p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.280D4, C42.732C23, C4.522- (1+4), D4.Q8⋊1C2, C8⋊2Q8⋊13C2, C8⋊3Q8⋊21C2, C4.30(C4○D8), C4.4D8⋊14C2, (C4×C8).77C22, C4⋊C8.313C22, C4⋊C4.167C23, C4.29(C8⋊C22), (C2×C8).166C23, (C2×C4).426C24, C23.296(C2×D4), (C22×C4).509D4, C4⋊Q8.310C22, C2.D8.36C22, C4.Q8.84C22, (C4×D4).113C22, (C2×D4).175C23, C23.19D4⋊3C2, C42.12C4⋊37C2, C4⋊1D4.171C22, C4⋊D4.198C22, C22⋊C8.195C22, (C2×C42).887C22, C22.686(C22×D4), D4⋊C4.111C22, (C22×C4).1091C23, C42.C2.129C22, C23.37C23⋊20C2, C42⋊C2.163C22, C22.26C24.45C2, C2.74(C23.38C23), C2.45(C2×C4○D8), (C2×C4).553(C2×D4), C2.60(C2×C8⋊C22), SmallGroup(128,1960)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 364 in 185 conjugacy classes, 88 normal (34 characteristic)
C1, C2 [×3], C2 [×3], C4 [×6], C4 [×9], C22, C22 [×9], C8 [×4], C2×C4 [×6], C2×C4 [×16], D4 [×12], Q8 [×6], C23, C23 [×2], C42 [×4], C42 [×2], C22⋊C4 [×6], C4⋊C4 [×6], C4⋊C4 [×7], C2×C8 [×4], C22×C4 [×3], C22×C4 [×2], C2×D4 [×2], C2×D4 [×4], C2×Q8 [×3], C4○D4 [×4], C4×C8 [×2], C22⋊C8 [×2], D4⋊C4 [×8], C4⋊C8 [×2], C4.Q8 [×4], C2.D8 [×4], C2×C42, C42⋊C2 [×2], C4×D4 [×2], C4×D4, C4×Q8 [×2], C4⋊D4 [×2], C4⋊D4, C22⋊Q8 [×2], C4.4D4, C42.C2 [×2], C4⋊1D4, C4⋊Q8 [×3], C2×C4○D4, C42.12C4, D4.Q8 [×4], C23.19D4 [×4], C4.4D8 [×2], C8⋊3Q8, C8⋊2Q8, C22.26C24, C23.37C23, C42.280D4
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C24, C4○D8 [×2], C8⋊C22 [×2], C22×D4, 2- (1+4) [×2], C23.38C23, C2×C4○D8, C2×C8⋊C22, C42.280D4
Generators and relations
G = < a,b,c,d | a4=b4=d2=1, c4=b2, ab=ba, ac=ca, dad=a-1, cbc-1=dbd=a2b, dcd=a2b2c3 >
►On 64 points
(1 61 29 41)(2 62 30 42)(3 63 31 43)(4 64 32 44)(5 57 25 45)(6 58 26 46)(7 59 27 47)(8 60 28 48)(9 20 37 56)(10 21 38 49)(11 22 39 50)(12 23 40 51)(13 24 33 52)(14 17 34 53)(15 18 35 54)(16 19 36 55)
(1 39 5 35)(2 12 6 16)(3 33 7 37)(4 14 8 10)(9 31 13 27)(11 25 15 29)(17 60 21 64)(18 41 22 45)(19 62 23 58)(20 43 24 47)(26 36 30 40)(28 38 32 34)(42 51 46 55)(44 53 48 49)(50 57 54 61)(52 59 56 63)
(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 28)(3 7)(4 26)(6 32)(8 30)(9 33)(10 12)(11 39)(13 37)(14 16)(15 35)(17 55)(19 53)(20 24)(21 51)(23 49)(27 31)(34 36)(38 40)(41 61)(42 48)(43 59)(44 46)(45 57)(47 63)(52 56)(58 64)(60 62)
G:=sub<Sym(64)| (1,61,29,41)(2,62,30,42)(3,63,31,43)(4,64,32,44)(5,57,25,45)(6,58,26,46)(7,59,27,47)(8,60,28,48)(9,20,37,56)(10,21,38,49)(11,22,39,50)(12,23,40,51)(13,24,33,52)(14,17,34,53)(15,18,35,54)(16,19,36,55), (1,39,5,35)(2,12,6,16)(3,33,7,37)(4,14,8,10)(9,31,13,27)(11,25,15,29)(17,60,21,64)(18,41,22,45)(19,62,23,58)(20,43,24,47)(26,36,30,40)(28,38,32,34)(42,51,46,55)(44,53,48,49)(50,57,54,61)(52,59,56,63), (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), (2,28)(3,7)(4,26)(6,32)(8,30)(9,33)(10,12)(11,39)(13,37)(14,16)(15,35)(17,55)(19,53)(20,24)(21,51)(23,49)(27,31)(34,36)(38,40)(41,61)(42,48)(43,59)(44,46)(45,57)(47,63)(52,56)(58,64)(60,62)>;
G:=Group( (1,61,29,41)(2,62,30,42)(3,63,31,43)(4,64,32,44)(5,57,25,45)(6,58,26,46)(7,59,27,47)(8,60,28,48)(9,20,37,56)(10,21,38,49)(11,22,39,50)(12,23,40,51)(13,24,33,52)(14,17,34,53)(15,18,35,54)(16,19,36,55), (1,39,5,35)(2,12,6,16)(3,33,7,37)(4,14,8,10)(9,31,13,27)(11,25,15,29)(17,60,21,64)(18,41,22,45)(19,62,23,58)(20,43,24,47)(26,36,30,40)(28,38,32,34)(42,51,46,55)(44,53,48,49)(50,57,54,61)(52,59,56,63), (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), (2,28)(3,7)(4,26)(6,32)(8,30)(9,33)(10,12)(11,39)(13,37)(14,16)(15,35)(17,55)(19,53)(20,24)(21,51)(23,49)(27,31)(34,36)(38,40)(41,61)(42,48)(43,59)(44,46)(45,57)(47,63)(52,56)(58,64)(60,62) );
G=PermutationGroup([(1,61,29,41),(2,62,30,42),(3,63,31,43),(4,64,32,44),(5,57,25,45),(6,58,26,46),(7,59,27,47),(8,60,28,48),(9,20,37,56),(10,21,38,49),(11,22,39,50),(12,23,40,51),(13,24,33,52),(14,17,34,53),(15,18,35,54),(16,19,36,55)], [(1,39,5,35),(2,12,6,16),(3,33,7,37),(4,14,8,10),(9,31,13,27),(11,25,15,29),(17,60,21,64),(18,41,22,45),(19,62,23,58),(20,43,24,47),(26,36,30,40),(28,38,32,34),(42,51,46,55),(44,53,48,49),(50,57,54,61),(52,59,56,63)], [(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)], [(2,28),(3,7),(4,26),(6,32),(8,30),(9,33),(10,12),(11,39),(13,37),(14,16),(15,35),(17,55),(19,53),(20,24),(21,51),(23,49),(27,31),(34,36),(38,40),(41,61),(42,48),(43,59),(44,46),(45,57),(47,63),(52,56),(58,64),(60,62)])
Matrix representation ►G ⊆ GL6(𝔽17)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 16 | 16 | 2 |
| 0 | 0 | 1 | 0 | 16 | 1 |
| 13 | 0 | 0 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 16 | 1 | 1 | 15 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 16 | 1 | 0 | 16 |
| 3 | 3 | 0 | 0 | 0 | 0 |
| 14 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 12 | 0 | 0 |
| 0 | 0 | 5 | 5 | 0 | 0 |
| 0 | 0 | 12 | 5 | 0 | 7 |
| 0 | 0 | 12 | 10 | 5 | 7 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 | 16 | 1 |
G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,1,1,0,0,16,0,16,0,0,0,0,0,16,16,0,0,0,0,2,1],[13,0,0,0,0,0,0,13,0,0,0,0,0,0,0,16,16,16,0,0,0,1,0,1,0,0,1,1,0,0,0,0,0,15,0,16],[3,14,0,0,0,0,3,3,0,0,0,0,0,0,5,5,12,12,0,0,12,5,5,10,0,0,0,0,0,5,0,0,0,0,7,7],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,16,0,0,0,0,16,16,0,0,0,0,0,1] >;
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | ··· | 4J | 4K | 4L | ··· | 4Q | 8A | ··· | 8H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 4 | 8 | 8 | 2 | ··· | 2 | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
32 irreducible representations
| dim | 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 | D4 | D4 | C4○D8 | C8⋊C22 | 2- (1+4) |
| kernel | C42.280D4 | C42.12C4 | D4.Q8 | C23.19D4 | C4.4D8 | C8⋊3Q8 | C8⋊2Q8 | C22.26C24 | C23.37C23 | C42 | C22×C4 | C4 | C4 | C4 |
| # reps | 1 | 1 | 4 | 4 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 2 | 2 |
In GAP, Magma, Sage, TeX
C_4^2._{280}D_4 % in TeX
G:=Group("C4^2.280D4"); // GroupNames label
G:=SmallGroup(128,1960);
// by ID
G=gap.SmallGroup(128,1960);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,120,758,219,100,675,248,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=d^2=1,c^4=b^2,a*b=b*a,a*c=c*a,d*a*d=a^-1,c*b*c^-1=d*b*d=a^2*b,d*c*d=a^2*b^2*c^3>;
// generators/relations