p-group, metabelian, nilpotent (class 2), monomial
Aliases: C42.172D4, C24.333C23, C23.462C24, C22.2472+ (1+4), C22.1882- (1+4), C4.59(C4.4D4), C23.54(C4○D4), (C2×C42).65C22, C23.7Q8⋊69C2, (C22×C4).543C23, (C23×C4).403C22, C22.313(C22×D4), C24.C22⋊88C2, C23.10D4.22C2, (C22×D4).531C22, C23.83C23⋊39C2, C2.63(C22.19C24), C2.C42.198C22, C2.64(C22.46C24), C2.15(C22.31C24), C2.57(C22.47C24), (C4×C4⋊C4)⋊96C2, (C2×C4×D4).62C2, (C2×C4).681(C2×D4), (C2×C42.C2)⋊13C2, C2.24(C2×C4.4D4), (C2×C4).824(C4○D4), (C2×C4⋊C4).311C22, C22.338(C2×C4○D4), (C2×C22⋊C4).185C22, SmallGroup(128,1294)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 500 in 266 conjugacy classes, 104 normal (20 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×4], C4 [×14], C22 [×3], C22 [×4], C22 [×20], C2×C4 [×10], C2×C4 [×42], D4 [×8], C23, C23 [×4], C23 [×12], C42 [×4], C42 [×2], C22⋊C4 [×16], C4⋊C4 [×18], C22×C4 [×3], C22×C4 [×10], C22×C4 [×12], C2×D4 [×6], C24 [×2], C2.C42 [×10], C2×C42, C2×C42 [×2], C2×C22⋊C4 [×10], C2×C4⋊C4 [×3], C2×C4⋊C4 [×6], C4×D4 [×4], C42.C2 [×4], C23×C4 [×2], C22×D4, C4×C4⋊C4, C23.7Q8 [×4], C24.C22 [×4], C23.10D4 [×2], C23.83C23 [×2], C2×C4×D4, C2×C42.C2, C42.172D4
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×8], C24, C4.4D4 [×4], C22×D4, C2×C4○D4 [×4], 2+ (1+4), 2- (1+4), C22.19C24, C2×C4.4D4, C22.31C24, C22.46C24 [×2], C22.47C24 [×2], C42.172D4
Generators and relations
G = < a,b,c,d | a4=b4=c4=1, d2=a2, ab=ba, cac-1=ab2, dad-1=a-1b2, cbc-1=dbd-1=b-1, dcd-1=a2c-1 >
(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 27 14 10)(2 28 15 11)(3 25 16 12)(4 26 13 9)(5 37 24 32)(6 38 21 29)(7 39 22 30)(8 40 23 31)(17 36 62 51)(18 33 63 52)(19 34 64 49)(20 35 61 50)(41 53 60 46)(42 54 57 47)(43 55 58 48)(44 56 59 45)
(1 52 8 42)(2 34 5 58)(3 50 6 44)(4 36 7 60)(9 62 30 46)(10 18 31 54)(11 64 32 48)(12 20 29 56)(13 51 22 41)(14 33 23 57)(15 49 24 43)(16 35 21 59)(17 39 53 26)(19 37 55 28)(25 61 38 45)(27 63 40 47)
(1 34 3 36)(2 52 4 50)(5 42 7 44)(6 60 8 58)(9 20 11 18)(10 64 12 62)(13 35 15 33)(14 49 16 51)(17 27 19 25)(21 41 23 43)(22 59 24 57)(26 61 28 63)(29 46 31 48)(30 56 32 54)(37 47 39 45)(38 53 40 55)
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), (1,27,14,10)(2,28,15,11)(3,25,16,12)(4,26,13,9)(5,37,24,32)(6,38,21,29)(7,39,22,30)(8,40,23,31)(17,36,62,51)(18,33,63,52)(19,34,64,49)(20,35,61,50)(41,53,60,46)(42,54,57,47)(43,55,58,48)(44,56,59,45), (1,52,8,42)(2,34,5,58)(3,50,6,44)(4,36,7,60)(9,62,30,46)(10,18,31,54)(11,64,32,48)(12,20,29,56)(13,51,22,41)(14,33,23,57)(15,49,24,43)(16,35,21,59)(17,39,53,26)(19,37,55,28)(25,61,38,45)(27,63,40,47), (1,34,3,36)(2,52,4,50)(5,42,7,44)(6,60,8,58)(9,20,11,18)(10,64,12,62)(13,35,15,33)(14,49,16,51)(17,27,19,25)(21,41,23,43)(22,59,24,57)(26,61,28,63)(29,46,31,48)(30,56,32,54)(37,47,39,45)(38,53,40,55)>;
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), (1,27,14,10)(2,28,15,11)(3,25,16,12)(4,26,13,9)(5,37,24,32)(6,38,21,29)(7,39,22,30)(8,40,23,31)(17,36,62,51)(18,33,63,52)(19,34,64,49)(20,35,61,50)(41,53,60,46)(42,54,57,47)(43,55,58,48)(44,56,59,45), (1,52,8,42)(2,34,5,58)(3,50,6,44)(4,36,7,60)(9,62,30,46)(10,18,31,54)(11,64,32,48)(12,20,29,56)(13,51,22,41)(14,33,23,57)(15,49,24,43)(16,35,21,59)(17,39,53,26)(19,37,55,28)(25,61,38,45)(27,63,40,47), (1,34,3,36)(2,52,4,50)(5,42,7,44)(6,60,8,58)(9,20,11,18)(10,64,12,62)(13,35,15,33)(14,49,16,51)(17,27,19,25)(21,41,23,43)(22,59,24,57)(26,61,28,63)(29,46,31,48)(30,56,32,54)(37,47,39,45)(38,53,40,55) );
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)], [(1,27,14,10),(2,28,15,11),(3,25,16,12),(4,26,13,9),(5,37,24,32),(6,38,21,29),(7,39,22,30),(8,40,23,31),(17,36,62,51),(18,33,63,52),(19,34,64,49),(20,35,61,50),(41,53,60,46),(42,54,57,47),(43,55,58,48),(44,56,59,45)], [(1,52,8,42),(2,34,5,58),(3,50,6,44),(4,36,7,60),(9,62,30,46),(10,18,31,54),(11,64,32,48),(12,20,29,56),(13,51,22,41),(14,33,23,57),(15,49,24,43),(16,35,21,59),(17,39,53,26),(19,37,55,28),(25,61,38,45),(27,63,40,47)], [(1,34,3,36),(2,52,4,50),(5,42,7,44),(6,60,8,58),(9,20,11,18),(10,64,12,62),(13,35,15,33),(14,49,16,51),(17,27,19,25),(21,41,23,43),(22,59,24,57),(26,61,28,63),(29,46,31,48),(30,56,32,54),(37,47,39,45),(38,53,40,55)])
Matrix representation ►G ⊆ GL6(𝔽5)
| 4 | 0 | 0 | 0 | 0 | 0 |
| 3 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 2 | 0 | 0 |
| 0 | 0 | 4 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 3 | 0 | 0 | 0 | 0 | 0 |
| 1 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 3 | 2 | 0 | 0 | 0 | 0 |
| 1 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 0 | 0 | 0 |
| 0 | 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 2 | 3 | 0 | 0 | 0 | 0 |
| 4 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 4 | 0 | 0 |
| 0 | 0 | 0 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(6,GF(5))| [4,3,0,0,0,0,0,1,0,0,0,0,0,0,4,4,0,0,0,0,2,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[3,1,0,0,0,0,0,2,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[3,1,0,0,0,0,2,2,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,0,4,0,0,0,0,1,0],[2,4,0,0,0,0,3,3,0,0,0,0,0,0,3,0,0,0,0,0,4,2,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;
38 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 4A | ··· | 4H | 4I | ··· | 4V | 4W | 4X | 4Y | 4Z |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 |
| size | 1 | 1 | ··· | 1 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | C4○D4 | C4○D4 | 2+ (1+4) | 2- (1+4) |
| kernel | C42.172D4 | C4×C4⋊C4 | C23.7Q8 | C24.C22 | C23.10D4 | C23.83C23 | C2×C4×D4 | C2×C42.C2 | C42 | C2×C4 | C23 | C22 | C22 |
| # reps | 1 | 1 | 4 | 4 | 2 | 2 | 1 | 1 | 4 | 8 | 8 | 1 | 1 |
In GAP, Magma, Sage, TeX
C_4^2._{172}D_4 % in TeX
G:=Group("C4^2.172D4"); // GroupNames label
G:=SmallGroup(128,1294);
// by ID
G=gap.SmallGroup(128,1294);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,672,253,456,758,723,675,80]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^4=1,d^2=a^2,a*b=b*a,c*a*c^-1=a*b^2,d*a*d^-1=a^-1*b^2,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=a^2*c^-1>;
// generators/relations