direct product, p-group, metabelian, nilpotent (class 4), monomial
Aliases: C2×C8.17D4, C23.37SD16, M5(2).20C22, C8.97(C2×D4), C4.65(C2×D8), (C2×C4).143D8, (C2×C8).124D4, C8.9(C22×C4), (C2×Q16).14C4, Q16.12(C2×C4), (C2×C4).54SD16, C8.11(C22⋊C4), (C2×C8).227C23, (C22×C4).337D4, C4.27(D4⋊C4), (C2×M5(2)).22C2, (C22×Q16).14C2, C22.17(C2×SD16), C8.C4.10C22, (C22×C8).236C22, (C2×Q16).148C22, C22.34(D4⋊C4), (C2×C8).86(C2×C4), (C2×C4).272(C2×D4), C4.59(C2×C22⋊C4), C2.37(C2×D4⋊C4), (C2×C8.C4).21C2, (C2×C4).275(C22⋊C4), SmallGroup(128,879)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C2×C8.17D4
G = < a,b,c,d | a2=b8=1, c4=b4, d2=cbc-1=b-1, ab=ba, ac=ca, ad=da, bd=db, dcd-1=b-1c3 >
Subgroups: 212 in 108 conjugacy classes, 52 normal (32 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, Q8, C23, C16, C2×C8, C2×C8, M4(2), Q16, Q16, C22×C4, C22×C4, C2×Q8, C8.C4, C8.C4, C2×C16, M5(2), M5(2), C22×C8, C2×M4(2), C2×Q16, C2×Q16, C22×Q8, C8.17D4, C2×C8.C4, C2×M5(2), C22×Q16, C2×C8.17D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, D8, SD16, C22×C4, C2×D4, D4⋊C4, C2×C22⋊C4, C2×D8, C2×SD16, C8.17D4, C2×D4⋊C4, C2×C8.17D4
►On 64 points
(1 20)(2 21)(3 22)(4 23)(5 24)(6 25)(7 26)(8 27)(9 28)(10 29)(11 30)(12 31)(13 32)(14 17)(15 18)(16 19)(33 55)(34 56)(35 57)(36 58)(37 59)(38 60)(39 61)(40 62)(41 63)(42 64)(43 49)(44 50)(45 51)(46 52)(47 53)(48 54)
(1 15 13 11 9 7 5 3)(2 16 14 12 10 8 6 4)(17 31 29 27 25 23 21 19)(18 32 30 28 26 24 22 20)(33 47 45 43 41 39 37 35)(34 48 46 44 42 40 38 36)(49 63 61 59 57 55 53 51)(50 64 62 60 58 56 54 52)
(1 42 13 46 9 34 5 38)(2 45 6 41 10 37 14 33)(3 40 15 44 11 48 7 36)(4 43 8 39 12 35 16 47)(17 55 21 51 25 63 29 59)(18 50 30 54 26 58 22 62)(19 53 23 49 27 61 31 57)(20 64 32 52 28 56 24 60)
(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)
G:=sub<Sym(64)| (1,20)(2,21)(3,22)(4,23)(5,24)(6,25)(7,26)(8,27)(9,28)(10,29)(11,30)(12,31)(13,32)(14,17)(15,18)(16,19)(33,55)(34,56)(35,57)(36,58)(37,59)(38,60)(39,61)(40,62)(41,63)(42,64)(43,49)(44,50)(45,51)(46,52)(47,53)(48,54), (1,15,13,11,9,7,5,3)(2,16,14,12,10,8,6,4)(17,31,29,27,25,23,21,19)(18,32,30,28,26,24,22,20)(33,47,45,43,41,39,37,35)(34,48,46,44,42,40,38,36)(49,63,61,59,57,55,53,51)(50,64,62,60,58,56,54,52), (1,42,13,46,9,34,5,38)(2,45,6,41,10,37,14,33)(3,40,15,44,11,48,7,36)(4,43,8,39,12,35,16,47)(17,55,21,51,25,63,29,59)(18,50,30,54,26,58,22,62)(19,53,23,49,27,61,31,57)(20,64,32,52,28,56,24,60), (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)>;
G:=Group( (1,20)(2,21)(3,22)(4,23)(5,24)(6,25)(7,26)(8,27)(9,28)(10,29)(11,30)(12,31)(13,32)(14,17)(15,18)(16,19)(33,55)(34,56)(35,57)(36,58)(37,59)(38,60)(39,61)(40,62)(41,63)(42,64)(43,49)(44,50)(45,51)(46,52)(47,53)(48,54), (1,15,13,11,9,7,5,3)(2,16,14,12,10,8,6,4)(17,31,29,27,25,23,21,19)(18,32,30,28,26,24,22,20)(33,47,45,43,41,39,37,35)(34,48,46,44,42,40,38,36)(49,63,61,59,57,55,53,51)(50,64,62,60,58,56,54,52), (1,42,13,46,9,34,5,38)(2,45,6,41,10,37,14,33)(3,40,15,44,11,48,7,36)(4,43,8,39,12,35,16,47)(17,55,21,51,25,63,29,59)(18,50,30,54,26,58,22,62)(19,53,23,49,27,61,31,57)(20,64,32,52,28,56,24,60), (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) );
G=PermutationGroup([[(1,20),(2,21),(3,22),(4,23),(5,24),(6,25),(7,26),(8,27),(9,28),(10,29),(11,30),(12,31),(13,32),(14,17),(15,18),(16,19),(33,55),(34,56),(35,57),(36,58),(37,59),(38,60),(39,61),(40,62),(41,63),(42,64),(43,49),(44,50),(45,51),(46,52),(47,53),(48,54)], [(1,15,13,11,9,7,5,3),(2,16,14,12,10,8,6,4),(17,31,29,27,25,23,21,19),(18,32,30,28,26,24,22,20),(33,47,45,43,41,39,37,35),(34,48,46,44,42,40,38,36),(49,63,61,59,57,55,53,51),(50,64,62,60,58,56,54,52)], [(1,42,13,46,9,34,5,38),(2,45,6,41,10,37,14,33),(3,40,15,44,11,48,7,36),(4,43,8,39,12,35,16,47),(17,55,21,51,25,63,29,59),(18,50,30,54,26,58,22,62),(19,53,23,49,27,61,31,57),(20,64,32,52,28,56,24,60)], [(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)]])
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 8I | 8J | 16A | ··· | 16H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 16 | ··· | 16 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 8 | 8 | 8 | 8 | 2 | 2 | 2 | 2 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | ··· | 4 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C4 | D4 | D4 | D8 | SD16 | SD16 | C8.17D4 |
| kernel | C2×C8.17D4 | C8.17D4 | C2×C8.C4 | C2×M5(2) | C22×Q16 | C2×Q16 | C2×C8 | C22×C4 | C2×C4 | C2×C4 | C23 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 8 | 3 | 1 | 4 | 2 | 2 | 4 |
Matrix representation of C2×C8.17D4 ►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 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 14 | 3 | 0 | 0 |
| 0 | 0 | 14 | 14 | 0 | 0 |
| 0 | 0 | 7 | 5 | 14 | 14 |
| 0 | 0 | 12 | 7 | 3 | 14 |
| 8 | 8 | 0 | 0 | 0 | 0 |
| 11 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 12 | 2 | 0 |
| 0 | 0 | 13 | 2 | 0 | 2 |
| 0 | 0 | 6 | 11 | 11 | 5 |
| 0 | 0 | 8 | 5 | 4 | 15 |
| 9 | 9 | 0 | 0 | 0 | 0 |
| 10 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 14 | 8 | 0 |
| 0 | 0 | 16 | 9 | 0 | 9 |
| 0 | 0 | 8 | 12 | 10 | 14 |
| 0 | 0 | 4 | 13 | 16 | 8 |
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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,14,14,7,12,0,0,3,14,5,7,0,0,0,0,14,3,0,0,0,0,14,14],[8,11,0,0,0,0,8,9,0,0,0,0,0,0,6,13,6,8,0,0,12,2,11,5,0,0,2,0,11,4,0,0,0,2,5,15],[9,10,0,0,0,0,9,8,0,0,0,0,0,0,7,16,8,4,0,0,14,9,12,13,0,0,8,0,10,16,0,0,0,9,14,8] >;
C2×C8.17D4 in GAP, Magma, Sage, TeX
C_2\times C_8._{17}D_4 % in TeX
G:=Group("C2xC8.17D4"); // GroupNames label
G:=SmallGroup(128,879);
// by ID
G=gap.SmallGroup(128,879);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,-2,112,141,456,1123,1466,136,1411,172,4037,2028,124]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^8=1,c^4=b^4,d^2=c*b*c^-1=b^-1,a*b=b*a,a*c=c*a,a*d=d*a,b*d=d*b,d*c*d^-1=b^-1*c^3>;
// generators/relations