p-group, metabelian, nilpotent (class 3), monomial
Aliases: (C2×C4)⋊3SD16, C4.9C22≀C2, (C2×D4).92D4, (C2×C8).154D4, (C2×Q8).83D4, C2.12(C8⋊5D4), C4.45(C4⋊D4), C23.901(C2×D4), (C22×C4).308D4, C2.19(Q8⋊D4), C2.13(C8.2D4), (C22×SD16).9C2, C22.91(C2×SD16), C2.12(D4.D4), C22.203C22≀C2, C2.18(C23⋊2D4), C22.74(C4⋊1D4), (C2×C42).349C22, (C22×C8).318C22, (C22×D4).65C22, (C22×Q8).54C22, C22.222(C4⋊D4), (C22×C4).1435C23, C22.7C42⋊30C2, C22.118(C8.C22), C24.3C22.12C2, (C2×C4⋊Q8)⋊2C2, (C2×C4).745(C2×D4), (C2×Q8⋊C4)⋊33C2, (C2×C4).874(C4○D4), (C2×C4⋊C4).108C22, SmallGroup(128,745)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for (C2×C4)⋊3SD16
G = < a,b,c,d | a2=b4=c8=d2=1, dbd=ab=ba, ac=ca, ad=da, cbc-1=ab-1, dcd=c3 >
Subgroups: 456 in 203 conjugacy classes, 58 normal (18 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, SD16, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C24, Q8⋊C4, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C4⋊Q8, C22×C8, C2×SD16, C22×D4, C22×Q8, C22.7C42, C24.3C22, C2×Q8⋊C4, C2×C4⋊Q8, C22×SD16, (C2×C4)⋊3SD16
Quotients: C1, C2, C22, D4, C23, SD16, C2×D4, C4○D4, C22≀C2, C4⋊D4, C4⋊1D4, C2×SD16, C8.C22, C23⋊2D4, Q8⋊D4, D4.D4, C8⋊5D4, C8.2D4, (C2×C4)⋊3SD16
►On 64 points
(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 9)(8 10)(17 57)(18 58)(19 59)(20 60)(21 61)(22 62)(23 63)(24 64)(25 44)(26 45)(27 46)(28 47)(29 48)(30 41)(31 42)(32 43)(33 54)(34 55)(35 56)(36 49)(37 50)(38 51)(39 52)(40 53)
(1 60 27 40)(2 54 28 21)(3 62 29 34)(4 56 30 23)(5 64 31 36)(6 50 32 17)(7 58 25 38)(8 52 26 19)(9 18 44 51)(10 39 45 59)(11 20 46 53)(12 33 47 61)(13 22 48 55)(14 35 41 63)(15 24 42 49)(16 37 43 57)
(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)
(2 4)(3 7)(6 8)(9 13)(10 16)(12 14)(17 59)(18 62)(19 57)(20 60)(21 63)(22 58)(23 61)(24 64)(25 29)(26 32)(28 30)(33 56)(34 51)(35 54)(36 49)(37 52)(38 55)(39 50)(40 53)(41 47)(43 45)(44 48)
G:=sub<Sym(64)| (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,9)(8,10)(17,57)(18,58)(19,59)(20,60)(21,61)(22,62)(23,63)(24,64)(25,44)(26,45)(27,46)(28,47)(29,48)(30,41)(31,42)(32,43)(33,54)(34,55)(35,56)(36,49)(37,50)(38,51)(39,52)(40,53), (1,60,27,40)(2,54,28,21)(3,62,29,34)(4,56,30,23)(5,64,31,36)(6,50,32,17)(7,58,25,38)(8,52,26,19)(9,18,44,51)(10,39,45,59)(11,20,46,53)(12,33,47,61)(13,22,48,55)(14,35,41,63)(15,24,42,49)(16,37,43,57), (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), (2,4)(3,7)(6,8)(9,13)(10,16)(12,14)(17,59)(18,62)(19,57)(20,60)(21,63)(22,58)(23,61)(24,64)(25,29)(26,32)(28,30)(33,56)(34,51)(35,54)(36,49)(37,52)(38,55)(39,50)(40,53)(41,47)(43,45)(44,48)>;
G:=Group( (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,9)(8,10)(17,57)(18,58)(19,59)(20,60)(21,61)(22,62)(23,63)(24,64)(25,44)(26,45)(27,46)(28,47)(29,48)(30,41)(31,42)(32,43)(33,54)(34,55)(35,56)(36,49)(37,50)(38,51)(39,52)(40,53), (1,60,27,40)(2,54,28,21)(3,62,29,34)(4,56,30,23)(5,64,31,36)(6,50,32,17)(7,58,25,38)(8,52,26,19)(9,18,44,51)(10,39,45,59)(11,20,46,53)(12,33,47,61)(13,22,48,55)(14,35,41,63)(15,24,42,49)(16,37,43,57), (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), (2,4)(3,7)(6,8)(9,13)(10,16)(12,14)(17,59)(18,62)(19,57)(20,60)(21,63)(22,58)(23,61)(24,64)(25,29)(26,32)(28,30)(33,56)(34,51)(35,54)(36,49)(37,52)(38,55)(39,50)(40,53)(41,47)(43,45)(44,48) );
G=PermutationGroup([[(1,11),(2,12),(3,13),(4,14),(5,15),(6,16),(7,9),(8,10),(17,57),(18,58),(19,59),(20,60),(21,61),(22,62),(23,63),(24,64),(25,44),(26,45),(27,46),(28,47),(29,48),(30,41),(31,42),(32,43),(33,54),(34,55),(35,56),(36,49),(37,50),(38,51),(39,52),(40,53)], [(1,60,27,40),(2,54,28,21),(3,62,29,34),(4,56,30,23),(5,64,31,36),(6,50,32,17),(7,58,25,38),(8,52,26,19),(9,18,44,51),(10,39,45,59),(11,20,46,53),(12,33,47,61),(13,22,48,55),(14,35,41,63),(15,24,42,49),(16,37,43,57)], [(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)], [(2,4),(3,7),(6,8),(9,13),(10,16),(12,14),(17,59),(18,62),(19,57),(20,60),(21,63),(22,58),(23,61),(24,64),(25,29),(26,32),(28,30),(33,56),(34,51),(35,54),(36,49),(37,52),(38,55),(39,50),(40,53),(41,47),(43,45),(44,48)]])
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 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D4 | SD16 | C4○D4 | C8.C22 |
| kernel | (C2×C4)⋊3SD16 | C22.7C42 | C24.3C22 | C2×Q8⋊C4 | C2×C4⋊Q8 | C22×SD16 | C2×C8 | C22×C4 | C2×D4 | C2×Q8 | C2×C4 | C2×C4 | C22 |
| # reps | 1 | 1 | 1 | 2 | 1 | 2 | 4 | 2 | 2 | 4 | 8 | 2 | 2 |
Matrix representation of (C2×C4)⋊3SD16 ►in GL6(𝔽17)
| 1 | 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 | 16 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 5 | 0 | 0 |
| 0 | 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[4,0,0,0,0,0,0,13,0,0,0,0,0,0,0,1,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,16,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,12,12,0,0,0,0,5,12,0,0,0,0,0,0,0,16,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,16] >;
(C2×C4)⋊3SD16 in GAP, Magma, Sage, TeX
(C_2\times C_4)\rtimes_3{\rm SD}_{16} % in TeX
G:=Group("(C2xC4):3SD16"); // GroupNames label
G:=SmallGroup(128,745);
// by ID
G=gap.SmallGroup(128,745);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,141,456,422,387,352,2019,1018,521,248,2804,1411,718,172]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^4=c^8=d^2=1,d*b*d=a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=a*b^-1,d*c*d=c^3>;
// generators/relations