p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.505C23, C4.262- 1+4, (C8×Q8)⋊12C2, C4⋊C4.278D4, Q8⋊3Q8⋊3C2, (C4×Q16)⋊16C2, Q8.Q8⋊10C2, C4⋊2Q16⋊16C2, C4.76(C4○D8), (C4×C8).94C22, (C2×Q8).184D4, C2.62(Q8○D8), C4⋊C8.325C22, C4⋊C4.432C23, (C2×C8).207C23, (C2×C4).556C24, Q8.34(C4○D4), D4⋊Q8.10C2, C4.SD16⋊32C2, (C4×SD16).13C2, C4⋊Q8.185C22, Q8.D4.1C2, C2.64(Q8⋊5D4), (C2×D4).268C23, (C4×D4).195C22, (C2×Q8).254C23, (C4×Q8).187C22, C4.Q8.176C22, C2.D8.202C22, Q8⋊C4.19C22, (C2×Q16).142C22, C4.4D4.76C22, C22.816(C22×D4), C42.C2.61C22, D4⋊C4.152C22, (C2×SD16).171C22, C22.50C24.8C2, C42.78C22.2C2, C2.74(C2×C4○D8), C4.257(C2×C4○D4), (C2×C4).176(C2×D4), SmallGroup(128,2096)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.505C23
G = < a,b,c,d,e | a4=b4=c2=1, d2=a2b2, e2=b2, ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc=ebe-1=b-1, bd=db, dcd-1=a2c, ece-1=bc, de=ed >
Subgroups: 280 in 168 conjugacy classes, 88 normal (38 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, Q8, Q8, C23, C42, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, SD16, Q16, C22×C4, C2×D4, C2×Q8, C2×Q8, C2×Q8, C4×C8, C4×C8, D4⋊C4, D4⋊C4, Q8⋊C4, Q8⋊C4, C4⋊C8, C4⋊C8, C4.Q8, C2.D8, C42⋊C2, C4×D4, C4×Q8, C4×Q8, C4×Q8, C22⋊Q8, C4.4D4, C42.C2, C42.C2, C42⋊2C2, C4⋊Q8, C4⋊Q8, C2×SD16, C2×Q16, C4×SD16, C4×Q16, C8×Q8, C4⋊2Q16, Q8.D4, D4⋊Q8, Q8.Q8, C4.SD16, C42.78C22, C22.50C24, Q8⋊3Q8, C42.505C23
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C4○D8, C22×D4, C2×C4○D4, 2- 1+4, Q8⋊5D4, C2×C4○D8, Q8○D8, C42.505C23
►On 64 points
(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 23 25 17)(2 24 26 18)(3 21 27 19)(4 22 28 20)(5 15 9 64)(6 16 10 61)(7 13 11 62)(8 14 12 63)(29 37 41 35)(30 38 42 36)(31 39 43 33)(32 40 44 34)(45 53 51 57)(46 54 52 58)(47 55 49 59)(48 56 50 60)
(5 13)(6 14)(7 15)(8 16)(9 62)(10 63)(11 64)(12 61)(17 23)(18 24)(19 21)(20 22)(29 37)(30 38)(31 39)(32 40)(33 43)(34 44)(35 41)(36 42)(45 49)(46 50)(47 51)(48 52)(53 55)(54 56)(57 59)(58 60)
(1 57 27 55)(2 60 28 54)(3 59 25 53)(4 58 26 56)(5 44 11 30)(6 43 12 29)(7 42 9 32)(8 41 10 31)(13 36 64 40)(14 35 61 39)(15 34 62 38)(16 33 63 37)(17 51 21 47)(18 50 22 46)(19 49 23 45)(20 52 24 48)
(1 41 25 29)(2 42 26 30)(3 43 27 31)(4 44 28 32)(5 60 9 56)(6 57 10 53)(7 58 11 54)(8 59 12 55)(13 52 62 46)(14 49 63 47)(15 50 64 48)(16 51 61 45)(17 35 23 37)(18 36 24 38)(19 33 21 39)(20 34 22 40)
G:=sub<Sym(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), (1,23,25,17)(2,24,26,18)(3,21,27,19)(4,22,28,20)(5,15,9,64)(6,16,10,61)(7,13,11,62)(8,14,12,63)(29,37,41,35)(30,38,42,36)(31,39,43,33)(32,40,44,34)(45,53,51,57)(46,54,52,58)(47,55,49,59)(48,56,50,60), (5,13)(6,14)(7,15)(8,16)(9,62)(10,63)(11,64)(12,61)(17,23)(18,24)(19,21)(20,22)(29,37)(30,38)(31,39)(32,40)(33,43)(34,44)(35,41)(36,42)(45,49)(46,50)(47,51)(48,52)(53,55)(54,56)(57,59)(58,60), (1,57,27,55)(2,60,28,54)(3,59,25,53)(4,58,26,56)(5,44,11,30)(6,43,12,29)(7,42,9,32)(8,41,10,31)(13,36,64,40)(14,35,61,39)(15,34,62,38)(16,33,63,37)(17,51,21,47)(18,50,22,46)(19,49,23,45)(20,52,24,48), (1,41,25,29)(2,42,26,30)(3,43,27,31)(4,44,28,32)(5,60,9,56)(6,57,10,53)(7,58,11,54)(8,59,12,55)(13,52,62,46)(14,49,63,47)(15,50,64,48)(16,51,61,45)(17,35,23,37)(18,36,24,38)(19,33,21,39)(20,34,22,40)>;
G:=Group( (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), (1,23,25,17)(2,24,26,18)(3,21,27,19)(4,22,28,20)(5,15,9,64)(6,16,10,61)(7,13,11,62)(8,14,12,63)(29,37,41,35)(30,38,42,36)(31,39,43,33)(32,40,44,34)(45,53,51,57)(46,54,52,58)(47,55,49,59)(48,56,50,60), (5,13)(6,14)(7,15)(8,16)(9,62)(10,63)(11,64)(12,61)(17,23)(18,24)(19,21)(20,22)(29,37)(30,38)(31,39)(32,40)(33,43)(34,44)(35,41)(36,42)(45,49)(46,50)(47,51)(48,52)(53,55)(54,56)(57,59)(58,60), (1,57,27,55)(2,60,28,54)(3,59,25,53)(4,58,26,56)(5,44,11,30)(6,43,12,29)(7,42,9,32)(8,41,10,31)(13,36,64,40)(14,35,61,39)(15,34,62,38)(16,33,63,37)(17,51,21,47)(18,50,22,46)(19,49,23,45)(20,52,24,48), (1,41,25,29)(2,42,26,30)(3,43,27,31)(4,44,28,32)(5,60,9,56)(6,57,10,53)(7,58,11,54)(8,59,12,55)(13,52,62,46)(14,49,63,47)(15,50,64,48)(16,51,61,45)(17,35,23,37)(18,36,24,38)(19,33,21,39)(20,34,22,40) );
G=PermutationGroup([[(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)], [(1,23,25,17),(2,24,26,18),(3,21,27,19),(4,22,28,20),(5,15,9,64),(6,16,10,61),(7,13,11,62),(8,14,12,63),(29,37,41,35),(30,38,42,36),(31,39,43,33),(32,40,44,34),(45,53,51,57),(46,54,52,58),(47,55,49,59),(48,56,50,60)], [(5,13),(6,14),(7,15),(8,16),(9,62),(10,63),(11,64),(12,61),(17,23),(18,24),(19,21),(20,22),(29,37),(30,38),(31,39),(32,40),(33,43),(34,44),(35,41),(36,42),(45,49),(46,50),(47,51),(48,52),(53,55),(54,56),(57,59),(58,60)], [(1,57,27,55),(2,60,28,54),(3,59,25,53),(4,58,26,56),(5,44,11,30),(6,43,12,29),(7,42,9,32),(8,41,10,31),(13,36,64,40),(14,35,61,39),(15,34,62,38),(16,33,63,37),(17,51,21,47),(18,50,22,46),(19,49,23,45),(20,52,24,48)], [(1,41,25,29),(2,42,26,30),(3,43,27,31),(4,44,28,32),(5,60,9,56),(6,57,10,53),(7,58,11,54),(8,59,12,55),(13,52,62,46),(14,49,63,47),(15,50,64,48),(16,51,61,45),(17,35,23,37),(18,36,24,38),(19,33,21,39),(20,34,22,40)]])
35 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | ··· | 4H | 4I | ··· | 4O | 4P | ··· | 4T | 8A | 8B | 8C | 8D | 8E | ··· | 8J |
| order | 1 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | 8 | 8 | 8 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 8 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
35 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | C4○D4 | C4○D8 | 2- 1+4 | Q8○D8 |
| kernel | C42.505C23 | C4×SD16 | C4×Q16 | C8×Q8 | C4⋊2Q16 | Q8.D4 | D4⋊Q8 | Q8.Q8 | C4.SD16 | C42.78C22 | C22.50C24 | Q8⋊3Q8 | C4⋊C4 | C2×Q8 | Q8 | C4 | C4 | C2 |
| # reps | 1 | 1 | 2 | 1 | 1 | 2 | 1 | 2 | 1 | 2 | 1 | 1 | 3 | 1 | 4 | 8 | 1 | 2 |
Matrix representation of C42.505C23 ►in GL4(𝔽17) generated by
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 13 | 0 |
| 0 | 0 | 0 | 4 |
| 0 | 1 | 0 | 0 |
| 16 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 16 |
| 13 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 16 | 0 |
| 5 | 5 | 0 | 0 |
| 5 | 12 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(17))| [16,0,0,0,0,16,0,0,0,0,13,0,0,0,0,4],[0,16,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,16,0,0,0,0,1,0,0,0,0,16],[13,0,0,0,0,13,0,0,0,0,0,16,0,0,1,0],[5,5,0,0,5,12,0,0,0,0,1,0,0,0,0,1] >;
C42.505C23 in GAP, Magma, Sage, TeX
C_4^2._{505}C_2^3 % in TeX
G:=Group("C4^2.505C2^3"); // GroupNames label
G:=SmallGroup(128,2096);
// by ID
G=gap.SmallGroup(128,2096);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,568,758,352,346,80,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=c^2=1,d^2=a^2*b^2,e^2=b^2,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,a*e=e*a,c*b*c=e*b*e^-1=b^-1,b*d=d*b,d*c*d^-1=a^2*c,e*c*e^-1=b*c,d*e=e*d>;
// generators/relations