p-group, metabelian, nilpotent (class 3), monomial
Aliases: (C2×C4)⋊6D8, (C2×D8)⋊7C4, C2.12(C4×D8), C4.19(C4×D4), C8⋊4(C22⋊C4), (C2×C8).245D4, C2.5(C8⋊7D4), C2.3(C8⋊4D4), (C22×D8).2C2, C22.47(C2×D8), C4.83(C4⋊D4), C4.3(C4.4D4), C22.183(C4×D4), (C22×C4).560D4, C23.803(C2×D4), C2.5(C8.12D4), C22.77(C4○D8), C22.39(C4⋊1D4), (C22×C8).494C22, C24.3C22⋊4C2, (C22×D4).49C22, C22.145(C4⋊D4), (C22×C4).1410C23, (C2×C42).1077C22, C2.22(C24.3C22), (C2×C4×C8)⋊23C2, (C2×C2.D8)⋊6C2, (C2×D4⋊C4)⋊8C2, (C2×C8).173(C2×C4), C4.38(C2×C22⋊C4), (C2×D4).110(C2×C4), (C2×C4).1355(C2×D4), (C2×C4⋊C4).91C22, (C2×C4).601(C4○D4), (C2×C4).424(C22×C4), SmallGroup(128,702)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for (C2×C4)⋊6D8
G = < a,b,c,d | a2=b4=c8=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=ab-1, dcd=c-1 >
Subgroups: 484 in 192 conjugacy classes, 68 normal (26 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C8, C8, C2×C4, C2×C4, C2×C4, D4, C23, C23, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, D8, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C24, C4×C8, D4⋊C4, C2.D8, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C22×C8, C2×D8, C2×D8, C22×D4, C24.3C22, C2×C4×C8, C2×D4⋊C4, C2×C2.D8, C22×D8, (C2×C4)⋊6D8
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, D8, C22×C4, C2×D4, C4○D4, C2×C22⋊C4, C4×D4, C4⋊D4, C4.4D4, C4⋊1D4, C2×D8, C4○D8, C24.3C22, C4×D8, C8⋊7D4, C8⋊4D4, C8.12D4, (C2×C4)⋊6D8
►On 64 points
(1 27)(2 28)(3 29)(4 30)(5 31)(6 32)(7 25)(8 26)(9 40)(10 33)(11 34)(12 35)(13 36)(14 37)(15 38)(16 39)(17 51)(18 52)(19 53)(20 54)(21 55)(22 56)(23 49)(24 50)(41 59)(42 60)(43 61)(44 62)(45 63)(46 64)(47 57)(48 58)
(1 19 33 46)(2 20 34 47)(3 21 35 48)(4 22 36 41)(5 23 37 42)(6 24 38 43)(7 17 39 44)(8 18 40 45)(9 63 26 52)(10 64 27 53)(11 57 28 54)(12 58 29 55)(13 59 30 56)(14 60 31 49)(15 61 32 50)(16 62 25 51)
(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 2)(3 8)(4 7)(5 6)(9 12)(10 11)(13 16)(14 15)(17 59)(18 58)(19 57)(20 64)(21 63)(22 62)(23 61)(24 60)(25 30)(26 29)(27 28)(31 32)(33 34)(35 40)(36 39)(37 38)(41 51)(42 50)(43 49)(44 56)(45 55)(46 54)(47 53)(48 52)
G:=sub<Sym(64)| (1,27)(2,28)(3,29)(4,30)(5,31)(6,32)(7,25)(8,26)(9,40)(10,33)(11,34)(12,35)(13,36)(14,37)(15,38)(16,39)(17,51)(18,52)(19,53)(20,54)(21,55)(22,56)(23,49)(24,50)(41,59)(42,60)(43,61)(44,62)(45,63)(46,64)(47,57)(48,58), (1,19,33,46)(2,20,34,47)(3,21,35,48)(4,22,36,41)(5,23,37,42)(6,24,38,43)(7,17,39,44)(8,18,40,45)(9,63,26,52)(10,64,27,53)(11,57,28,54)(12,58,29,55)(13,59,30,56)(14,60,31,49)(15,61,32,50)(16,62,25,51), (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), (1,2)(3,8)(4,7)(5,6)(9,12)(10,11)(13,16)(14,15)(17,59)(18,58)(19,57)(20,64)(21,63)(22,62)(23,61)(24,60)(25,30)(26,29)(27,28)(31,32)(33,34)(35,40)(36,39)(37,38)(41,51)(42,50)(43,49)(44,56)(45,55)(46,54)(47,53)(48,52)>;
G:=Group( (1,27)(2,28)(3,29)(4,30)(5,31)(6,32)(7,25)(8,26)(9,40)(10,33)(11,34)(12,35)(13,36)(14,37)(15,38)(16,39)(17,51)(18,52)(19,53)(20,54)(21,55)(22,56)(23,49)(24,50)(41,59)(42,60)(43,61)(44,62)(45,63)(46,64)(47,57)(48,58), (1,19,33,46)(2,20,34,47)(3,21,35,48)(4,22,36,41)(5,23,37,42)(6,24,38,43)(7,17,39,44)(8,18,40,45)(9,63,26,52)(10,64,27,53)(11,57,28,54)(12,58,29,55)(13,59,30,56)(14,60,31,49)(15,61,32,50)(16,62,25,51), (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), (1,2)(3,8)(4,7)(5,6)(9,12)(10,11)(13,16)(14,15)(17,59)(18,58)(19,57)(20,64)(21,63)(22,62)(23,61)(24,60)(25,30)(26,29)(27,28)(31,32)(33,34)(35,40)(36,39)(37,38)(41,51)(42,50)(43,49)(44,56)(45,55)(46,54)(47,53)(48,52) );
G=PermutationGroup([[(1,27),(2,28),(3,29),(4,30),(5,31),(6,32),(7,25),(8,26),(9,40),(10,33),(11,34),(12,35),(13,36),(14,37),(15,38),(16,39),(17,51),(18,52),(19,53),(20,54),(21,55),(22,56),(23,49),(24,50),(41,59),(42,60),(43,61),(44,62),(45,63),(46,64),(47,57),(48,58)], [(1,19,33,46),(2,20,34,47),(3,21,35,48),(4,22,36,41),(5,23,37,42),(6,24,38,43),(7,17,39,44),(8,18,40,45),(9,63,26,52),(10,64,27,53),(11,57,28,54),(12,58,29,55),(13,59,30,56),(14,60,31,49),(15,61,32,50),(16,62,25,51)], [(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)], [(1,2),(3,8),(4,7),(5,6),(9,12),(10,11),(13,16),(14,15),(17,59),(18,58),(19,57),(20,64),(21,63),(22,62),(23,61),(24,60),(25,30),(26,29),(27,28),(31,32),(33,34),(35,40),(36,39),(37,38),(41,51),(42,50),(43,49),(44,56),(45,55),(46,54),(47,53),(48,52)]])
44 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 4A | ··· | 4L | 4M | 4N | 4O | 4P | 8A | ··· | 8P |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 8 | 8 | 8 | 8 | 2 | ··· | 2 | 8 | 8 | 8 | 8 | 2 | ··· | 2 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | D4 | D8 | C4○D4 | C4○D8 |
| kernel | (C2×C4)⋊6D8 | C24.3C22 | C2×C4×C8 | C2×D4⋊C4 | C2×C2.D8 | C22×D8 | C2×D8 | C2×C8 | C22×C4 | C2×C4 | C2×C4 | C22 |
| # reps | 1 | 2 | 1 | 2 | 1 | 1 | 8 | 6 | 2 | 8 | 4 | 8 |
Matrix representation of (C2×C4)⋊6D8 ►in GL5(𝔽17)
| 16 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 16 |
| 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 |
| 0 | 13 | 0 | 0 | 0 |
| 0 | 0 | 0 | 13 | 0 |
| 0 | 0 | 0 | 0 | 13 |
| 16 | 0 | 0 | 0 | 0 |
| 0 | 14 | 3 | 0 | 0 |
| 0 | 14 | 14 | 0 | 0 |
| 0 | 0 | 0 | 14 | 3 |
| 0 | 0 | 0 | 14 | 14 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 14 | 14 | 0 | 0 |
| 0 | 14 | 3 | 0 | 0 |
| 0 | 0 | 0 | 3 | 3 |
| 0 | 0 | 0 | 3 | 14 |
G:=sub<GL(5,GF(17))| [16,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,16],[4,0,0,0,0,0,0,13,0,0,0,4,0,0,0,0,0,0,13,0,0,0,0,0,13],[16,0,0,0,0,0,14,14,0,0,0,3,14,0,0,0,0,0,14,14,0,0,0,3,14],[1,0,0,0,0,0,14,14,0,0,0,14,3,0,0,0,0,0,3,3,0,0,0,3,14] >;
(C2×C4)⋊6D8 in GAP, Magma, Sage, TeX
(C_2\times C_4)\rtimes_6D_8
% in TeX
G:=Group("(C2xC4):6D8"); // GroupNames label
G:=SmallGroup(128,702);
// by ID
G=gap.SmallGroup(128,702);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,736,422,436,2019,248,2804,172]);
// 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,b*c=c*b,d*b*d=a*b^-1,d*c*d=c^-1>;
// generators/relations