p-group, metabelian, nilpotent (class 4), monomial
Aliases: C23.39D8, C4○D8⋊2C4, D8⋊8(C2×C4), Q16⋊8(C2×C4), C8.93(C2×D4), (C2×C8).117D4, (C2×C4).137D8, C2.D16⋊14C2, C8.4(C22⋊C4), C8.30(C22×C4), (C2×C4).47SD16, C4.11(C2×SD16), C22.54(C2×D8), C2.Q32⋊14C2, C2.1(C16⋊C22), (C2×M5(2))⋊12C2, (C2×C8).495C23, (C2×C16).50C22, C2.1(Q32⋊C2), (C22×C4).330D4, C4.37(D4⋊C4), (C2×D8).101C22, C2.D8.144C22, C22.4(D4⋊C4), (C22×C8).229C22, (C2×Q16).100C22, (C2×C8).79(C2×C4), (C2×C4○D8).9C2, (C2×C2.D8)⋊35C2, (C2×C4).757(C2×D4), C4.51(C2×C22⋊C4), C2.29(C2×D4⋊C4), (C2×C4).148(C22⋊C4), SmallGroup(128,871)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C23.39D8
G = < a,b,c,d,e | a2=b2=c2=1, d8=c, e2=b, ab=ba, dad-1=ac=ca, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=bcd7 >
Subgroups: 276 in 116 conjugacy classes, 52 normal (22 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C8, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C16, C4⋊C4, C2×C8, C2×C8, D8, D8, SD16, Q16, Q16, C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, C2.D8, C2.D8, C2×C16, M5(2), C2×C4⋊C4, C22×C8, C2×D8, C2×SD16, C2×Q16, C4○D8, C4○D8, C2×C4○D4, C2.D16, C2.Q32, C2×C2.D8, C2×M5(2), C2×C4○D8, C23.39D8
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, C2×D4⋊C4, C16⋊C22, Q32⋊C2, C23.39D8
►On 64 points
(1 48)(2 41)(3 34)(4 43)(5 36)(6 45)(7 38)(8 47)(9 40)(10 33)(11 42)(12 35)(13 44)(14 37)(15 46)(16 39)(17 56)(18 49)(19 58)(20 51)(21 60)(22 53)(23 62)(24 55)(25 64)(26 57)(27 50)(28 59)(29 52)(30 61)(31 54)(32 63)
(1 17)(2 18)(3 19)(4 20)(5 21)(6 22)(7 23)(8 24)(9 25)(10 26)(11 27)(12 28)(13 29)(14 30)(15 31)(16 32)(33 57)(34 58)(35 59)(36 60)(37 61)(38 62)(39 63)(40 64)(41 49)(42 50)(43 51)(44 52)(45 53)(46 54)(47 55)(48 56)
(1 9)(2 10)(3 11)(4 12)(5 13)(6 14)(7 15)(8 16)(17 25)(18 26)(19 27)(20 28)(21 29)(22 30)(23 31)(24 32)(33 41)(34 42)(35 43)(36 44)(37 45)(38 46)(39 47)(40 48)(49 57)(50 58)(51 59)(52 60)(53 61)(54 62)(55 63)(56 64)
(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 32 17 16)(2 15 18 31)(3 30 19 14)(4 13 20 29)(5 28 21 12)(6 11 22 27)(7 26 23 10)(8 9 24 25)(33 38 57 62)(34 61 58 37)(35 36 59 60)(39 48 63 56)(40 55 64 47)(41 46 49 54)(42 53 50 45)(43 44 51 52)
G:=sub<Sym(64)| (1,48)(2,41)(3,34)(4,43)(5,36)(6,45)(7,38)(8,47)(9,40)(10,33)(11,42)(12,35)(13,44)(14,37)(15,46)(16,39)(17,56)(18,49)(19,58)(20,51)(21,60)(22,53)(23,62)(24,55)(25,64)(26,57)(27,50)(28,59)(29,52)(30,61)(31,54)(32,63), (1,17)(2,18)(3,19)(4,20)(5,21)(6,22)(7,23)(8,24)(9,25)(10,26)(11,27)(12,28)(13,29)(14,30)(15,31)(16,32)(33,57)(34,58)(35,59)(36,60)(37,61)(38,62)(39,63)(40,64)(41,49)(42,50)(43,51)(44,52)(45,53)(46,54)(47,55)(48,56), (1,9)(2,10)(3,11)(4,12)(5,13)(6,14)(7,15)(8,16)(17,25)(18,26)(19,27)(20,28)(21,29)(22,30)(23,31)(24,32)(33,41)(34,42)(35,43)(36,44)(37,45)(38,46)(39,47)(40,48)(49,57)(50,58)(51,59)(52,60)(53,61)(54,62)(55,63)(56,64), (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), (1,32,17,16)(2,15,18,31)(3,30,19,14)(4,13,20,29)(5,28,21,12)(6,11,22,27)(7,26,23,10)(8,9,24,25)(33,38,57,62)(34,61,58,37)(35,36,59,60)(39,48,63,56)(40,55,64,47)(41,46,49,54)(42,53,50,45)(43,44,51,52)>;
G:=Group( (1,48)(2,41)(3,34)(4,43)(5,36)(6,45)(7,38)(8,47)(9,40)(10,33)(11,42)(12,35)(13,44)(14,37)(15,46)(16,39)(17,56)(18,49)(19,58)(20,51)(21,60)(22,53)(23,62)(24,55)(25,64)(26,57)(27,50)(28,59)(29,52)(30,61)(31,54)(32,63), (1,17)(2,18)(3,19)(4,20)(5,21)(6,22)(7,23)(8,24)(9,25)(10,26)(11,27)(12,28)(13,29)(14,30)(15,31)(16,32)(33,57)(34,58)(35,59)(36,60)(37,61)(38,62)(39,63)(40,64)(41,49)(42,50)(43,51)(44,52)(45,53)(46,54)(47,55)(48,56), (1,9)(2,10)(3,11)(4,12)(5,13)(6,14)(7,15)(8,16)(17,25)(18,26)(19,27)(20,28)(21,29)(22,30)(23,31)(24,32)(33,41)(34,42)(35,43)(36,44)(37,45)(38,46)(39,47)(40,48)(49,57)(50,58)(51,59)(52,60)(53,61)(54,62)(55,63)(56,64), (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), (1,32,17,16)(2,15,18,31)(3,30,19,14)(4,13,20,29)(5,28,21,12)(6,11,22,27)(7,26,23,10)(8,9,24,25)(33,38,57,62)(34,61,58,37)(35,36,59,60)(39,48,63,56)(40,55,64,47)(41,46,49,54)(42,53,50,45)(43,44,51,52) );
G=PermutationGroup([[(1,48),(2,41),(3,34),(4,43),(5,36),(6,45),(7,38),(8,47),(9,40),(10,33),(11,42),(12,35),(13,44),(14,37),(15,46),(16,39),(17,56),(18,49),(19,58),(20,51),(21,60),(22,53),(23,62),(24,55),(25,64),(26,57),(27,50),(28,59),(29,52),(30,61),(31,54),(32,63)], [(1,17),(2,18),(3,19),(4,20),(5,21),(6,22),(7,23),(8,24),(9,25),(10,26),(11,27),(12,28),(13,29),(14,30),(15,31),(16,32),(33,57),(34,58),(35,59),(36,60),(37,61),(38,62),(39,63),(40,64),(41,49),(42,50),(43,51),(44,52),(45,53),(46,54),(47,55),(48,56)], [(1,9),(2,10),(3,11),(4,12),(5,13),(6,14),(7,15),(8,16),(17,25),(18,26),(19,27),(20,28),(21,29),(22,30),(23,31),(24,32),(33,41),(34,42),(35,43),(36,44),(37,45),(38,46),(39,47),(40,48),(49,57),(50,58),(51,59),(52,60),(53,61),(54,62),(55,63),(56,64)], [(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)], [(1,32,17,16),(2,15,18,31),(3,30,19,14),(4,13,20,29),(5,28,21,12),(6,11,22,27),(7,26,23,10),(8,9,24,25),(33,38,57,62),(34,61,58,37),(35,36,59,60),(39,48,63,56),(40,55,64,47),(41,46,49,54),(42,53,50,45),(43,44,51,52)]])
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | ··· | 4J | 8A | 8B | 8C | 8D | 8E | 8F | 16A | ··· | 16H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 8 | 8 | 8 | 8 | 8 | 8 | 16 | ··· | 16 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 8 | 2 | 2 | 2 | 2 | 8 | ··· | 8 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | ··· | 4 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | D4 | D8 | SD16 | D8 | C16⋊C22 | Q32⋊C2 |
| kernel | C23.39D8 | C2.D16 | C2.Q32 | C2×C2.D8 | C2×M5(2) | C2×C4○D8 | C4○D8 | C2×C8 | C22×C4 | C2×C4 | C2×C4 | C23 | C2 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 8 | 3 | 1 | 2 | 4 | 2 | 2 | 2 |
Matrix representation of C23.39D8 ►in GL6(𝔽17)
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 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 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 9 | 4 | 0 | 0 | 0 | 0 |
| 14 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 13 | 7 | 6 | 2 |
| 0 | 0 | 5 | 6 | 16 | 4 |
| 0 | 0 | 11 | 15 | 4 | 10 |
| 0 | 0 | 1 | 13 | 12 | 11 |
| 9 | 4 | 0 | 0 | 0 | 0 |
| 5 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 13 | 7 | 6 | 2 |
| 0 | 0 | 16 | 4 | 12 | 11 |
| 0 | 0 | 6 | 2 | 13 | 7 |
| 0 | 0 | 12 | 11 | 16 | 4 |
G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0],[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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[9,14,0,0,0,0,4,8,0,0,0,0,0,0,13,5,11,1,0,0,7,6,15,13,0,0,6,16,4,12,0,0,2,4,10,11],[9,5,0,0,0,0,4,8,0,0,0,0,0,0,13,16,6,12,0,0,7,4,2,11,0,0,6,12,13,16,0,0,2,11,7,4] >;
C23.39D8 in GAP, Magma, Sage, TeX
C_2^3._{39}D_8 % in TeX
G:=Group("C2^3.39D8"); // GroupNames label
G:=SmallGroup(128,871);
// by ID
G=gap.SmallGroup(128,871);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,-2,112,141,723,1123,570,360,4037,2028,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=1,d^8=c,e^2=b,a*b=b*a,d*a*d^-1=a*c=c*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=b*c*d^7>;
// generators/relations