p-group, metabelian, nilpotent (class 3), monomial
Aliases: C24.83D4, C23.11D8, (C2×C8)⋊7D4, C4⋊C4.88D4, (C2×D4).99D4, C22.83(C2×D8), C2.17(C8⋊D4), C2.13(C8⋊7D4), C23.909(C2×D4), (C22×C4).145D4, C2.21(C22⋊D8), C23.7Q8⋊9C2, C4.142(C4⋊D4), C22.4Q16⋊21C2, C4.36(C4.4D4), (C22×C8).69C22, C22.215C22≀C2, C2.31(D4.7D4), C22.107(C4○D8), (C23×C4).271C22, C2.6(C22.D8), (C22×D4).76C22, C22.226(C4⋊D4), C22.135(C8⋊C22), (C22×C4).1443C23, C4.17(C22.D4), C2.8(C23.19D4), C2.6(C23.10D4), C22.124(C8.C22), C22.112(C22.D4), (C2×C2.D8)⋊7C2, (C2×C22⋊C8)⋊19C2, (C2×D4⋊C4)⋊12C2, (C2×C4).1035(C2×D4), (C2×C4⋊D4).13C2, (C2×C4).770(C4○D4), (C2×C4⋊C4).118C22, SmallGroup(128,765)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C24.83D4
G = < a,b,c,d,e | a2=b2=c2=d8=e2=1, eae=ab=ba, ac=ca, dad-1=abc, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=bd-1 >
Subgroups: 464 in 184 conjugacy classes, 52 normal (44 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, D4, C23, C23, C23, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C22×C4, C2×D4, C2×D4, C24, C24, C2.C42, C22⋊C8, D4⋊C4, C2.D8, C2×C22⋊C4, C2×C4⋊C4, C4⋊D4, C22×C8, C23×C4, C22×D4, C22×D4, C22.4Q16, C23.7Q8, C2×C22⋊C8, C2×D4⋊C4, C2×C2.D8, C2×C4⋊D4, C24.83D4
Quotients: C1, C2, C22, D4, C23, D8, C2×D4, C4○D4, C22≀C2, C4⋊D4, C22.D4, C4.4D4, C2×D8, C4○D8, C8⋊C22, C8.C22, C23.10D4, C22⋊D8, D4.7D4, C8⋊7D4, C8⋊D4, C22.D8, C23.19D4, C24.83D4
►On 64 points
(1 26)(2 63)(3 28)(4 57)(5 30)(6 59)(7 32)(8 61)(9 43)(10 51)(11 45)(12 53)(13 47)(14 55)(15 41)(16 49)(17 54)(18 48)(19 56)(20 42)(21 50)(22 44)(23 52)(24 46)(25 38)(27 40)(29 34)(31 36)(33 64)(35 58)(37 60)(39 62)
(1 41)(2 42)(3 43)(4 44)(5 45)(6 46)(7 47)(8 48)(9 28)(10 29)(11 30)(12 31)(13 32)(14 25)(15 26)(16 27)(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 56)(2 49)(3 50)(4 51)(5 52)(6 53)(7 54)(8 55)(9 64)(10 57)(11 58)(12 59)(13 60)(14 61)(15 62)(16 63)(17 32)(18 25)(19 26)(20 27)(21 28)(22 29)(23 30)(24 31)(33 43)(34 44)(35 45)(36 46)(37 47)(38 48)(39 41)(40 42)
(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)
(2 48)(3 7)(4 46)(6 44)(8 42)(9 32)(10 12)(11 30)(13 28)(14 16)(15 26)(17 64)(18 20)(19 62)(21 60)(22 24)(23 58)(25 27)(29 31)(33 37)(34 53)(36 51)(38 49)(40 55)(43 47)(50 54)(57 59)(61 63)
G:=sub<Sym(64)| (1,26)(2,63)(3,28)(4,57)(5,30)(6,59)(7,32)(8,61)(9,43)(10,51)(11,45)(12,53)(13,47)(14,55)(15,41)(16,49)(17,54)(18,48)(19,56)(20,42)(21,50)(22,44)(23,52)(24,46)(25,38)(27,40)(29,34)(31,36)(33,64)(35,58)(37,60)(39,62), (1,41)(2,42)(3,43)(4,44)(5,45)(6,46)(7,47)(8,48)(9,28)(10,29)(11,30)(12,31)(13,32)(14,25)(15,26)(16,27)(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), (1,56)(2,49)(3,50)(4,51)(5,52)(6,53)(7,54)(8,55)(9,64)(10,57)(11,58)(12,59)(13,60)(14,61)(15,62)(16,63)(17,32)(18,25)(19,26)(20,27)(21,28)(22,29)(23,30)(24,31)(33,43)(34,44)(35,45)(36,46)(37,47)(38,48)(39,41)(40,42), (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), (2,48)(3,7)(4,46)(6,44)(8,42)(9,32)(10,12)(11,30)(13,28)(14,16)(15,26)(17,64)(18,20)(19,62)(21,60)(22,24)(23,58)(25,27)(29,31)(33,37)(34,53)(36,51)(38,49)(40,55)(43,47)(50,54)(57,59)(61,63)>;
G:=Group( (1,26)(2,63)(3,28)(4,57)(5,30)(6,59)(7,32)(8,61)(9,43)(10,51)(11,45)(12,53)(13,47)(14,55)(15,41)(16,49)(17,54)(18,48)(19,56)(20,42)(21,50)(22,44)(23,52)(24,46)(25,38)(27,40)(29,34)(31,36)(33,64)(35,58)(37,60)(39,62), (1,41)(2,42)(3,43)(4,44)(5,45)(6,46)(7,47)(8,48)(9,28)(10,29)(11,30)(12,31)(13,32)(14,25)(15,26)(16,27)(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), (1,56)(2,49)(3,50)(4,51)(5,52)(6,53)(7,54)(8,55)(9,64)(10,57)(11,58)(12,59)(13,60)(14,61)(15,62)(16,63)(17,32)(18,25)(19,26)(20,27)(21,28)(22,29)(23,30)(24,31)(33,43)(34,44)(35,45)(36,46)(37,47)(38,48)(39,41)(40,42), (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), (2,48)(3,7)(4,46)(6,44)(8,42)(9,32)(10,12)(11,30)(13,28)(14,16)(15,26)(17,64)(18,20)(19,62)(21,60)(22,24)(23,58)(25,27)(29,31)(33,37)(34,53)(36,51)(38,49)(40,55)(43,47)(50,54)(57,59)(61,63) );
G=PermutationGroup([[(1,26),(2,63),(3,28),(4,57),(5,30),(6,59),(7,32),(8,61),(9,43),(10,51),(11,45),(12,53),(13,47),(14,55),(15,41),(16,49),(17,54),(18,48),(19,56),(20,42),(21,50),(22,44),(23,52),(24,46),(25,38),(27,40),(29,34),(31,36),(33,64),(35,58),(37,60),(39,62)], [(1,41),(2,42),(3,43),(4,44),(5,45),(6,46),(7,47),(8,48),(9,28),(10,29),(11,30),(12,31),(13,32),(14,25),(15,26),(16,27),(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)], [(1,56),(2,49),(3,50),(4,51),(5,52),(6,53),(7,54),(8,55),(9,64),(10,57),(11,58),(12,59),(13,60),(14,61),(15,62),(16,63),(17,32),(18,25),(19,26),(20,27),(21,28),(22,29),(23,30),(24,31),(33,43),(34,44),(35,45),(36,46),(37,47),(38,48),(39,41),(40,42)], [(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)], [(2,48),(3,7),(4,46),(6,44),(8,42),(9,32),(10,12),(11,30),(13,28),(14,16),(15,26),(17,64),(18,20),(19,62),(21,60),(22,24),(23,58),(25,27),(29,31),(33,37),(34,53),(36,51),(38,49),(40,55),(43,47),(50,54),(57,59),(61,63)]])
32 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 4A | 4B | 4C | 4D | 4E | 4F | 4G | ··· | 4L | 8A | ··· | 8H |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 4 | 4 | 8 | 8 | 2 | 2 | 2 | 2 | 4 | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D4 | D4 | C4○D4 | D8 | C4○D8 | C8⋊C22 | C8.C22 |
| kernel | C24.83D4 | C22.4Q16 | C23.7Q8 | C2×C22⋊C8 | C2×D4⋊C4 | C2×C2.D8 | C2×C4⋊D4 | C4⋊C4 | C2×C8 | C22×C4 | C2×D4 | C24 | C2×C4 | C23 | C22 | C22 | C22 |
| # reps | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 1 | 2 | 1 | 6 | 4 | 4 | 1 | 1 |
Matrix representation of C24.83D4 ►in GL6(𝔽17)
| 16 | 16 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 16 | 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 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 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 |
| 13 | 0 | 0 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 13 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 11 |
| 0 | 0 | 0 | 0 | 3 | 11 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 15 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 | 16 |
G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,16,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[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],[13,0,0,0,0,0,0,13,0,0,0,0,0,0,4,0,0,0,0,0,0,13,0,0,0,0,0,0,0,3,0,0,0,0,11,11],[1,15,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,1,0,0,0,0,0,16] >;
C24.83D4 in GAP, Magma, Sage, TeX
C_2^4._{83}D_4 % in TeX
G:=Group("C2^4.83D4"); // GroupNames label
G:=SmallGroup(128,765);
// by ID
G=gap.SmallGroup(128,765);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,141,422,387,58,2804,718,172]);
// 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,d*a*d^-1=a*b*c,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=b*d^-1>;
// generators/relations