p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.284D4, C42.734C23, C4.562- (1+4), D4.Q8⋊2C2, Q8.Q8⋊2C2, C8.5Q8⋊3C2, C4⋊C4.171C23, C4⋊C8.315C22, (C2×C4).430C24, (C4×C8).115C22, (C2×C8).334C23, C23.701(C2×D4), (C22×C4).173D4, C2.D8.39C22, C4.Q8.87C22, (C4×D4).114C22, D4⋊C4.4C22, (C2×D4).176C23, C22.34(C4○D8), C23.48D4⋊6C2, (C4×Q8).111C22, (C2×Q8).164C23, C22.D8.2C2, C42.12C4⋊38C2, C23.47D4⋊30C2, C4⋊D4.199C22, C22⋊C8.182C22, (C2×C42).891C22, C23.46D4.5C2, C22.690(C22×D4), C22⋊Q8.204C22, C2.61(D8⋊C22), C42.78C22⋊9C2, (C22×C4).1095C23, Q8⋊C4.105C22, C4.4D4.158C22, C42.C2.131C22, C23.36C23.25C2, C2.78(C23.38C23), C2.47(C2×C4○D8), (C2×C4).555(C2×D4), (C2×C42.C2)⋊36C2, (C2×C4⋊C4).649C22, SmallGroup(128,1964)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 292 in 167 conjugacy classes, 86 normal (42 characteristic)
C1, C2 [×3], C2 [×3], C4 [×2], C4 [×12], C22, C22 [×2], C22 [×5], C8 [×4], C2×C4 [×6], C2×C4 [×18], D4 [×4], Q8 [×2], C23, C23, C42 [×4], C42, C22⋊C4 [×5], C4⋊C4 [×2], C4⋊C4 [×4], C4⋊C4 [×14], C2×C8 [×4], C22×C4 [×3], C22×C4 [×3], C2×D4, C2×D4, C2×Q8, C4×C8 [×2], C22⋊C8 [×2], D4⋊C4 [×4], Q8⋊C4 [×4], C4⋊C8 [×2], C4.Q8 [×4], C2.D8 [×4], C2×C42, C2×C4⋊C4 [×2], C2×C4⋊C4 [×2], C42⋊C2, C4×D4, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C42.C2, C42.C2 [×4], C42.C2 [×2], C42⋊2C2, C42.12C4, D4.Q8 [×2], Q8.Q8 [×2], C22.D8, C23.46D4, C23.47D4, C23.48D4, C42.78C22 [×2], C8.5Q8 [×2], C2×C42.C2, C23.36C23, C42.284D4
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C24, C4○D8 [×2], C22×D4, 2- (1+4) [×2], C23.38C23, C2×C4○D8, D8⋊C22, C42.284D4
Generators and relations
G = < a,b,c,d | a4=b4=d2=1, c4=b2, ab=ba, ac=ca, dad=a-1b2, cbc-1=a2b, dbd=a2b-1, dcd=a2b2c3 >
►On 64 points
(1 45 29 63)(2 46 30 64)(3 47 31 57)(4 48 32 58)(5 41 25 59)(6 42 26 60)(7 43 27 61)(8 44 28 62)(9 50 39 24)(10 51 40 17)(11 52 33 18)(12 53 34 19)(13 54 35 20)(14 55 36 21)(15 56 37 22)(16 49 38 23)
(1 11 5 15)(2 34 6 38)(3 13 7 9)(4 36 8 40)(10 32 14 28)(12 26 16 30)(17 48 21 44)(18 59 22 63)(19 42 23 46)(20 61 24 57)(25 37 29 33)(27 39 31 35)(41 56 45 52)(43 50 47 54)(49 64 53 60)(51 58 55 62)
(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 39)(10 16)(11 37)(12 14)(13 35)(15 33)(17 53)(19 51)(20 24)(21 49)(23 55)(27 31)(34 36)(38 40)(41 63)(42 44)(43 61)(45 59)(46 48)(47 57)(50 54)(58 64)(60 62)
G:=sub<Sym(64)| (1,45,29,63)(2,46,30,64)(3,47,31,57)(4,48,32,58)(5,41,25,59)(6,42,26,60)(7,43,27,61)(8,44,28,62)(9,50,39,24)(10,51,40,17)(11,52,33,18)(12,53,34,19)(13,54,35,20)(14,55,36,21)(15,56,37,22)(16,49,38,23), (1,11,5,15)(2,34,6,38)(3,13,7,9)(4,36,8,40)(10,32,14,28)(12,26,16,30)(17,48,21,44)(18,59,22,63)(19,42,23,46)(20,61,24,57)(25,37,29,33)(27,39,31,35)(41,56,45,52)(43,50,47,54)(49,64,53,60)(51,58,55,62), (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)(9,39)(10,16)(11,37)(12,14)(13,35)(15,33)(17,53)(19,51)(20,24)(21,49)(23,55)(27,31)(34,36)(38,40)(41,63)(42,44)(43,61)(45,59)(46,48)(47,57)(50,54)(58,64)(60,62)>;
G:=Group( (1,45,29,63)(2,46,30,64)(3,47,31,57)(4,48,32,58)(5,41,25,59)(6,42,26,60)(7,43,27,61)(8,44,28,62)(9,50,39,24)(10,51,40,17)(11,52,33,18)(12,53,34,19)(13,54,35,20)(14,55,36,21)(15,56,37,22)(16,49,38,23), (1,11,5,15)(2,34,6,38)(3,13,7,9)(4,36,8,40)(10,32,14,28)(12,26,16,30)(17,48,21,44)(18,59,22,63)(19,42,23,46)(20,61,24,57)(25,37,29,33)(27,39,31,35)(41,56,45,52)(43,50,47,54)(49,64,53,60)(51,58,55,62), (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)(9,39)(10,16)(11,37)(12,14)(13,35)(15,33)(17,53)(19,51)(20,24)(21,49)(23,55)(27,31)(34,36)(38,40)(41,63)(42,44)(43,61)(45,59)(46,48)(47,57)(50,54)(58,64)(60,62) );
G=PermutationGroup([(1,45,29,63),(2,46,30,64),(3,47,31,57),(4,48,32,58),(5,41,25,59),(6,42,26,60),(7,43,27,61),(8,44,28,62),(9,50,39,24),(10,51,40,17),(11,52,33,18),(12,53,34,19),(13,54,35,20),(14,55,36,21),(15,56,37,22),(16,49,38,23)], [(1,11,5,15),(2,34,6,38),(3,13,7,9),(4,36,8,40),(10,32,14,28),(12,26,16,30),(17,48,21,44),(18,59,22,63),(19,42,23,46),(20,61,24,57),(25,37,29,33),(27,39,31,35),(41,56,45,52),(43,50,47,54),(49,64,53,60),(51,58,55,62)], [(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),(9,39),(10,16),(11,37),(12,14),(13,35),(15,33),(17,53),(19,51),(20,24),(21,49),(23,55),(27,31),(34,36),(38,40),(41,63),(42,44),(43,61),(45,59),(46,48),(47,57),(50,54),(58,64),(60,62)])
Matrix representation ►G ⊆ GL6(𝔽17)
| 0 | 4 | 0 | 0 | 0 | 0 |
| 13 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 15 |
| 0 | 0 | 1 | 0 | 2 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 14 | 3 | 0 | 0 | 0 | 0 |
| 14 | 14 | 0 | 0 | 0 | 0 |
| 0 | 0 | 10 | 16 | 9 | 9 |
| 0 | 0 | 1 | 10 | 8 | 9 |
| 0 | 0 | 7 | 1 | 7 | 1 |
| 0 | 0 | 16 | 7 | 16 | 7 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 16 | 0 | 16 | 0 |
| 0 | 0 | 0 | 1 | 0 | 1 |
G:=sub<GL(6,GF(17))| [0,13,0,0,0,0,4,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,16,0,0,0,0,0,0,2,0,16,0,0,15,0,1,0],[14,14,0,0,0,0,3,14,0,0,0,0,0,0,10,1,7,16,0,0,16,10,1,7,0,0,9,8,7,16,0,0,9,9,1,7],[1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,16,0,0,0,0,16,0,1,0,0,0,0,16,0,0,0,0,0,0,1] >;
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | ··· | 4H | 4I | 4J | 4K | ··· | 4Q | 8A | ··· | 8H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 2 | ··· | 2 | 4 | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
32 irreducible representations
| dim | 1 | 1 | 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 | C2 | C2 | C2 | D4 | D4 | C4○D8 | 2- (1+4) | D8⋊C22 |
| kernel | C42.284D4 | C42.12C4 | D4.Q8 | Q8.Q8 | C22.D8 | C23.46D4 | C23.47D4 | C23.48D4 | C42.78C22 | C8.5Q8 | C2×C42.C2 | C23.36C23 | C42 | C22×C4 | C22 | C4 | C2 |
| # reps | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 2 | 8 | 2 | 2 |
In GAP, Magma, Sage, TeX
C_4^2._{284}D_4 % in TeX
G:=Group("C4^2.284D4"); // GroupNames label
G:=SmallGroup(128,1964);
// by ID
G=gap.SmallGroup(128,1964);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,568,758,219,100,675,4037,1027,124]);
// 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*b^2,c*b*c^-1=a^2*b,d*b*d=a^2*b^-1,d*c*d=a^2*b^2*c^3>;
// generators/relations