p-group, metabelian, nilpotent (class 2), monomial
Aliases: D4⋊4C42, C23.157C24, C24.525C23, C22.282+ (1+4), C22.162- (1+4), (C4×D4)⋊16C4, C42⋊13(C2×C4), C42⋊4C4⋊8C2, C4.10(C2×C42), D4○(C2.C42), C22.1(C2×C42), C2.10(C22×C42), (C23×C4).32C22, C22.29(C23×C4), (C2×C42).400C22, C23.115(C22×C4), (C22×C4).1233C23, C2.3(C22.11C24), (C22×D4).606C22, C2.C42.568C22, C2.2(C23.33C23), (C4×C4⋊C4)⋊12C2, C4⋊C4⋊50(C2×C4), (C2×C4×D4).24C2, (C4×C22⋊C4)⋊3C2, C22⋊C4⋊46(C2×C4), (C22×C4)⋊12(C2×C4), (C2×D4).242(C2×C4), C4⋊C4○(C2.C42), (C2×C4⋊C4).971C22, (C2×C2.C42)⋊5C2, (C2×C4).481(C22×C4), (C2×D4)○(C2.C42), C2.C42○(C22×D4), C22⋊C4○(C2.C42), (C2×C22⋊C4).553C22, C2.C42○(C2×C22⋊C4), SmallGroup(128,1007)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 604 in 390 conjugacy classes, 260 normal (9 characteristic)
C1, C2, C2 [×6], C2 [×8], C4 [×4], C4 [×24], C22, C22 [×14], C22 [×24], C2×C4 [×30], C2×C4 [×48], D4 [×16], C23, C23 [×12], C23 [×8], C42 [×12], C42 [×6], C22⋊C4 [×24], C4⋊C4 [×12], C22×C4, C22×C4 [×36], C22×C4 [×12], C2×D4 [×12], C24 [×2], C2.C42, C2.C42 [×9], C2×C42 [×9], C2×C22⋊C4 [×6], C2×C4⋊C4 [×3], C4×D4 [×24], C23×C4 [×6], C22×D4, C2×C2.C42 [×2], C42⋊4C4, C4×C22⋊C4 [×6], C4×C4⋊C4 [×3], C2×C4×D4 [×3], D4⋊4C42
Quotients:
C1, C2 [×15], C4 [×24], C22 [×35], C2×C4 [×84], C23 [×15], C42 [×16], C22×C4 [×42], C24, C2×C42 [×12], C23×C4 [×3], 2+ (1+4) [×3], 2- (1+4), C22×C42, C22.11C24 [×3], C23.33C23 [×3], D4⋊4C42
Generators and relations
G = < a,b,c,d | a4=b2=c4=d4=1, bab=cac-1=dad-1=a-1, cbc-1=a2b, bd=db, cd=dc >
(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 17)(2 20)(3 19)(4 18)(5 45)(6 48)(7 47)(8 46)(9 36)(10 35)(11 34)(12 33)(13 40)(14 39)(15 38)(16 37)(21 53)(22 56)(23 55)(24 54)(25 49)(26 52)(27 51)(28 50)(29 58)(30 57)(31 60)(32 59)(41 64)(42 63)(43 62)(44 61)
(1 31 11 47)(2 30 12 46)(3 29 9 45)(4 32 10 48)(5 17 58 34)(6 20 59 33)(7 19 60 36)(8 18 57 35)(13 41 55 50)(14 44 56 49)(15 43 53 52)(16 42 54 51)(21 28 38 64)(22 27 39 63)(23 26 40 62)(24 25 37 61)
(1 21 18 56)(2 24 19 55)(3 23 20 54)(4 22 17 53)(5 43 48 63)(6 42 45 62)(7 41 46 61)(8 44 47 64)(9 40 33 16)(10 39 34 15)(11 38 35 14)(12 37 36 13)(25 60 50 30)(26 59 51 29)(27 58 52 32)(28 57 49 31)
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), (1,17)(2,20)(3,19)(4,18)(5,45)(6,48)(7,47)(8,46)(9,36)(10,35)(11,34)(12,33)(13,40)(14,39)(15,38)(16,37)(21,53)(22,56)(23,55)(24,54)(25,49)(26,52)(27,51)(28,50)(29,58)(30,57)(31,60)(32,59)(41,64)(42,63)(43,62)(44,61), (1,31,11,47)(2,30,12,46)(3,29,9,45)(4,32,10,48)(5,17,58,34)(6,20,59,33)(7,19,60,36)(8,18,57,35)(13,41,55,50)(14,44,56,49)(15,43,53,52)(16,42,54,51)(21,28,38,64)(22,27,39,63)(23,26,40,62)(24,25,37,61), (1,21,18,56)(2,24,19,55)(3,23,20,54)(4,22,17,53)(5,43,48,63)(6,42,45,62)(7,41,46,61)(8,44,47,64)(9,40,33,16)(10,39,34,15)(11,38,35,14)(12,37,36,13)(25,60,50,30)(26,59,51,29)(27,58,52,32)(28,57,49,31)>;
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), (1,17)(2,20)(3,19)(4,18)(5,45)(6,48)(7,47)(8,46)(9,36)(10,35)(11,34)(12,33)(13,40)(14,39)(15,38)(16,37)(21,53)(22,56)(23,55)(24,54)(25,49)(26,52)(27,51)(28,50)(29,58)(30,57)(31,60)(32,59)(41,64)(42,63)(43,62)(44,61), (1,31,11,47)(2,30,12,46)(3,29,9,45)(4,32,10,48)(5,17,58,34)(6,20,59,33)(7,19,60,36)(8,18,57,35)(13,41,55,50)(14,44,56,49)(15,43,53,52)(16,42,54,51)(21,28,38,64)(22,27,39,63)(23,26,40,62)(24,25,37,61), (1,21,18,56)(2,24,19,55)(3,23,20,54)(4,22,17,53)(5,43,48,63)(6,42,45,62)(7,41,46,61)(8,44,47,64)(9,40,33,16)(10,39,34,15)(11,38,35,14)(12,37,36,13)(25,60,50,30)(26,59,51,29)(27,58,52,32)(28,57,49,31) );
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)], [(1,17),(2,20),(3,19),(4,18),(5,45),(6,48),(7,47),(8,46),(9,36),(10,35),(11,34),(12,33),(13,40),(14,39),(15,38),(16,37),(21,53),(22,56),(23,55),(24,54),(25,49),(26,52),(27,51),(28,50),(29,58),(30,57),(31,60),(32,59),(41,64),(42,63),(43,62),(44,61)], [(1,31,11,47),(2,30,12,46),(3,29,9,45),(4,32,10,48),(5,17,58,34),(6,20,59,33),(7,19,60,36),(8,18,57,35),(13,41,55,50),(14,44,56,49),(15,43,53,52),(16,42,54,51),(21,28,38,64),(22,27,39,63),(23,26,40,62),(24,25,37,61)], [(1,21,18,56),(2,24,19,55),(3,23,20,54),(4,22,17,53),(5,43,48,63),(6,42,45,62),(7,41,46,61),(8,44,47,64),(9,40,33,16),(10,39,34,15),(11,38,35,14),(12,37,36,13),(25,60,50,30),(26,59,51,29),(27,58,52,32),(28,57,49,31)])
Matrix representation ►G ⊆ GL6(𝔽5)
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 2 |
| 0 | 0 | 0 | 4 | 3 | 0 |
| 0 | 0 | 0 | 1 | 1 | 0 |
| 0 | 0 | 4 | 0 | 0 | 4 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 3 |
| 0 | 0 | 0 | 1 | 2 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 0 | 0 | 0 |
| 0 | 0 | 0 | 2 | 0 | 0 |
| 0 | 0 | 0 | 3 | 3 | 0 |
| 0 | 0 | 2 | 0 | 0 | 2 |
| 2 | 0 | 0 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 2 | 0 |
| 0 | 0 | 1 | 0 | 0 | 2 |
| 0 | 0 | 4 | 0 | 0 | 4 |
| 0 | 0 | 0 | 4 | 4 | 0 |
G:=sub<GL(6,GF(5))| [4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,4,0,0,0,4,1,0,0,0,0,3,1,0,0,0,2,0,0,4],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,2,4,0,0,0,3,0,0,1],[4,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,2,0,0,0,2,3,0,0,0,0,0,3,0,0,0,0,0,0,2],[2,0,0,0,0,0,0,3,0,0,0,0,0,0,0,1,4,0,0,0,1,0,0,4,0,0,2,0,0,4,0,0,0,2,4,0] >;
68 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | ··· | 2O | 4A | ··· | 4AZ |
| order | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
| size | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
68 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | 2+ (1+4) | 2- (1+4) |
| kernel | D4⋊4C42 | C2×C2.C42 | C42⋊4C4 | C4×C22⋊C4 | C4×C4⋊C4 | C2×C4×D4 | C4×D4 | C22 | C22 |
| # reps | 1 | 2 | 1 | 6 | 3 | 3 | 48 | 3 | 1 |
In GAP, Magma, Sage, TeX
D_4\rtimes_4C_4^2
% in TeX
G:=Group("D4:4C4^2"); // GroupNames label
G:=SmallGroup(128,1007);
// by ID
G=gap.SmallGroup(128,1007);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,224,253,456,219,184,675]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=c^4=d^4=1,b*a*b=c*a*c^-1=d*a*d^-1=a^-1,c*b*c^-1=a^2*b,b*d=d*b,c*d=d*c>;
// generators/relations