Copied to
clipboard

G = D42M4(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: D42M4(2), C42.643C23, C8⋊C85C2, D4⋊C834C2, (C8×D4)⋊35C2, C82C812C2, (C2×D8).7C4, C86D429C2, (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(C89D4), 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

C1C2×C4 — D42M4(2)
C1C2C22C2×C4C42C4×C8C8×D4 — D42M4(2)
C1C2C2×C4 — D42M4(2)
C1C2×C4C4×C8 — D42M4(2)
C1C22C22C42 — D42M4(2)

Generators and relations for D42M4(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 [×3], C2 [×3], C4 [×4], C4 [×3], C22, C22 [×7], C8 [×8], C2×C4 [×3], C2×C4 [×6], D4 [×2], D4 [×3], C23 [×2], C42, C22⋊C4 [×2], C4⋊C4 [×2], C2×C8 [×4], C2×C8 [×5], M4(2) [×2], D8 [×2], C22×C4 [×2], C2×D4 [×2], C4×C8 [×3], C22⋊C8 [×2], D4⋊C4 [×2], C4⋊C8 [×2], C2.D8, C4×D4 [×2], C22×C8, C2×M4(2), C2×D8, C8⋊C8, D4⋊C8 [×2], C82C8, C8×D4, C86D4, C4×D8, D42M4(2)
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], C23, M4(2) [×2], C22×C4, C2×D4, C4○D4, C4×D4, C2×M4(2), C8○D4, C8⋊C22 [×2], C89D4, D8⋊C4, C8○D8, D42M4(2)

Smallest permutation representation of D42M4(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 39 53 60)(18 61 54 40)(19 33 55 62)(20 63 56 34)(21 35 49 64)(22 57 50 36)(23 37 51 58)(24 59 52 38)
(1 63)(2 21)(3 57)(4 23)(5 59)(6 17)(7 61)(8 19)(9 35)(10 22)(11 37)(12 24)(13 39)(14 18)(15 33)(16 20)(25 40)(26 55)(27 34)(28 49)(29 36)(30 51)(31 38)(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)(17,39,53,60)(18,61,54,40)(19,33,55,62)(20,63,56,34)(21,35,49,64)(22,57,50,36)(23,37,51,58)(24,59,52,38), (1,63)(2,21)(3,57)(4,23)(5,59)(6,17)(7,61)(8,19)(9,35)(10,22)(11,37)(12,24)(13,39)(14,18)(15,33)(16,20)(25,40)(26,55)(27,34)(28,49)(29,36)(30,51)(31,38)(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)(17,39,53,60)(18,61,54,40)(19,33,55,62)(20,63,56,34)(21,35,49,64)(22,57,50,36)(23,37,51,58)(24,59,52,38), (1,63)(2,21)(3,57)(4,23)(5,59)(6,17)(7,61)(8,19)(9,35)(10,22)(11,37)(12,24)(13,39)(14,18)(15,33)(16,20)(25,40)(26,55)(27,34)(28,49)(29,36)(30,51)(31,38)(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),(17,39,53,60),(18,61,54,40),(19,33,55,62),(20,63,56,34),(21,35,49,64),(22,57,50,36),(23,37,51,58),(24,59,52,38)], [(1,63),(2,21),(3,57),(4,23),(5,59),(6,17),(7,61),(8,19),(9,35),(10,22),(11,37),(12,24),(13,39),(14,18),(15,33),(16,20),(25,40),(26,55),(27,34),(28,49),(29,36),(30,51),(31,38),(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
kernelD42M4(2)C8⋊C8D4⋊C8C82C8C8×D4C86D4C4×D8D4⋊C4C2.D8C2×D8C2×C8C2×C4D4C4C2C4
# reps1121111422224482

Matrix representation of D42M4(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] >;

D42M4(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

׿
×
𝔽