Copied to
clipboard

G = D4.M4(2)  order 128 = 27

2nd non-split extension by D4 of M4(2) acting via M4(2)/C8=C2

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

Aliases: D4.2M4(2), C42.642C23, C8⋊C84C2, Q8⋊C839C2, C81C816C2, C4.Q8.9C4, D4⋊C8.13C2, C84Q829C2, (C8×D4).16C2, C2.6(C8○D8), (C2×C8).307D4, D4⋊C4.4C4, C4.34(C8○D4), Q8⋊C4.4C4, (C2×SD16).5C4, (C4×SD16).1C2, C2.12(C89D4), C4⋊C8.276C22, (C4×C8).314C22, C22.133(C4×D4), C4.28(C2×M4(2)), (C4×Q8).11C22, C4.143(C8⋊C22), (C4×D4).273C22, C2.5(SD16⋊C4), C4.137(C8.C22), (C2×C8).32(C2×C4), C4⋊C4.136(C2×C4), (C2×Q8).52(C2×C4), (C2×D4).154(C2×C4), (C2×C4).1478(C2×D4), (C2×C4).503(C4○D4), (C2×C4).334(C22×C4), SmallGroup(128,317)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — D4.M4(2)
C1C2C22C2×C4C42C4×C8C8×D4 — D4.M4(2)
C1C2C2×C4 — D4.M4(2)
C1C2×C4C4×C8 — D4.M4(2)
C1C22C22C42 — D4.M4(2)

Generators and relations for D4.M4(2)
 G = < a,b,c,d | a4=b2=c8=1, d2=a2, bab=cac-1=dad-1=a-1, cbc-1=dbd-1=ab, dcd-1=a2c5 >

Subgroups: 152 in 84 conjugacy classes, 42 normal (40 characteristic)
C1, C2 [×3], C2 [×2], C4 [×4], C4 [×4], C22, C22 [×4], C8 [×8], C2×C4 [×3], C2×C4 [×5], D4 [×2], D4, Q8 [×2], C23, C42, C42, C22⋊C4, C4⋊C4 [×2], C4⋊C4, C2×C8 [×4], C2×C8 [×5], SD16 [×2], C22×C4, C2×D4, C2×Q8, C4×C8 [×3], C8⋊C4, C22⋊C8, D4⋊C4, Q8⋊C4, C4⋊C8 [×2], C4⋊C8, C4.Q8, C4×D4, C4×Q8, C22×C8, C2×SD16, C8⋊C8, D4⋊C8, Q8⋊C8, C81C8, C8×D4, C4×SD16, C84Q8, D4.M4(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, C8.C22, C89D4, SD16⋊C4, C8○D8, D4.M4(2)

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

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

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

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

38 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G4H4I4J4K4L8A8B8C8D8E···8R8S8T
order12222244444444444488888···888
size11114411112222448822224···488

38 irreducible representations

dim1111111111112222244
type++++++++++-
imageC1C2C2C2C2C2C2C2C4C4C4C4D4C4○D4M4(2)C8○D4C8○D8C8⋊C22C8.C22
kernelD4.M4(2)C8⋊C8D4⋊C8Q8⋊C8C81C8C8×D4C4×SD16C84Q8D4⋊C4Q8⋊C4C4.Q8C2×SD16C2×C8C2×C4D4C4C2C4C4
# reps1111111122222244811

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

0100
16000
00160
00016
,
01600
16000
0010
00316
,
10700
7700
0058
00812
,
51200
121200
001016
00147
G:=sub<GL(4,GF(17))| [0,16,0,0,1,0,0,0,0,0,16,0,0,0,0,16],[0,16,0,0,16,0,0,0,0,0,1,3,0,0,0,16],[10,7,0,0,7,7,0,0,0,0,5,8,0,0,8,12],[5,12,0,0,12,12,0,0,0,0,10,14,0,0,16,7] >;

D4.M4(2) in GAP, Magma, Sage, TeX

D_4.M_4(2)
% in TeX

G:=Group("D4.M4(2)");
// GroupNames label

G:=SmallGroup(128,317);
// by ID

G=gap.SmallGroup(128,317);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,2102,100,1123,570,136,172]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^2=c^8=1,d^2=a^2,b*a*b=c*a*c^-1=d*a*d^-1=a^-1,c*b*c^-1=d*b*d^-1=a*b,d*c*d^-1=a^2*c^5>;
// generators/relations

׿
×
𝔽