p-group, metabelian, nilpotent (class 3), monomial
Aliases: C23.22D8, C24.134D4, C23.12Q16, C8⋊7(C22⋊C4), (C22×C8)⋊14C4, (C2×C8).347D4, C2.1(C8⋊7D4), (C23×C8).13C2, C22.32(C2×D8), C22⋊2(C2.D8), (C22×C4).90Q8, C23.68(C4⋊C4), C22.4Q16⋊4C2, (C22×C4).545D4, C23.745(C2×D4), C4.70(C22⋊Q8), C22.25(C2×Q16), C2.1(C8.18D4), C22.44(C4○D8), C4.53(C42⋊C2), C23.7Q8.7C2, (C23×C4).671C22, (C22×C8).523C22, C22.110(C4⋊D4), (C22×C4).1330C23, C2.8(C23.25D4), C2.17(C23.7Q8), (C2×C2.D8)⋊1C2, C2.6(C2×C2.D8), (C2×C4).84(C4⋊C4), (C2×C8).209(C2×C4), C4.87(C2×C22⋊C4), C22.91(C2×C4⋊C4), (C2×C4).188(C2×Q8), (C2×C4).1319(C2×D4), (C2×C4⋊C4).37C22, (C2×C4).552(C4○D4), (C22×C4).481(C2×C4), (C2×C4).528(C22×C4), SmallGroup(128,540)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C23.22D8
G = < a,b,c,d,e | a2=b2=c2=d8=1, e2=c, eae-1=ab=ba, ac=ca, ad=da, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >
Subgroups: 316 in 168 conjugacy classes, 76 normal (20 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C8, C8, C2×C4, C2×C4, C2×C4, C23, C23, C23, C22⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C22×C4, C22×C4, C24, C2.C42, C2.D8, C2×C22⋊C4, C2×C4⋊C4, C22×C8, C22×C8, C22×C8, C23×C4, C22.4Q16, C23.7Q8, C2×C2.D8, C23×C8, C23.22D8
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C22⋊C4, C4⋊C4, D8, Q16, C22×C4, C2×D4, C2×Q8, C4○D4, C2.D8, C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C4⋊D4, C22⋊Q8, C2×D8, C2×Q16, C4○D8, C23.7Q8, C2×C2.D8, C23.25D4, C8⋊7D4, C8.18D4, C23.22D8
►On 64 points
(1 46)(2 47)(3 48)(4 41)(5 42)(6 43)(7 44)(8 45)(9 20)(10 21)(11 22)(12 23)(13 24)(14 17)(15 18)(16 19)(25 58)(26 59)(27 60)(28 61)(29 62)(30 63)(31 64)(32 57)(33 50)(34 51)(35 52)(36 53)(37 54)(38 55)(39 56)(40 49)
(1 52)(2 53)(3 54)(4 55)(5 56)(6 49)(7 50)(8 51)(9 63)(10 64)(11 57)(12 58)(13 59)(14 60)(15 61)(16 62)(17 27)(18 28)(19 29)(20 30)(21 31)(22 32)(23 25)(24 26)(33 44)(34 45)(35 46)(36 47)(37 48)(38 41)(39 42)(40 43)
(1 46)(2 47)(3 48)(4 41)(5 42)(6 43)(7 44)(8 45)(9 30)(10 31)(11 32)(12 25)(13 26)(14 27)(15 28)(16 29)(17 60)(18 61)(19 62)(20 63)(21 64)(22 57)(23 58)(24 59)(33 50)(34 51)(35 52)(36 53)(37 54)(38 55)(39 56)(40 49)
(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 19 46 62)(2 18 47 61)(3 17 48 60)(4 24 41 59)(5 23 42 58)(6 22 43 57)(7 21 44 64)(8 20 45 63)(9 51 30 34)(10 50 31 33)(11 49 32 40)(12 56 25 39)(13 55 26 38)(14 54 27 37)(15 53 28 36)(16 52 29 35)
G:=sub<Sym(64)| (1,46)(2,47)(3,48)(4,41)(5,42)(6,43)(7,44)(8,45)(9,20)(10,21)(11,22)(12,23)(13,24)(14,17)(15,18)(16,19)(25,58)(26,59)(27,60)(28,61)(29,62)(30,63)(31,64)(32,57)(33,50)(34,51)(35,52)(36,53)(37,54)(38,55)(39,56)(40,49), (1,52)(2,53)(3,54)(4,55)(5,56)(6,49)(7,50)(8,51)(9,63)(10,64)(11,57)(12,58)(13,59)(14,60)(15,61)(16,62)(17,27)(18,28)(19,29)(20,30)(21,31)(22,32)(23,25)(24,26)(33,44)(34,45)(35,46)(36,47)(37,48)(38,41)(39,42)(40,43), (1,46)(2,47)(3,48)(4,41)(5,42)(6,43)(7,44)(8,45)(9,30)(10,31)(11,32)(12,25)(13,26)(14,27)(15,28)(16,29)(17,60)(18,61)(19,62)(20,63)(21,64)(22,57)(23,58)(24,59)(33,50)(34,51)(35,52)(36,53)(37,54)(38,55)(39,56)(40,49), (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), (1,19,46,62)(2,18,47,61)(3,17,48,60)(4,24,41,59)(5,23,42,58)(6,22,43,57)(7,21,44,64)(8,20,45,63)(9,51,30,34)(10,50,31,33)(11,49,32,40)(12,56,25,39)(13,55,26,38)(14,54,27,37)(15,53,28,36)(16,52,29,35)>;
G:=Group( (1,46)(2,47)(3,48)(4,41)(5,42)(6,43)(7,44)(8,45)(9,20)(10,21)(11,22)(12,23)(13,24)(14,17)(15,18)(16,19)(25,58)(26,59)(27,60)(28,61)(29,62)(30,63)(31,64)(32,57)(33,50)(34,51)(35,52)(36,53)(37,54)(38,55)(39,56)(40,49), (1,52)(2,53)(3,54)(4,55)(5,56)(6,49)(7,50)(8,51)(9,63)(10,64)(11,57)(12,58)(13,59)(14,60)(15,61)(16,62)(17,27)(18,28)(19,29)(20,30)(21,31)(22,32)(23,25)(24,26)(33,44)(34,45)(35,46)(36,47)(37,48)(38,41)(39,42)(40,43), (1,46)(2,47)(3,48)(4,41)(5,42)(6,43)(7,44)(8,45)(9,30)(10,31)(11,32)(12,25)(13,26)(14,27)(15,28)(16,29)(17,60)(18,61)(19,62)(20,63)(21,64)(22,57)(23,58)(24,59)(33,50)(34,51)(35,52)(36,53)(37,54)(38,55)(39,56)(40,49), (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), (1,19,46,62)(2,18,47,61)(3,17,48,60)(4,24,41,59)(5,23,42,58)(6,22,43,57)(7,21,44,64)(8,20,45,63)(9,51,30,34)(10,50,31,33)(11,49,32,40)(12,56,25,39)(13,55,26,38)(14,54,27,37)(15,53,28,36)(16,52,29,35) );
G=PermutationGroup([[(1,46),(2,47),(3,48),(4,41),(5,42),(6,43),(7,44),(8,45),(9,20),(10,21),(11,22),(12,23),(13,24),(14,17),(15,18),(16,19),(25,58),(26,59),(27,60),(28,61),(29,62),(30,63),(31,64),(32,57),(33,50),(34,51),(35,52),(36,53),(37,54),(38,55),(39,56),(40,49)], [(1,52),(2,53),(3,54),(4,55),(5,56),(6,49),(7,50),(8,51),(9,63),(10,64),(11,57),(12,58),(13,59),(14,60),(15,61),(16,62),(17,27),(18,28),(19,29),(20,30),(21,31),(22,32),(23,25),(24,26),(33,44),(34,45),(35,46),(36,47),(37,48),(38,41),(39,42),(40,43)], [(1,46),(2,47),(3,48),(4,41),(5,42),(6,43),(7,44),(8,45),(9,30),(10,31),(11,32),(12,25),(13,26),(14,27),(15,28),(16,29),(17,60),(18,61),(19,62),(20,63),(21,64),(22,57),(23,58),(24,59),(33,50),(34,51),(35,52),(36,53),(37,54),(38,55),(39,56),(40,49)], [(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)], [(1,19,46,62),(2,18,47,61),(3,17,48,60),(4,24,41,59),(5,23,42,58),(6,22,43,57),(7,21,44,64),(8,20,45,63),(9,51,30,34),(10,50,31,33),(11,49,32,40),(12,56,25,39),(13,55,26,38),(14,54,27,37),(15,53,28,36),(16,52,29,35)]])
44 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 4A | ··· | 4H | 4I | ··· | 4P | 8A | ··· | 8P |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 8 | ··· | 8 | 2 | ··· | 2 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | - | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C4 | D4 | D4 | Q8 | D4 | C4○D4 | D8 | Q16 | C4○D8 |
| kernel | C23.22D8 | C22.4Q16 | C23.7Q8 | C2×C2.D8 | C23×C8 | C22×C8 | C2×C8 | C22×C4 | C22×C4 | C24 | C2×C4 | C23 | C23 | C22 |
| # reps | 1 | 2 | 2 | 2 | 1 | 8 | 4 | 1 | 2 | 1 | 4 | 4 | 4 | 8 |
Matrix representation of C23.22D8 ►in GL5(𝔽17)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 16 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 16 |
| 16 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 3 | 14 | 0 | 0 |
| 0 | 3 | 3 | 0 | 0 |
| 0 | 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 0 | 9 |
| 4 | 0 | 0 | 0 | 0 |
| 0 | 6 | 13 | 0 | 0 |
| 0 | 13 | 11 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 |
| 0 | 0 | 0 | 2 | 0 |
G:=sub<GL(5,GF(17))| [1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,16],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,16],[16,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,3,3,0,0,0,14,3,0,0,0,0,0,2,0,0,0,0,0,9],[4,0,0,0,0,0,6,13,0,0,0,13,11,0,0,0,0,0,0,2,0,0,0,9,0] >;
C23.22D8 in GAP, Magma, Sage, TeX
C_2^3._{22}D_8 % in TeX
G:=Group("C2^3.22D8"); // GroupNames label
G:=SmallGroup(128,540);
// by ID
G=gap.SmallGroup(128,540);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,512,422,2019,248]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^8=1,e^2=c,e*a*e^-1=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^-1=d^-1>;
// generators/relations