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G = D4⋊2M4(2)  order 128 = 27

2nd semidirect product of D4 and M4(2) acting via M4(2)/C8=C2

p-group, metabelian, nilpotent (class 3), monomial

Aliases: D4⋊2M4(2), C42.643C23, C8⋊C8⋊5C2, D4⋊C8⋊34C2, (C8×D4)⋊35C2, C8⋊2C8⋊12C2, (C2×D8).7C4, C8⋊6D4⋊29C2, (C4×D8).12C2, C2.7(C8○D8), (C2×C8).308D4, C2.D8.11C4, D4⋊C4.8C4, C4.35(C8○D4), C2.4(D8⋊C4), C2.13(C8⋊9D4), C4⋊C8.277C22, (C4×C8).315C22, (C4×D4).11C22, C22.134(C4×D4), C4.29(C2×M4(2)), C4.144(C8⋊C22), C4⋊C4.137(C2×C4), (C2×C8).104(C2×C4), (C2×D4).155(C2×C4), (C2×C4).1479(C2×D4), (C2×C4).504(C4○D4), (C2×C4).335(C22×C4), SmallGroup(128,318)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series
C1 — C2×C4 — D4⋊2M4(2)
Chief series
C1 — C2 — C22 — C2×C4 — C42 — C4×C8 — C8×D4 — D4⋊2M4(2)
Lower central
C1 — C2 — C2×C4 — D4⋊2M4(2)
Upper central
C1 — C2×C4 — C4×C8 — D4⋊2M4(2)
Jennings
C1 — C22 — C22 — C42 — D4⋊2M4(2)

Generators and relations for D4⋊2M4(2)
 G = < a,b,c,d | a4=b2=c8=d2=1, bab=cac-1=dad=a-1, cbc-1=ab, dbd=a-1b, dcd=c5 >

Subgroups: 184 in 91 conjugacy classes, 42 normal (40 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, D4, D4, C23, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), D8, C22×C4, C2×D4, C4×C8, C22⋊C8, D4⋊C4, C4⋊C8, C2.D8, C4×D4, C22×C8, C2×M4(2), C2×D8, C8⋊C8, D4⋊C8, C8⋊2C8, C8×D4, C8⋊6D4, C4×D8, D4⋊2M4(2)
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, M4(2), C22×C4, C2×D4, C4○D4, C4×D4, C2×M4(2), C8○D4, C8⋊C22, C8⋊9D4, D8⋊C4, C8○D8, D4⋊2M4(2)

Smallest permutation representation of D4⋊2M4(2)
►On 64 points
Generators in S64
(1 16 27 43)(2 44 28 9)(3 10 29 45)(4 46 30 11)(5 12 31 47)(6 48 32 13)(7 14 25 41)(8 42 26 15)(17 33 53 60)(18 61 54 34)(19 35 55 62)(20 63 56 36)(21 37 49 64)(22 57 50 38)(23 39 51 58)(24 59 52 40)

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

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

38 conjugacy classes

class 1 2A2B2C2D2E2F4A4B4C4D4E4F4G4H4I4J4K8A8B8C8D8E···8R8S8T
order12222224444444444488888···888
size11114481111222244822224···488

38 irreducible representations

dim1111111111222224
type+++++++++
imageC1C2C2C2C2C2C2C4C4C4D4C4○D4M4(2)C8○D4C8○D8C8⋊C22
kernelD4⋊2M4(2)C8⋊C8D4⋊C8C8⋊2C8C8×D4C8⋊6D4C4×D8D4⋊C4C2.D8C2×D8C2×C8C2×C4D4C4C2C4
# reps1121111422224482

Matrix representation of D4⋊2M4(2) ►in GL4(𝔽17) generated by

0100
16000
0010
0001
,
14300
3300
00160
00016
,
15000
0200
00152
0002
,
01600
16000
0010
00216
G:=sub<GL(4,GF(17))| [0,16,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[14,3,0,0,3,3,0,0,0,0,16,0,0,0,0,16],[15,0,0,0,0,2,0,0,0,0,15,0,0,0,2,2],[0,16,0,0,16,0,0,0,0,0,1,2,0,0,0,16] >;
 

D4⋊2M4(2) in GAP, Magma, Sage, TeX 

D_4\rtimes_2M_4(2)
 
% in TeX
 
G:=Group("D4:2M4(2)");
 
// GroupNames label
 
G:=SmallGroup(128,318);
 
// by ID
 
G=gap.SmallGroup(128,318);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,2102,268,1123,570,136,172]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^4=b^2=c^8=d^2=1,b*a*b=c*a*c^-1=d*a*d=a^-1,c*b*c^-1=a*b,d*b*d=a^-1*b,d*c*d=c^5>;
 
// generators/relations
 

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