p-group, metabelian, nilpotent (class 3), monomial
Aliases: C23.23D8, C24.136D4, C23.26SD16, (C2×C8)⋊36D4, (C23×C8)⋊2C2, C4⋊D4⋊12C4, C4.113(C4×D4), C4.90C22≀C2, C2.3(C8⋊8D4), C2.2(C8⋊7D4), C22.40(C2×D8), C23.773(C2×D4), (C22×C4).551D4, C23.7Q8⋊6C2, C22⋊2(D4⋊C4), C22.4Q16⋊15C2, C22.60(C4○D8), C22.61(C2×SD16), (C22×C8).482C22, (C23×C4).674C22, (C22×D4).24C22, C22.119(C4⋊D4), C23.122(C22⋊C4), (C22×C4).1369C23, C4.85(C22.D4), C2.32(C23.23D4), C2.27(C23.24D4), C4⋊C4.72(C2×C4), (C2×D4⋊C4)⋊6C2, (C2×D4).81(C2×C4), (C2×C4⋊D4).8C2, C2.21(C2×D4⋊C4), (C2×C4).1332(C2×D4), (C2×C4⋊C4).61C22, (C2×C4).566(C4○D4), (C2×C4).387(C22×C4), (C22×C4).405(C2×C4), (C2×C4).194(C22⋊C4), C22.268(C2×C22⋊C4), SmallGroup(128,625)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C23.23D8
G = < a,b,c,d,e | a2=b2=c2=d8=e2=1, eae=ab=ba, ac=ca, ad=da, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=cd-1 >
Subgroups: 484 in 216 conjugacy classes, 72 normal (22 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, C23, C23, C23, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C24, C24, C2.C42, D4⋊C4, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C4⋊D4, C4⋊D4, C22×C8, C22×C8, C23×C4, C22×D4, C22×D4, C22.4Q16, C23.7Q8, C2×D4⋊C4, C2×C4⋊D4, C23×C8, C23.23D8
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, D8, SD16, C22×C4, C2×D4, C4○D4, D4⋊C4, C2×C22⋊C4, C4×D4, C22≀C2, C4⋊D4, C22.D4, C2×D8, C2×SD16, C4○D8, C23.23D4, C2×D4⋊C4, C23.24D4, C8⋊8D4, C8⋊7D4, C23.23D8
►On 64 points
(1 19)(2 20)(3 21)(4 22)(5 23)(6 24)(7 17)(8 18)(9 62)(10 63)(11 64)(12 57)(13 58)(14 59)(15 60)(16 61)(25 50)(26 51)(27 52)(28 53)(29 54)(30 55)(31 56)(32 49)(33 42)(34 43)(35 44)(36 45)(37 46)(38 47)(39 48)(40 41)
(1 37)(2 38)(3 39)(4 40)(5 33)(6 34)(7 35)(8 36)(9 49)(10 50)(11 51)(12 52)(13 53)(14 54)(15 55)(16 56)(17 44)(18 45)(19 46)(20 47)(21 48)(22 41)(23 42)(24 43)(25 63)(26 64)(27 57)(28 58)(29 59)(30 60)(31 61)(32 62)
(1 31)(2 32)(3 25)(4 26)(5 27)(6 28)(7 29)(8 30)(9 47)(10 48)(11 41)(12 42)(13 43)(14 44)(15 45)(16 46)(17 54)(18 55)(19 56)(20 49)(21 50)(22 51)(23 52)(24 53)(33 57)(34 58)(35 59)(36 60)(37 61)(38 62)(39 63)(40 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 29)(2 6)(3 27)(5 25)(7 31)(9 53)(10 23)(11 51)(12 21)(13 49)(14 19)(15 55)(16 17)(18 45)(20 43)(22 41)(24 47)(28 32)(33 63)(34 38)(35 61)(37 59)(39 57)(42 50)(44 56)(46 54)(48 52)(58 62)
G:=sub<Sym(64)| (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,17)(8,18)(9,62)(10,63)(11,64)(12,57)(13,58)(14,59)(15,60)(16,61)(25,50)(26,51)(27,52)(28,53)(29,54)(30,55)(31,56)(32,49)(33,42)(34,43)(35,44)(36,45)(37,46)(38,47)(39,48)(40,41), (1,37)(2,38)(3,39)(4,40)(5,33)(6,34)(7,35)(8,36)(9,49)(10,50)(11,51)(12,52)(13,53)(14,54)(15,55)(16,56)(17,44)(18,45)(19,46)(20,47)(21,48)(22,41)(23,42)(24,43)(25,63)(26,64)(27,57)(28,58)(29,59)(30,60)(31,61)(32,62), (1,31)(2,32)(3,25)(4,26)(5,27)(6,28)(7,29)(8,30)(9,47)(10,48)(11,41)(12,42)(13,43)(14,44)(15,45)(16,46)(17,54)(18,55)(19,56)(20,49)(21,50)(22,51)(23,52)(24,53)(33,57)(34,58)(35,59)(36,60)(37,61)(38,62)(39,63)(40,64), (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), (1,29)(2,6)(3,27)(5,25)(7,31)(9,53)(10,23)(11,51)(12,21)(13,49)(14,19)(15,55)(16,17)(18,45)(20,43)(22,41)(24,47)(28,32)(33,63)(34,38)(35,61)(37,59)(39,57)(42,50)(44,56)(46,54)(48,52)(58,62)>;
G:=Group( (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,17)(8,18)(9,62)(10,63)(11,64)(12,57)(13,58)(14,59)(15,60)(16,61)(25,50)(26,51)(27,52)(28,53)(29,54)(30,55)(31,56)(32,49)(33,42)(34,43)(35,44)(36,45)(37,46)(38,47)(39,48)(40,41), (1,37)(2,38)(3,39)(4,40)(5,33)(6,34)(7,35)(8,36)(9,49)(10,50)(11,51)(12,52)(13,53)(14,54)(15,55)(16,56)(17,44)(18,45)(19,46)(20,47)(21,48)(22,41)(23,42)(24,43)(25,63)(26,64)(27,57)(28,58)(29,59)(30,60)(31,61)(32,62), (1,31)(2,32)(3,25)(4,26)(5,27)(6,28)(7,29)(8,30)(9,47)(10,48)(11,41)(12,42)(13,43)(14,44)(15,45)(16,46)(17,54)(18,55)(19,56)(20,49)(21,50)(22,51)(23,52)(24,53)(33,57)(34,58)(35,59)(36,60)(37,61)(38,62)(39,63)(40,64), (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), (1,29)(2,6)(3,27)(5,25)(7,31)(9,53)(10,23)(11,51)(12,21)(13,49)(14,19)(15,55)(16,17)(18,45)(20,43)(22,41)(24,47)(28,32)(33,63)(34,38)(35,61)(37,59)(39,57)(42,50)(44,56)(46,54)(48,52)(58,62) );
G=PermutationGroup([[(1,19),(2,20),(3,21),(4,22),(5,23),(6,24),(7,17),(8,18),(9,62),(10,63),(11,64),(12,57),(13,58),(14,59),(15,60),(16,61),(25,50),(26,51),(27,52),(28,53),(29,54),(30,55),(31,56),(32,49),(33,42),(34,43),(35,44),(36,45),(37,46),(38,47),(39,48),(40,41)], [(1,37),(2,38),(3,39),(4,40),(5,33),(6,34),(7,35),(8,36),(9,49),(10,50),(11,51),(12,52),(13,53),(14,54),(15,55),(16,56),(17,44),(18,45),(19,46),(20,47),(21,48),(22,41),(23,42),(24,43),(25,63),(26,64),(27,57),(28,58),(29,59),(30,60),(31,61),(32,62)], [(1,31),(2,32),(3,25),(4,26),(5,27),(6,28),(7,29),(8,30),(9,47),(10,48),(11,41),(12,42),(13,43),(14,44),(15,45),(16,46),(17,54),(18,55),(19,56),(20,49),(21,50),(22,51),(23,52),(24,53),(33,57),(34,58),(35,59),(36,60),(37,61),(38,62),(39,63),(40,64)], [(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)], [(1,29),(2,6),(3,27),(5,25),(7,31),(9,53),(10,23),(11,51),(12,21),(13,49),(14,19),(15,55),(16,17),(18,45),(20,43),(22,41),(24,47),(28,32),(33,63),(34,38),(35,61),(37,59),(39,57),(42,50),(44,56),(46,54),(48,52),(58,62)]])
44 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 2L | 2M | 4A | ··· | 4H | 4I | ··· | 4N | 8A | ··· | 8P |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 8 | 8 | 2 | ··· | 2 | 8 | ··· | 8 | 2 | ··· | 2 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | D4 | D4 | C4○D4 | D8 | SD16 | C4○D8 |
| kernel | C23.23D8 | C22.4Q16 | C23.7Q8 | C2×D4⋊C4 | C2×C4⋊D4 | C23×C8 | C4⋊D4 | C2×C8 | C22×C4 | C24 | C2×C4 | C23 | C23 | C22 |
| # reps | 1 | 2 | 1 | 2 | 1 | 1 | 8 | 4 | 3 | 1 | 4 | 4 | 4 | 8 |
Matrix representation of C23.23D8 ►in GL5(𝔽17)
| 16 | 0 | 0 | 0 | 0 |
| 0 | 4 | 9 | 0 | 0 |
| 0 | 4 | 13 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 16 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 16 |
| 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 0 | 0 |
| 0 | 3 | 11 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 |
| 0 | 0 | 0 | 5 | 7 |
| 16 | 0 | 0 | 0 | 0 |
| 0 | 16 | 2 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 16 | 2 |
| 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(5,GF(17))| [16,0,0,0,0,0,4,4,0,0,0,9,13,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,1,0,0,0,0,0,1],[16,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,16],[4,0,0,0,0,0,0,3,0,0,0,11,11,0,0,0,0,0,0,5,0,0,0,7,7],[16,0,0,0,0,0,16,0,0,0,0,2,1,0,0,0,0,0,16,0,0,0,0,2,1] >;
C23.23D8 in GAP, Magma, Sage, TeX
C_2^3._{23}D_8 % in TeX
G:=Group("C2^3.23D8"); // GroupNames label
G:=SmallGroup(128,625);
// by ID
G=gap.SmallGroup(128,625);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,736,422,2019,248]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^8=e^2=1,e*a*e=a*b=b*a,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=c*d^-1>;
// generators/relations