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G = D4⋊4C42  order 128 = 27

The semidirect product of D4 and C42 acting through Inn(D4)

p-group, metabelian, nilpotent (class 2), monomial

Aliases: D4⋊4C42, C24.525C23, C23.157C24, 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), C22.29(C23×C4), C2.10(C22×C42), (C23×C4).32C22, C23.115(C22×C4), (C2×C42).400C22, (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×C4).481(C22×C4), (C2×C2.C42)⋊5C2, (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

Derived series
C1 — C2 — D4⋊4C42
Chief series
C1 — C2 — C22 — C23 — C24 — C23×C4 — C2×C2.C42 — D4⋊4C42
Lower central
C1 — C2 — D4⋊4C42
Upper central
C1 — C23 — D4⋊4C42
Jennings
C1 — C23 — D4⋊4C42

Generators and relations for D4⋊4C42
 G = < a,b,c,d | a4=b2=c4=d4=1, bab=cac-1=dad-1=a-1, cbc-1=a2b, bd=db, cd=dc >

Subgroups: 604 in 390 conjugacy classes, 260 normal (9 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, D4, C23, C23, C23, C42, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C24, C2.C42, C2.C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C4×D4, C23×C4, C22×D4, C2×C2.C42, C42⋊4C4, C4×C22⋊C4, C4×C4⋊C4, C2×C4×D4, D4⋊4C42
Quotients: C1, C2, C4, C22, C2×C4, C23, C42, C22×C4, C24, C2×C42, C23×C4, 2+ 1+4, 2- 1+4, C22×C42, C22.11C24, C23.33C23, D4⋊4C42

Smallest permutation representation of D4⋊4C42
►On 64 points
Generators in S64
(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 14)(6 13)(7 16)(8 15)(9 56)(10 55)(11 54)(12 53)(21 26)(22 25)(23 28)(24 27)(29 41)(30 44)(31 43)(32 42)(33 57)(34 60)(35 59)(36 58)(37 48)(38 47)(39 46)(40 45)(49 62)(50 61)(51 64)(52 63)

(1 30 24 47)(2 29 21 46)(3 32 22 45)(4 31 23 48)(5 33 52 54)(6 36 49 53)(7 35 50 56)(8 34 51 55)(9 14 59 63)(10 13 60 62)(11 16 57 61)(12 15 58 64)(17 42 27 40)(18 41 28 39)(19 44 25 38)(20 43 26 37)

(1 58 18 35)(2 57 19 34)(3 60 20 33)(4 59 17 36)(5 45 13 37)(6 48 14 40)(7 47 15 39)(8 46 16 38)(9 27 53 23)(10 26 54 22)(11 25 55 21)(12 28 56 24)(29 61 44 51)(30 64 41 50)(31 63 42 49)(32 62 43 52)
 
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)(5,14)(6,13)(7,16)(8,15)(9,56)(10,55)(11,54)(12,53)(21,26)(22,25)(23,28)(24,27)(29,41)(30,44)(31,43)(32,42)(33,57)(34,60)(35,59)(36,58)(37,48)(38,47)(39,46)(40,45)(49,62)(50,61)(51,64)(52,63), (1,30,24,47)(2,29,21,46)(3,32,22,45)(4,31,23,48)(5,33,52,54)(6,36,49,53)(7,35,50,56)(8,34,51,55)(9,14,59,63)(10,13,60,62)(11,16,57,61)(12,15,58,64)(17,42,27,40)(18,41,28,39)(19,44,25,38)(20,43,26,37), (1,58,18,35)(2,57,19,34)(3,60,20,33)(4,59,17,36)(5,45,13,37)(6,48,14,40)(7,47,15,39)(8,46,16,38)(9,27,53,23)(10,26,54,22)(11,25,55,21)(12,28,56,24)(29,61,44,51)(30,64,41,50)(31,63,42,49)(32,62,43,52)>;
 
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)(5,14)(6,13)(7,16)(8,15)(9,56)(10,55)(11,54)(12,53)(21,26)(22,25)(23,28)(24,27)(29,41)(30,44)(31,43)(32,42)(33,57)(34,60)(35,59)(36,58)(37,48)(38,47)(39,46)(40,45)(49,62)(50,61)(51,64)(52,63), (1,30,24,47)(2,29,21,46)(3,32,22,45)(4,31,23,48)(5,33,52,54)(6,36,49,53)(7,35,50,56)(8,34,51,55)(9,14,59,63)(10,13,60,62)(11,16,57,61)(12,15,58,64)(17,42,27,40)(18,41,28,39)(19,44,25,38)(20,43,26,37), (1,58,18,35)(2,57,19,34)(3,60,20,33)(4,59,17,36)(5,45,13,37)(6,48,14,40)(7,47,15,39)(8,46,16,38)(9,27,53,23)(10,26,54,22)(11,25,55,21)(12,28,56,24)(29,61,44,51)(30,64,41,50)(31,63,42,49)(32,62,43,52) );
 
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),(5,14),(6,13),(7,16),(8,15),(9,56),(10,55),(11,54),(12,53),(21,26),(22,25),(23,28),(24,27),(29,41),(30,44),(31,43),(32,42),(33,57),(34,60),(35,59),(36,58),(37,48),(38,47),(39,46),(40,45),(49,62),(50,61),(51,64),(52,63)], [(1,30,24,47),(2,29,21,46),(3,32,22,45),(4,31,23,48),(5,33,52,54),(6,36,49,53),(7,35,50,56),(8,34,51,55),(9,14,59,63),(10,13,60,62),(11,16,57,61),(12,15,58,64),(17,42,27,40),(18,41,28,39),(19,44,25,38),(20,43,26,37)], [(1,58,18,35),(2,57,19,34),(3,60,20,33),(4,59,17,36),(5,45,13,37),(6,48,14,40),(7,47,15,39),(8,46,16,38),(9,27,53,23),(10,26,54,22),(11,25,55,21),(12,28,56,24),(29,61,44,51),(30,64,41,50),(31,63,42,49),(32,62,43,52)]])
 

68 conjugacy classes

class 1 2A···2G2H···2O4A···4AZ
order12···22···24···4
size11···12···22···2

68 irreducible representations

dim111111144
type+++++++-
imageC1C2C2C2C2C2C42+ 1+42- 1+4
kernelD4⋊4C42C2×C2.C42C42⋊4C4C4×C22⋊C4C4×C4⋊C4C2×C4×D4C4×D4C22C22
# reps1216334831

Matrix representation of D4⋊4C42 ►in GL6(𝔽5)

400000
040000
001002
000430
000110
004004
,
400000
040000
004003
000120
000040
000001
,
400000
030000
003000
000200
000330
002002
,
200000
030000
000120
001002
004004
000440

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] >;
 

D4⋊4C42 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
 

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