p-group, metabelian, nilpotent (class 2), monomial
Aliases: C43⋊13C2, C42⋊37D4, C24.124C23, C23.760C24, C4⋊1(C42⋊2C2), C22.470(C22×D4), (C2×C42).1095C22, (C22×C4).1266C23, (C22×D4).315C22, C24.C22⋊186C2, C24.3C22.81C2, C23.65C23⋊171C2, C2.C42.455C22, C2.60(C22.26C24), C2.114(C23.36C23), (C2×C4).689(C2×D4), (C2×C42⋊2C2)⋊30C2, (C2×C4).530(C4○D4), (C2×C4⋊C4).563C22, C2.26(C2×C42⋊2C2), C22.601(C2×C4○D4), (C2×C22⋊C4).369C22, SmallGroup(128,1592)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 468 in 258 conjugacy classes, 108 normal (9 characteristic)
C1, C2, C2 [×6], C2 [×2], C4 [×4], C4 [×18], C22, C22 [×6], C22 [×14], C2×C4 [×18], C2×C4 [×30], D4 [×4], C23, C23 [×14], C42 [×4], C42 [×12], C22⋊C4 [×24], C4⋊C4 [×18], C22×C4, C22×C4 [×12], C2×D4 [×6], C24 [×2], C2.C42 [×6], C2×C42, C2×C42 [×6], C2×C22⋊C4 [×12], C2×C4⋊C4 [×9], C42⋊2C2 [×8], C22×D4, C43, C24.C22 [×6], C23.65C23 [×3], C24.3C22 [×3], C2×C42⋊2C2 [×2], C43⋊13C2
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×12], C24, C42⋊2C2 [×4], C22×D4, C2×C4○D4 [×6], C2×C42⋊2C2, C23.36C23 [×3], C22.26C24 [×3], C43⋊13C2
Generators and relations
G = < a,b,c,d | a4=b4=c4=d2=1, ab=ba, ac=ca, dad=ac2, bc=cb, dbd=b-1, dcd=a2c-1 >
►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 5 58 52)(2 6 59 49)(3 7 60 50)(4 8 57 51)(9 18 34 28)(10 19 35 25)(11 20 36 26)(12 17 33 27)(13 37 31 21)(14 38 32 22)(15 39 29 23)(16 40 30 24)(41 46 64 56)(42 47 61 53)(43 48 62 54)(44 45 63 55)
(1 20 45 13)(2 17 46 14)(3 18 47 15)(4 19 48 16)(5 36 63 37)(6 33 64 38)(7 34 61 39)(8 35 62 40)(9 42 23 50)(10 43 24 51)(11 44 21 52)(12 41 22 49)(25 54 30 57)(26 55 31 58)(27 56 32 59)(28 53 29 60)
(2 46)(4 48)(5 52)(6 41)(7 50)(8 43)(9 37)(10 33)(11 39)(12 35)(13 18)(14 16)(15 20)(17 19)(21 34)(22 40)(23 36)(24 38)(25 27)(26 29)(28 31)(30 32)(42 61)(44 63)(49 64)(51 62)(54 57)(56 59)
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), (1,5,58,52)(2,6,59,49)(3,7,60,50)(4,8,57,51)(9,18,34,28)(10,19,35,25)(11,20,36,26)(12,17,33,27)(13,37,31,21)(14,38,32,22)(15,39,29,23)(16,40,30,24)(41,46,64,56)(42,47,61,53)(43,48,62,54)(44,45,63,55), (1,20,45,13)(2,17,46,14)(3,18,47,15)(4,19,48,16)(5,36,63,37)(6,33,64,38)(7,34,61,39)(8,35,62,40)(9,42,23,50)(10,43,24,51)(11,44,21,52)(12,41,22,49)(25,54,30,57)(26,55,31,58)(27,56,32,59)(28,53,29,60), (2,46)(4,48)(5,52)(6,41)(7,50)(8,43)(9,37)(10,33)(11,39)(12,35)(13,18)(14,16)(15,20)(17,19)(21,34)(22,40)(23,36)(24,38)(25,27)(26,29)(28,31)(30,32)(42,61)(44,63)(49,64)(51,62)(54,57)(56,59)>;
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), (1,5,58,52)(2,6,59,49)(3,7,60,50)(4,8,57,51)(9,18,34,28)(10,19,35,25)(11,20,36,26)(12,17,33,27)(13,37,31,21)(14,38,32,22)(15,39,29,23)(16,40,30,24)(41,46,64,56)(42,47,61,53)(43,48,62,54)(44,45,63,55), (1,20,45,13)(2,17,46,14)(3,18,47,15)(4,19,48,16)(5,36,63,37)(6,33,64,38)(7,34,61,39)(8,35,62,40)(9,42,23,50)(10,43,24,51)(11,44,21,52)(12,41,22,49)(25,54,30,57)(26,55,31,58)(27,56,32,59)(28,53,29,60), (2,46)(4,48)(5,52)(6,41)(7,50)(8,43)(9,37)(10,33)(11,39)(12,35)(13,18)(14,16)(15,20)(17,19)(21,34)(22,40)(23,36)(24,38)(25,27)(26,29)(28,31)(30,32)(42,61)(44,63)(49,64)(51,62)(54,57)(56,59) );
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)], [(1,5,58,52),(2,6,59,49),(3,7,60,50),(4,8,57,51),(9,18,34,28),(10,19,35,25),(11,20,36,26),(12,17,33,27),(13,37,31,21),(14,38,32,22),(15,39,29,23),(16,40,30,24),(41,46,64,56),(42,47,61,53),(43,48,62,54),(44,45,63,55)], [(1,20,45,13),(2,17,46,14),(3,18,47,15),(4,19,48,16),(5,36,63,37),(6,33,64,38),(7,34,61,39),(8,35,62,40),(9,42,23,50),(10,43,24,51),(11,44,21,52),(12,41,22,49),(25,54,30,57),(26,55,31,58),(27,56,32,59),(28,53,29,60)], [(2,46),(4,48),(5,52),(6,41),(7,50),(8,43),(9,37),(10,33),(11,39),(12,35),(13,18),(14,16),(15,20),(17,19),(21,34),(22,40),(23,36),(24,38),(25,27),(26,29),(28,31),(30,32),(42,61),(44,63),(49,64),(51,62),(54,57),(56,59)])
Matrix representation ►G ⊆ GL6(𝔽5)
| 0 | 1 | 0 | 0 | 0 | 0 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 3 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 0 | 0 | 3 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 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 |
| 2 | 0 | 0 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 0 | 0 | 0 |
| 0 | 0 | 0 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 4 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
G:=sub<GL(6,GF(5))| [0,4,0,0,0,0,1,0,0,0,0,0,0,0,1,3,0,0,0,0,1,4,0,0,0,0,0,0,3,0,0,0,0,0,0,3],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,2,0,0,0,0,4,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[2,0,0,0,0,0,0,2,0,0,0,0,0,0,2,0,0,0,0,0,0,2,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,4,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4] >;
44 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 4A | ··· | 4AB | 4AC | ··· | 4AH |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
| size | 1 | 1 | ··· | 1 | 8 | 8 | 2 | ··· | 2 | 8 | ··· | 8 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | C4○D4 |
| kernel | C43⋊13C2 | C43 | C24.C22 | C23.65C23 | C24.3C22 | C2×C42⋊2C2 | C42 | C2×C4 |
| # reps | 1 | 1 | 6 | 3 | 3 | 2 | 4 | 24 |
In GAP, Magma, Sage, TeX
C_4^3\rtimes_{13}C_2 % in TeX
G:=Group("C4^3:13C2"); // GroupNames label
G:=SmallGroup(128,1592);
// by ID
G=gap.SmallGroup(128,1592);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,253,568,758,184,2019,80]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^4=d^2=1,a*b=b*a,a*c=c*a,d*a*d=a*c^2,b*c=c*b,d*b*d=b^-1,d*c*d=a^2*c^-1>;
// generators/relations