p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.244D4, C42.369C23, C4⋊C4.95C23, (C4×M4(2))⋊43C2, (C2×C4).340C24, (C4×C8).353C22, (C2×C8).461C23, C23.682(C2×D4), (C22×C4).464D4, C4⋊Q8.277C22, (C2×Q8).95C23, C4.62(C4.4D4), (C2×D4).107C23, C8⋊C4.171C22, C23.36D4⋊46C2, C4⋊1D4.149C22, (C2×C42).851C22, C22.600(C22×D4), D4⋊C4.135C22, C2.37(D8⋊C22), (C22×C4).1038C23, Q8⋊C4.127C22, C4.4D4.138C22, C22.42(C4.4D4), C42.C2.113C22, C42.30C22⋊19C2, C42.29C22⋊19C2, C42.78C22⋊28C2, (C2×M4(2)).377C22, C22.26C24.36C2, C4.49(C2×C4○D4), (C2×C4).518(C2×D4), (C2×C42.C2)⋊35C2, C2.51(C2×C4.4D4), (C2×C4).304(C4○D4), (C2×C4⋊C4).627C22, (C2×C4○D4).152C22, SmallGroup(128,1874)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 372 in 196 conjugacy classes, 92 normal (16 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×4], C4 [×10], C22, C22 [×2], C22 [×8], C8 [×4], C2×C4 [×2], C2×C4 [×8], C2×C4 [×16], D4 [×12], Q8 [×4], C23, C23 [×2], C42 [×2], C42 [×2], C22⋊C4 [×4], C4⋊C4 [×4], C4⋊C4 [×12], C2×C8 [×4], M4(2) [×4], C22×C4, C22×C4 [×2], C22×C4 [×4], C2×D4 [×2], C2×D4 [×4], C2×Q8 [×2], C4○D4 [×8], C4×C8 [×2], C8⋊C4 [×2], D4⋊C4 [×8], Q8⋊C4 [×8], C2×C42, C2×C4⋊C4 [×2], C2×C4⋊C4 [×2], C4×D4 [×2], C4⋊D4 [×2], C4.4D4 [×2], C42.C2 [×4], C42.C2 [×2], C4⋊1D4, C4⋊Q8, C2×M4(2) [×2], C2×C4○D4 [×2], C4×M4(2), C23.36D4 [×4], C42.78C22 [×4], C42.29C22 [×2], C42.30C22 [×2], C2×C42.C2, C22.26C24, C42.244D4
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×4], C24, C4.4D4 [×4], C22×D4, C2×C4○D4 [×2], C2×C4.4D4, D8⋊C22 [×2], C42.244D4
Generators and relations
G = < a,b,c,d | a4=b4=1, c4=b2, d2=a2, ab=ba, ac=ca, dad-1=a-1b2, cbc-1=dbd-1=b-1, dcd-1=a2b2c3 >
►On 64 points
(1 64 51 23)(2 57 52 24)(3 58 53 17)(4 59 54 18)(5 60 55 19)(6 61 56 20)(7 62 49 21)(8 63 50 22)(9 47 36 26)(10 48 37 27)(11 41 38 28)(12 42 39 29)(13 43 40 30)(14 44 33 31)(15 45 34 32)(16 46 35 25)
(1 41 5 45)(2 46 6 42)(3 43 7 47)(4 48 8 44)(9 17 13 21)(10 22 14 18)(11 19 15 23)(12 24 16 20)(25 56 29 52)(26 53 30 49)(27 50 31 54)(28 55 32 51)(33 59 37 63)(34 64 38 60)(35 61 39 57)(36 58 40 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)
(1 54 51 4)(2 3 52 53)(5 50 55 8)(6 7 56 49)(9 12 36 39)(10 38 37 11)(13 16 40 35)(14 34 33 15)(17 20 58 61)(18 60 59 19)(21 24 62 57)(22 64 63 23)(25 26 46 47)(27 32 48 45)(28 44 41 31)(29 30 42 43)
G:=sub<Sym(64)| (1,64,51,23)(2,57,52,24)(3,58,53,17)(4,59,54,18)(5,60,55,19)(6,61,56,20)(7,62,49,21)(8,63,50,22)(9,47,36,26)(10,48,37,27)(11,41,38,28)(12,42,39,29)(13,43,40,30)(14,44,33,31)(15,45,34,32)(16,46,35,25), (1,41,5,45)(2,46,6,42)(3,43,7,47)(4,48,8,44)(9,17,13,21)(10,22,14,18)(11,19,15,23)(12,24,16,20)(25,56,29,52)(26,53,30,49)(27,50,31,54)(28,55,32,51)(33,59,37,63)(34,64,38,60)(35,61,39,57)(36,58,40,62), (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), (1,54,51,4)(2,3,52,53)(5,50,55,8)(6,7,56,49)(9,12,36,39)(10,38,37,11)(13,16,40,35)(14,34,33,15)(17,20,58,61)(18,60,59,19)(21,24,62,57)(22,64,63,23)(25,26,46,47)(27,32,48,45)(28,44,41,31)(29,30,42,43)>;
G:=Group( (1,64,51,23)(2,57,52,24)(3,58,53,17)(4,59,54,18)(5,60,55,19)(6,61,56,20)(7,62,49,21)(8,63,50,22)(9,47,36,26)(10,48,37,27)(11,41,38,28)(12,42,39,29)(13,43,40,30)(14,44,33,31)(15,45,34,32)(16,46,35,25), (1,41,5,45)(2,46,6,42)(3,43,7,47)(4,48,8,44)(9,17,13,21)(10,22,14,18)(11,19,15,23)(12,24,16,20)(25,56,29,52)(26,53,30,49)(27,50,31,54)(28,55,32,51)(33,59,37,63)(34,64,38,60)(35,61,39,57)(36,58,40,62), (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), (1,54,51,4)(2,3,52,53)(5,50,55,8)(6,7,56,49)(9,12,36,39)(10,38,37,11)(13,16,40,35)(14,34,33,15)(17,20,58,61)(18,60,59,19)(21,24,62,57)(22,64,63,23)(25,26,46,47)(27,32,48,45)(28,44,41,31)(29,30,42,43) );
G=PermutationGroup([(1,64,51,23),(2,57,52,24),(3,58,53,17),(4,59,54,18),(5,60,55,19),(6,61,56,20),(7,62,49,21),(8,63,50,22),(9,47,36,26),(10,48,37,27),(11,41,38,28),(12,42,39,29),(13,43,40,30),(14,44,33,31),(15,45,34,32),(16,46,35,25)], [(1,41,5,45),(2,46,6,42),(3,43,7,47),(4,48,8,44),(9,17,13,21),(10,22,14,18),(11,19,15,23),(12,24,16,20),(25,56,29,52),(26,53,30,49),(27,50,31,54),(28,55,32,51),(33,59,37,63),(34,64,38,60),(35,61,39,57),(36,58,40,62)], [(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)], [(1,54,51,4),(2,3,52,53),(5,50,55,8),(6,7,56,49),(9,12,36,39),(10,38,37,11),(13,16,40,35),(14,34,33,15),(17,20,58,61),(18,60,59,19),(21,24,62,57),(22,64,63,23),(25,26,46,47),(27,32,48,45),(28,44,41,31),(29,30,42,43)])
Matrix representation ►G ⊆ GL6(𝔽17)
| 16 | 8 | 0 | 0 | 0 | 0 |
| 4 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 13 | 0 | 0 | 0 |
| 0 | 0 | 0 | 13 | 0 | 0 |
| 0 | 0 | 0 | 0 | 13 | 0 |
| 0 | 0 | 0 | 0 | 0 | 13 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 4 | 2 | 0 | 0 | 0 | 0 |
| 1 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 15 | 2 | 2 | 15 |
| 0 | 0 | 15 | 15 | 2 | 2 |
| 0 | 0 | 2 | 15 | 2 | 15 |
| 0 | 0 | 2 | 2 | 2 | 2 |
| 4 | 2 | 0 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 15 | 15 | 2 |
| 0 | 0 | 15 | 15 | 2 | 2 |
| 0 | 0 | 15 | 2 | 15 | 2 |
| 0 | 0 | 2 | 2 | 2 | 2 |
G:=sub<GL(6,GF(17))| [16,4,0,0,0,0,8,1,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,13],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,1,0,0,0,0,0,0,1,0,0],[4,1,0,0,0,0,2,13,0,0,0,0,0,0,15,15,2,2,0,0,2,15,15,2,0,0,2,2,2,2,0,0,15,2,15,2],[4,0,0,0,0,0,2,13,0,0,0,0,0,0,2,15,15,2,0,0,15,15,2,2,0,0,15,2,15,2,0,0,2,2,2,2] >;
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | ··· | 4H | 4I | 4J | 4K | ··· | 4P | 8A | ··· | 8H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 8 | 2 | ··· | 2 | 4 | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | C4○D4 | D8⋊C22 |
| kernel | C42.244D4 | C4×M4(2) | C23.36D4 | C42.78C22 | C42.29C22 | C42.30C22 | C2×C42.C2 | C22.26C24 | C42 | C22×C4 | C2×C4 | C2 |
| # reps | 1 | 1 | 4 | 4 | 2 | 2 | 1 | 1 | 2 | 2 | 8 | 4 |
In GAP, Magma, Sage, TeX
C_4^2._{244}D_4 % in TeX
G:=Group("C4^2.244D4"); // GroupNames label
G:=SmallGroup(128,1874);
// by ID
G=gap.SmallGroup(128,1874);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,680,758,100,1018,521,248,2804,172,4037,124]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^4=b^2,d^2=a^2,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1*b^2,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=a^2*b^2*c^3>;
// generators/relations