p-group, metabelian, nilpotent (class 3), monomial
Aliases: (C2×C4)⋊9SD16, C4.17(C4×D4), C8⋊5(C22⋊C4), (C2×C8).276D4, (C2×SD16)⋊16C4, C2.7(C8⋊8D4), C2.4(C8⋊5D4), C2.19(C4×SD16), C4.81(C4⋊D4), C4.1(C4.4D4), C22.181(C4×D4), C23.801(C2×D4), (C22×C4).558D4, C2.3(C8.12D4), C22.75(C4○D8), (C22×SD16).8C2, C22.72(C2×SD16), C22.37(C4⋊1D4), (C22×C8).492C22, (C22×D4).48C22, (C22×Q8).38C22, C22.143(C4⋊D4), (C22×C4).1408C23, C23.67C23⋊4C2, (C2×C42).1075C22, C24.3C22.10C2, C2.20(C24.3C22), (C2×C4×C8)⋊33C2, (C2×C4.Q8)⋊19C2, (C2×C8).185(C2×C4), (C2×Q8⋊C4)⋊9C2, C4.36(C2×C22⋊C4), (C2×Q8).94(C2×C4), (C2×D4⋊C4).9C2, (C2×D4).109(C2×C4), (C2×C4).1353(C2×D4), (C2×C4⋊C4).89C22, (C2×C4).599(C4○D4), (C2×C4).422(C22×C4), SmallGroup(128,700)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for (C2×C4)⋊9SD16
G = < a,b,c,d | a2=b4=c8=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=ab-1, dcd=c3 >
Subgroups: 388 in 174 conjugacy classes, 68 normal (34 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C8, 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, C2.C42, C4×C8, D4⋊C4, Q8⋊C4, C4.Q8, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C22×C8, C2×SD16, C2×SD16, C22×D4, C22×Q8, C24.3C22, C23.67C23, C2×C4×C8, C2×D4⋊C4, C2×Q8⋊C4, C2×C4.Q8, C22×SD16, (C2×C4)⋊9SD16
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, SD16, C22×C4, C2×D4, C4○D4, C2×C22⋊C4, C4×D4, C4⋊D4, C4.4D4, C4⋊1D4, C2×SD16, C4○D8, C24.3C22, C4×SD16, C8⋊8D4, C8⋊5D4, C8.12D4, (C2×C4)⋊9SD16
►On 64 points
(1 31)(2 32)(3 25)(4 26)(5 27)(6 28)(7 29)(8 30)(9 34)(10 35)(11 36)(12 37)(13 38)(14 39)(15 40)(16 33)(17 42)(18 43)(19 44)(20 45)(21 46)(22 47)(23 48)(24 41)(49 58)(50 59)(51 60)(52 61)(53 62)(54 63)(55 64)(56 57)
(1 24 33 55)(2 17 34 56)(3 18 35 49)(4 19 36 50)(5 20 37 51)(6 21 38 52)(7 22 39 53)(8 23 40 54)(9 57 32 42)(10 58 25 43)(11 59 26 44)(12 60 27 45)(13 61 28 46)(14 62 29 47)(15 63 30 48)(16 64 31 41)
(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 7)(3 5)(4 8)(10 12)(11 15)(14 16)(17 57)(18 60)(19 63)(20 58)(21 61)(22 64)(23 59)(24 62)(25 27)(26 30)(29 31)(33 39)(35 37)(36 40)(41 53)(42 56)(43 51)(44 54)(45 49)(46 52)(47 55)(48 50)
G:=sub<Sym(64)| (1,31)(2,32)(3,25)(4,26)(5,27)(6,28)(7,29)(8,30)(9,34)(10,35)(11,36)(12,37)(13,38)(14,39)(15,40)(16,33)(17,42)(18,43)(19,44)(20,45)(21,46)(22,47)(23,48)(24,41)(49,58)(50,59)(51,60)(52,61)(53,62)(54,63)(55,64)(56,57), (1,24,33,55)(2,17,34,56)(3,18,35,49)(4,19,36,50)(5,20,37,51)(6,21,38,52)(7,22,39,53)(8,23,40,54)(9,57,32,42)(10,58,25,43)(11,59,26,44)(12,60,27,45)(13,61,28,46)(14,62,29,47)(15,63,30,48)(16,64,31,41), (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), (1,7)(3,5)(4,8)(10,12)(11,15)(14,16)(17,57)(18,60)(19,63)(20,58)(21,61)(22,64)(23,59)(24,62)(25,27)(26,30)(29,31)(33,39)(35,37)(36,40)(41,53)(42,56)(43,51)(44,54)(45,49)(46,52)(47,55)(48,50)>;
G:=Group( (1,31)(2,32)(3,25)(4,26)(5,27)(6,28)(7,29)(8,30)(9,34)(10,35)(11,36)(12,37)(13,38)(14,39)(15,40)(16,33)(17,42)(18,43)(19,44)(20,45)(21,46)(22,47)(23,48)(24,41)(49,58)(50,59)(51,60)(52,61)(53,62)(54,63)(55,64)(56,57), (1,24,33,55)(2,17,34,56)(3,18,35,49)(4,19,36,50)(5,20,37,51)(6,21,38,52)(7,22,39,53)(8,23,40,54)(9,57,32,42)(10,58,25,43)(11,59,26,44)(12,60,27,45)(13,61,28,46)(14,62,29,47)(15,63,30,48)(16,64,31,41), (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), (1,7)(3,5)(4,8)(10,12)(11,15)(14,16)(17,57)(18,60)(19,63)(20,58)(21,61)(22,64)(23,59)(24,62)(25,27)(26,30)(29,31)(33,39)(35,37)(36,40)(41,53)(42,56)(43,51)(44,54)(45,49)(46,52)(47,55)(48,50) );
G=PermutationGroup([[(1,31),(2,32),(3,25),(4,26),(5,27),(6,28),(7,29),(8,30),(9,34),(10,35),(11,36),(12,37),(13,38),(14,39),(15,40),(16,33),(17,42),(18,43),(19,44),(20,45),(21,46),(22,47),(23,48),(24,41),(49,58),(50,59),(51,60),(52,61),(53,62),(54,63),(55,64),(56,57)], [(1,24,33,55),(2,17,34,56),(3,18,35,49),(4,19,36,50),(5,20,37,51),(6,21,38,52),(7,22,39,53),(8,23,40,54),(9,57,32,42),(10,58,25,43),(11,59,26,44),(12,60,27,45),(13,61,28,46),(14,62,29,47),(15,63,30,48),(16,64,31,41)], [(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)], [(1,7),(3,5),(4,8),(10,12),(11,15),(14,16),(17,57),(18,60),(19,63),(20,58),(21,61),(22,64),(23,59),(24,62),(25,27),(26,30),(29,31),(33,39),(35,37),(36,40),(41,53),(42,56),(43,51),(44,54),(45,49),(46,52),(47,55),(48,50)]])
44 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 4A | ··· | 4L | 4M | ··· | 4R | 8A | ··· | 8P |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 8 | 8 | 2 | ··· | 2 | 8 | ··· | 8 | 2 | ··· | 2 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | D4 | SD16 | C4○D4 | C4○D8 |
| kernel | (C2×C4)⋊9SD16 | C24.3C22 | C23.67C23 | C2×C4×C8 | C2×D4⋊C4 | C2×Q8⋊C4 | C2×C4.Q8 | C22×SD16 | C2×SD16 | C2×C8 | C22×C4 | C2×C4 | C2×C4 | C22 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 8 | 6 | 2 | 8 | 4 | 8 |
Matrix representation of (C2×C4)⋊9SD16 ►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 | 16 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 15 |
| 0 | 0 | 0 | 0 | 16 | 13 |
| 0 | 10 | 0 | 0 | 0 | 0 |
| 12 | 10 | 0 | 0 | 0 | 0 |
| 0 | 0 | 10 | 10 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 1 | 15 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 4 | 16 |
G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,16,0,0,0,0,15,13],[0,12,0,0,0,0,10,10,0,0,0,0,0,0,10,12,0,0,0,0,10,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[1,0,0,0,0,0,15,16,0,0,0,0,0,0,16,1,0,0,0,0,0,1,0,0,0,0,0,0,1,4,0,0,0,0,0,16] >;
(C2×C4)⋊9SD16 in GAP, Magma, Sage, TeX
(C_2\times C_4)\rtimes_9{\rm SD}_{16} % in TeX
G:=Group("(C2xC4):9SD16"); // GroupNames label
G:=SmallGroup(128,700);
// by ID
G=gap.SmallGroup(128,700);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,288,422,100,2019,248,2804,172]);
// 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^3>;
// generators/relations