p-group, metabelian, nilpotent (class 4), monomial
Aliases: C23.25D8, (C2×C16)⋊14C4, C4○(C16⋊3C4), C4○(C16⋊4C4), C8.31(C4⋊C4), C8.16(C2×Q8), C4.7(C2×Q16), (C2×C8).51Q8, C16.20(C2×C4), C16⋊3C4⋊13C2, C16⋊4C4⋊13C2, (C2×C8).278D4, (C2×C4).170D8, (C2×C4).41Q16, C2.2(C4○D16), C8.53(C22×C4), C4.24(C2.D8), C22.59(C2×D8), (C2×C8).500C23, (C2×C16).80C22, (C22×C16).13C2, (C22×C4).588D4, C2.D8.148C22, C22.13(C2.D8), (C22×C8).553C22, C23.25D4.4C2, C4.52(C2×C4⋊C4), (C2×C4)○(C16⋊4C4), (C2×C4)○(C16⋊3C4), C2.13(C2×C2.D8), (C2×C8).226(C2×C4), (C2×C4).764(C2×D4), (C2×C4).144(C4⋊C4), SmallGroup(128,890)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C23.25D8
G = < a,b,c,d,e | a2=b2=c2=1, d8=c, e2=b, ab=ba, eae-1=ac=ca, ad=da, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=cd7 >
Subgroups: 140 in 76 conjugacy classes, 52 normal (18 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C8, C8, C2×C4, C2×C4, C2×C4, C23, C16, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C4.Q8, C2.D8, C2×C16, C2×C16, C42⋊C2, C22×C8, C16⋊3C4, C16⋊4C4, C23.25D4, C22×C16, C23.25D8
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C4⋊C4, D8, Q16, C22×C4, C2×D4, C2×Q8, C2.D8, C2×C4⋊C4, C2×D8, C2×Q16, C2×C2.D8, C4○D16, C23.25D8
►On 64 points
(17 25)(18 26)(19 27)(20 28)(21 29)(22 30)(23 31)(24 32)(49 57)(50 58)(51 59)(52 60)(53 61)(54 62)(55 63)(56 64)
(1 36)(2 37)(3 38)(4 39)(5 40)(6 41)(7 42)(8 43)(9 44)(10 45)(11 46)(12 47)(13 48)(14 33)(15 34)(16 35)(17 59)(18 60)(19 61)(20 62)(21 63)(22 64)(23 49)(24 50)(25 51)(26 52)(27 53)(28 54)(29 55)(30 56)(31 57)(32 58)
(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 56 36 30)(2 55 37 29)(3 54 38 28)(4 53 39 27)(5 52 40 26)(6 51 41 25)(7 50 42 24)(8 49 43 23)(9 64 44 22)(10 63 45 21)(11 62 46 20)(12 61 47 19)(13 60 48 18)(14 59 33 17)(15 58 34 32)(16 57 35 31)
G:=sub<Sym(64)| (17,25)(18,26)(19,27)(20,28)(21,29)(22,30)(23,31)(24,32)(49,57)(50,58)(51,59)(52,60)(53,61)(54,62)(55,63)(56,64), (1,36)(2,37)(3,38)(4,39)(5,40)(6,41)(7,42)(8,43)(9,44)(10,45)(11,46)(12,47)(13,48)(14,33)(15,34)(16,35)(17,59)(18,60)(19,61)(20,62)(21,63)(22,64)(23,49)(24,50)(25,51)(26,52)(27,53)(28,54)(29,55)(30,56)(31,57)(32,58), (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), (1,56,36,30)(2,55,37,29)(3,54,38,28)(4,53,39,27)(5,52,40,26)(6,51,41,25)(7,50,42,24)(8,49,43,23)(9,64,44,22)(10,63,45,21)(11,62,46,20)(12,61,47,19)(13,60,48,18)(14,59,33,17)(15,58,34,32)(16,57,35,31)>;
G:=Group( (17,25)(18,26)(19,27)(20,28)(21,29)(22,30)(23,31)(24,32)(49,57)(50,58)(51,59)(52,60)(53,61)(54,62)(55,63)(56,64), (1,36)(2,37)(3,38)(4,39)(5,40)(6,41)(7,42)(8,43)(9,44)(10,45)(11,46)(12,47)(13,48)(14,33)(15,34)(16,35)(17,59)(18,60)(19,61)(20,62)(21,63)(22,64)(23,49)(24,50)(25,51)(26,52)(27,53)(28,54)(29,55)(30,56)(31,57)(32,58), (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), (1,56,36,30)(2,55,37,29)(3,54,38,28)(4,53,39,27)(5,52,40,26)(6,51,41,25)(7,50,42,24)(8,49,43,23)(9,64,44,22)(10,63,45,21)(11,62,46,20)(12,61,47,19)(13,60,48,18)(14,59,33,17)(15,58,34,32)(16,57,35,31) );
G=PermutationGroup([[(17,25),(18,26),(19,27),(20,28),(21,29),(22,30),(23,31),(24,32),(49,57),(50,58),(51,59),(52,60),(53,61),(54,62),(55,63),(56,64)], [(1,36),(2,37),(3,38),(4,39),(5,40),(6,41),(7,42),(8,43),(9,44),(10,45),(11,46),(12,47),(13,48),(14,33),(15,34),(16,35),(17,59),(18,60),(19,61),(20,62),(21,63),(22,64),(23,49),(24,50),(25,51),(26,52),(27,53),(28,54),(29,55),(30,56),(31,57),(32,58)], [(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)], [(1,56,36,30),(2,55,37,29),(3,54,38,28),(4,53,39,27),(5,52,40,26),(6,51,41,25),(7,50,42,24),(8,49,43,23),(9,64,44,22),(10,63,45,21),(11,62,46,20),(12,61,47,19),(13,60,48,18),(14,59,33,17),(15,58,34,32),(16,57,35,31)]])
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | ··· | 4N | 8A | ··· | 8H | 16A | ··· | 16P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 | 16 | ··· | 16 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 8 | ··· | 8 | 2 | ··· | 2 | 2 | ··· | 2 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | - | + | + | - | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C4 | D4 | Q8 | D4 | D8 | Q16 | D8 | C4○D16 |
| kernel | C23.25D8 | C16⋊3C4 | C16⋊4C4 | C23.25D4 | C22×C16 | C2×C16 | C2×C8 | C2×C8 | C22×C4 | C2×C4 | C2×C4 | C23 | C2 |
| # reps | 1 | 2 | 2 | 2 | 1 | 8 | 1 | 2 | 1 | 2 | 4 | 2 | 16 |
Matrix representation of C23.25D8 ►in GL3(𝔽17) generated by
| 16 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 16 |
| 16 | 0 | 0 |
| 0 | 16 | 0 |
| 0 | 0 | 16 |
| 1 | 0 | 0 |
| 0 | 16 | 0 |
| 0 | 0 | 16 |
| 1 | 0 | 0 |
| 0 | 7 | 0 |
| 0 | 0 | 5 |
| 4 | 0 | 0 |
| 0 | 0 | 5 |
| 0 | 10 | 0 |
G:=sub<GL(3,GF(17))| [16,0,0,0,1,0,0,0,16],[16,0,0,0,16,0,0,0,16],[1,0,0,0,16,0,0,0,16],[1,0,0,0,7,0,0,0,5],[4,0,0,0,0,10,0,5,0] >;
C23.25D8 in GAP, Magma, Sage, TeX
C_2^3._{25}D_8 % in TeX
G:=Group("C2^3.25D8"); // GroupNames label
G:=SmallGroup(128,890);
// by ID
G=gap.SmallGroup(128,890);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,-2,112,141,288,352,1123,360,4037,124]);
// 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,e*a*e^-1=a*c=c*a,a*d=d*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=c*d^7>;
// generators/relations