p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.280C23, C4⋊C4○D8, C4⋊C4○Q16, C4○D8⋊6C4, (C4×D8)⋊39C2, D8⋊16(C2×C4), C4⋊C4.403D4, (C4×Q16)⋊39C2, Q16⋊15(C2×C4), SD16⋊8(C2×C4), C2.5(D4○D8), C4.153(C4×D4), C2.5(Q8○D8), C4.28(C23×C4), C8.23(C22×C4), C22.21(C4×D4), SD16⋊C4⋊4C2, C4⋊C4.368C23, C8○2M4(2)⋊8C2, (C2×C4).208C24, (C2×C8).419C23, (C4×C8).222C22, C22⋊C4.190D4, D4.10(C22×C4), (C4×D4).59C22, C23.440(C2×D4), Q8.10(C22×C4), (C4×Q8).55C22, (C2×D8).173C22, (C2×D4).376C23, (C2×Q8).349C23, C2.D8.214C22, C8⋊C4.115C22, C23.36D4⋊38C2, (C22×C4).929C23, (C22×C8).251C22, (C2×Q16).169C22, C22.152(C22×D4), D4⋊C4.198C22, C23.33C23⋊5C2, Q8⋊C4.199C22, (C2×SD16).112C22, C42⋊C2.299C22, (C2×M4(2)).355C22, C2.68(C2×C4×D4), (C2×C8)⋊16(C2×C4), C4○D4⋊5(C2×C4), (C2×C2.D8)⋊39C2, C4.16(C2×C4○D4), (C2×C4○D8).15C2, (C2×C4).915(C2×D4), (C2×C4).267(C4○D4), (C2×C4⋊C4).576C22, (C2×C4).267(C22×C4), (C2×C4○D4).89C22, SmallGroup(128,1683)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.280C23
G = < a,b,c,d,e | a4=b4=d2=1, c2=a2, e2=b2, ab=ba, ac=ca, ad=da, eae-1=ab2, cbc-1=dbd=b-1, be=eb, dcd=bc, ece-1=b2c, de=ed >
Subgroups: 404 in 242 conjugacy classes, 140 normal (24 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C8, C8, C2×C4, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), D8, SD16, Q16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C4○D4, C4×C8, C8⋊C4, D4⋊C4, Q8⋊C4, C2.D8, C2×C4⋊C4, C2×C4⋊C4, C42⋊C2, C42⋊C2, C4×D4, C4×D4, C4×Q8, C22×C8, C2×M4(2), C2×D8, C2×SD16, C2×Q16, C4○D8, C2×C4○D4, C8○2M4(2), C23.36D4, C2×C2.D8, C4×D8, C4×Q16, SD16⋊C4, C23.33C23, C2×C4○D8, C42.280C23
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22×C4, C2×D4, C4○D4, C24, C4×D4, C23×C4, C22×D4, C2×C4○D4, C2×C4×D4, D4○D8, Q8○D8, C42.280C23
►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 15 27 23)(2 16 28 24)(3 13 25 21)(4 14 26 22)(5 11 35 31)(6 12 36 32)(7 9 33 29)(8 10 34 30)(17 42 49 62)(18 43 50 63)(19 44 51 64)(20 41 52 61)(37 53 57 45)(38 54 58 46)(39 55 59 47)(40 56 60 48)
(1 37 3 39)(2 38 4 40)(5 52 7 50)(6 49 8 51)(9 43 11 41)(10 44 12 42)(13 47 15 45)(14 48 16 46)(17 34 19 36)(18 35 20 33)(21 55 23 53)(22 56 24 54)(25 59 27 57)(26 60 28 58)(29 63 31 61)(30 64 32 62)
(1 3)(2 4)(5 33)(6 34)(7 35)(8 36)(9 11)(10 12)(13 23)(14 24)(15 21)(16 22)(17 44)(18 41)(19 42)(20 43)(25 27)(26 28)(29 31)(30 32)(37 47)(38 48)(39 45)(40 46)(49 64)(50 61)(51 62)(52 63)(53 59)(54 60)(55 57)(56 58)
(1 31 27 11)(2 12 28 32)(3 29 25 9)(4 10 26 30)(5 23 35 15)(6 16 36 24)(7 21 33 13)(8 14 34 22)(17 46 49 54)(18 55 50 47)(19 48 51 56)(20 53 52 45)(37 41 57 61)(38 62 58 42)(39 43 59 63)(40 64 60 44)
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), (1,15,27,23)(2,16,28,24)(3,13,25,21)(4,14,26,22)(5,11,35,31)(6,12,36,32)(7,9,33,29)(8,10,34,30)(17,42,49,62)(18,43,50,63)(19,44,51,64)(20,41,52,61)(37,53,57,45)(38,54,58,46)(39,55,59,47)(40,56,60,48), (1,37,3,39)(2,38,4,40)(5,52,7,50)(6,49,8,51)(9,43,11,41)(10,44,12,42)(13,47,15,45)(14,48,16,46)(17,34,19,36)(18,35,20,33)(21,55,23,53)(22,56,24,54)(25,59,27,57)(26,60,28,58)(29,63,31,61)(30,64,32,62), (1,3)(2,4)(5,33)(6,34)(7,35)(8,36)(9,11)(10,12)(13,23)(14,24)(15,21)(16,22)(17,44)(18,41)(19,42)(20,43)(25,27)(26,28)(29,31)(30,32)(37,47)(38,48)(39,45)(40,46)(49,64)(50,61)(51,62)(52,63)(53,59)(54,60)(55,57)(56,58), (1,31,27,11)(2,12,28,32)(3,29,25,9)(4,10,26,30)(5,23,35,15)(6,16,36,24)(7,21,33,13)(8,14,34,22)(17,46,49,54)(18,55,50,47)(19,48,51,56)(20,53,52,45)(37,41,57,61)(38,62,58,42)(39,43,59,63)(40,64,60,44)>;
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), (1,15,27,23)(2,16,28,24)(3,13,25,21)(4,14,26,22)(5,11,35,31)(6,12,36,32)(7,9,33,29)(8,10,34,30)(17,42,49,62)(18,43,50,63)(19,44,51,64)(20,41,52,61)(37,53,57,45)(38,54,58,46)(39,55,59,47)(40,56,60,48), (1,37,3,39)(2,38,4,40)(5,52,7,50)(6,49,8,51)(9,43,11,41)(10,44,12,42)(13,47,15,45)(14,48,16,46)(17,34,19,36)(18,35,20,33)(21,55,23,53)(22,56,24,54)(25,59,27,57)(26,60,28,58)(29,63,31,61)(30,64,32,62), (1,3)(2,4)(5,33)(6,34)(7,35)(8,36)(9,11)(10,12)(13,23)(14,24)(15,21)(16,22)(17,44)(18,41)(19,42)(20,43)(25,27)(26,28)(29,31)(30,32)(37,47)(38,48)(39,45)(40,46)(49,64)(50,61)(51,62)(52,63)(53,59)(54,60)(55,57)(56,58), (1,31,27,11)(2,12,28,32)(3,29,25,9)(4,10,26,30)(5,23,35,15)(6,16,36,24)(7,21,33,13)(8,14,34,22)(17,46,49,54)(18,55,50,47)(19,48,51,56)(20,53,52,45)(37,41,57,61)(38,62,58,42)(39,43,59,63)(40,64,60,44) );
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)], [(1,15,27,23),(2,16,28,24),(3,13,25,21),(4,14,26,22),(5,11,35,31),(6,12,36,32),(7,9,33,29),(8,10,34,30),(17,42,49,62),(18,43,50,63),(19,44,51,64),(20,41,52,61),(37,53,57,45),(38,54,58,46),(39,55,59,47),(40,56,60,48)], [(1,37,3,39),(2,38,4,40),(5,52,7,50),(6,49,8,51),(9,43,11,41),(10,44,12,42),(13,47,15,45),(14,48,16,46),(17,34,19,36),(18,35,20,33),(21,55,23,53),(22,56,24,54),(25,59,27,57),(26,60,28,58),(29,63,31,61),(30,64,32,62)], [(1,3),(2,4),(5,33),(6,34),(7,35),(8,36),(9,11),(10,12),(13,23),(14,24),(15,21),(16,22),(17,44),(18,41),(19,42),(20,43),(25,27),(26,28),(29,31),(30,32),(37,47),(38,48),(39,45),(40,46),(49,64),(50,61),(51,62),(52,63),(53,59),(54,60),(55,57),(56,58)], [(1,31,27,11),(2,12,28,32),(3,29,25,9),(4,10,26,30),(5,23,35,15),(6,16,36,24),(7,21,33,13),(8,14,34,22),(17,46,49,54),(18,55,50,47),(19,48,51,56),(20,53,52,45),(37,41,57,61),(38,62,58,42),(39,43,59,63),(40,64,60,44)]])
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | ··· | 4L | 4M | ··· | 4X | 8A | 8B | 8C | 8D | 8E | ··· | 8J |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | 8 | 8 | 8 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | D4 | C4○D4 | D4○D8 | Q8○D8 |
| kernel | C42.280C23 | C8○2M4(2) | C23.36D4 | C2×C2.D8 | C4×D8 | C4×Q16 | SD16⋊C4 | C23.33C23 | C2×C4○D8 | C4○D8 | C22⋊C4 | C4⋊C4 | C2×C4 | C2 | C2 |
| # reps | 1 | 1 | 2 | 1 | 2 | 2 | 4 | 2 | 1 | 16 | 2 | 2 | 4 | 2 | 2 |
Matrix representation of C42.280C23 ►in GL6(𝔽17)
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 15 | 0 |
| 0 | 0 | 0 | 16 | 0 | 15 |
| 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 | 0 | 1 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 4 | 15 | 0 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 14 | 6 | 11 |
| 0 | 0 | 14 | 14 | 11 | 11 |
| 0 | 0 | 0 | 0 | 14 | 3 |
| 0 | 0 | 0 | 0 | 3 | 3 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 13 | 1 | 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 | 0 | 16 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 13 | 0 | 13 | 0 |
| 0 | 0 | 0 | 13 | 0 | 13 |
G:=sub<GL(6,GF(17))| [4,0,0,0,0,0,0,4,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,15,0,1,0,0,0,0,15,0,1],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0],[4,0,0,0,0,0,15,13,0,0,0,0,0,0,3,14,0,0,0,0,14,14,0,0,0,0,6,11,14,3,0,0,11,11,3,3],[16,13,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,13,0,0,0,0,4,0,13,0,0,0,0,13,0,0,0,0,0,0,13] >;
C42.280C23 in GAP, Magma, Sage, TeX
C_4^2._{280}C_2^3 % in TeX
G:=Group("C4^2.280C2^3"); // GroupNames label
G:=SmallGroup(128,1683);
// by ID
G=gap.SmallGroup(128,1683);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,184,2019,521,248,2804,1411,172]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=d^2=1,c^2=a^2,e^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,e*a*e^-1=a*b^2,c*b*c^-1=d*b*d=b^-1,b*e=e*b,d*c*d=b*c,e*c*e^-1=b^2*c,d*e=e*d>;
// generators/relations