(4 63 47 50)(5 20 48 36)(6 57 41 52)(7 22 42 38)(8 59 43 54)(9 60 27 55)(10 17 28 33)(11 62 29 49)(12 19 30 35)(13 64 31 51)(14 21 32 37)(15 58 25 53)(16 23 26 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 5)(2 4)(6 8)(9 13)(10 12)(14 16)(17 63)(18 62)(19 61)(20 60)(21 59)(22 58)(23 57)(24 64)(26 32)(27 31)(28 30)(33 50)(34 49)(35 56)(36 55)(37 54)(38 53)(39 52)(40 51)(41 43)(44 48)(45 47)
G:=sub<Sym(64)| (1,27)(2,28)(3,29)(4,30)(5,31)(6,32)(7,25)(8,26)(9,44)(10,45)(11,46)(12,47)(13,48)(14,41)(15,42)(16,43)(17,56)(18,49)(19,50)(20,51)(21,52)(22,53)(23,54)(24,55)(33,61)(34,62)(35,63)(36,64)(37,57)(38,58)(39,59)(40,60), (1,24,44,40)(2,61,45,56)(3,18,46,34)(4,63,47,50)(5,20,48,36)(6,57,41,52)(7,22,42,38)(8,59,43,54)(9,60,27,55)(10,17,28,33)(11,62,29,49)(12,19,30,35)(13,64,31,51)(14,21,32,37)(15,58,25,53)(16,23,26,39), (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), (1,5)(2,4)(6,8)(9,13)(10,12)(14,16)(17,63)(18,62)(19,61)(20,60)(21,59)(22,58)(23,57)(24,64)(26,32)(27,31)(28,30)(33,50)(34,49)(35,56)(36,55)(37,54)(38,53)(39,52)(40,51)(41,43)(44,48)(45,47)>;
G:=Group( (1,27)(2,28)(3,29)(4,30)(5,31)(6,32)(7,25)(8,26)(9,44)(10,45)(11,46)(12,47)(13,48)(14,41)(15,42)(16,43)(17,56)(18,49)(19,50)(20,51)(21,52)(22,53)(23,54)(24,55)(33,61)(34,62)(35,63)(36,64)(37,57)(38,58)(39,59)(40,60), (1,24,44,40)(2,61,45,56)(3,18,46,34)(4,63,47,50)(5,20,48,36)(6,57,41,52)(7,22,42,38)(8,59,43,54)(9,60,27,55)(10,17,28,33)(11,62,29,49)(12,19,30,35)(13,64,31,51)(14,21,32,37)(15,58,25,53)(16,23,26,39), (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), (1,5)(2,4)(6,8)(9,13)(10,12)(14,16)(17,63)(18,62)(19,61)(20,60)(21,59)(22,58)(23,57)(24,64)(26,32)(27,31)(28,30)(33,50)(34,49)(35,56)(36,55)(37,54)(38,53)(39,52)(40,51)(41,43)(44,48)(45,47) );
G=PermutationGroup([[(1,27),(2,28),(3,29),(4,30),(5,31),(6,32),(7,25),(8,26),(9,44),(10,45),(11,46),(12,47),(13,48),(14,41),(15,42),(16,43),(17,56),(18,49),(19,50),(20,51),(21,52),(22,53),(23,54),(24,55),(33,61),(34,62),(35,63),(36,64),(37,57),(38,58),(39,59),(40,60)], [(1,24,44,40),(2,61,45,56),(3,18,46,34),(4,63,47,50),(5,20,48,36),(6,57,41,52),(7,22,42,38),(8,59,43,54),(9,60,27,55),(10,17,28,33),(11,62,29,49),(12,19,30,35),(13,64,31,51),(14,21,32,37),(15,58,25,53),(16,23,26,39)], [(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)], [(1,5),(2,4),(6,8),(9,13),(10,12),(14,16),(17,63),(18,62),(19,61),(20,60),(21,59),(22,58),(23,57),(24,64),(26,32),(27,31),(28,30),(33,50),(34,49),(35,56),(36,55),(37,54),(38,53),(39,52),(40,51),(41,43),(44,48),(45,47)]])
38 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 2L | 2M | 4A | ··· | 4H | 4I | ··· | 4N | 4O | 4P | 8A | ··· | 8H |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | 4 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 4 | 4 | 4 | 4 | 8 | 8 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 | 4 | ··· | 4 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | D4 | D4 | D8 | C4○D4 | C4○D8 | C8⋊C22 |
| kernel | (C2×C4)⋊9D8 | C22.7C42 | C22.4Q16 | C24.3C22 | C2×D4⋊C4 | C2×C4×D4 | C22×D8 | C2×D8 | C4⋊C4 | C22×C4 | C2×D4 | C2×C4 | C2×C4 | C22 | C22 |
| # reps | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 8 | 2 | 2 | 4 | 4 | 4 | 4 | 2 |
Matrix representation of (C2×C4)⋊9D8 ►in GL6(𝔽17)
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 13 | 0 | 0 |
| 0 | 0 | 13 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 3 |
| 0 | 0 | 0 | 0 | 14 | 3 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,16,0,0,0,0,16,0,0,0,0,0,0,0,0,13,0,0,0,0,13,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,16,0,0,0,0,0,0,0,3,14,0,0,0,0,3,3],[1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1] >;
(C2×C4)⋊9D8 in GAP, Magma, Sage, TeX
(C_2\times C_4)\rtimes_9D_8
% in TeX
G:=Group("(C2xC4):9D8"); // GroupNames label
G:=SmallGroup(128,611);
// by ID
G=gap.SmallGroup(128,611);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,736,422,1018,521,248,1411,718,172,2028,1027,124]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^4=c^8=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=a*b^-1,d*c*d=c^-1>;
// generators/relations