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G = M4(2).49D4order 128 = 27

13rd non-split extension by M4(2) of D4 acting via D4/C22=C2

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

Aliases: M4(2).49D4, C4○D4.47D4, (C2×C8).324D4, C4.120(C4×D4), C8.C224C4, C4.98C22≀C2, (C2×D4).207D4, (C2×Q8).165D4, C22.10(C4×D4), C4.10(C4⋊D4), C4.C426C2, M4(2).6(C2×C4), C22.C426C2, D4.10(C22⋊C4), C2.2(D4.5D4), C2.3(D4.3D4), Q8.10(C22⋊C4), C23.264(C4○D4), (C22×C8).392C22, (C22×C4).692C23, C23.36D4.9C2, (C22×Q8).23C22, C22.123(C4⋊D4), C2.47(C23.23D4), (C2×M4(2)).319C22, C22.8(C22.D4), (C2×C8○D4).15C2, C4○D4.17(C2×C4), (C2×C4).243(C2×D4), C4.24(C2×C22⋊C4), (C2×Q8).75(C2×C4), (C2×Q8⋊C4)⋊46C2, (C2×C4).60(C4○D4), (C2×C4⋊C4).67C22, (C2×C4).14(C22×C4), (C2×C8.C22).4C2, (C2×C4.10D4)⋊19C2, (C2×C4○D4).266C22, SmallGroup(128,640)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — M4(2).49D4
C1C2C4C2×C4C22×C4C2×C4○D4C2×C8○D4 — M4(2).49D4
C1C2C2×C4 — M4(2).49D4
C1C22C22×C4 — M4(2).49D4
C1C2C2C22×C4 — M4(2).49D4

Generators and relations for M4(2).49D4
 G = < a,b,c,d | a8=b2=c4=1, d2=a2b, bab=a5, cac-1=a5b, dad-1=ab, bc=cb, dbd-1=a4b, dcd-1=a2bc-1 >

Subgroups: 292 in 156 conjugacy classes, 56 normal (38 characteristic)
C1, C2 [×3], C2 [×4], C4 [×4], C4 [×5], C22 [×3], C22 [×6], C8 [×7], C2×C4 [×6], C2×C4 [×11], D4 [×2], D4 [×5], Q8 [×2], Q8 [×7], C23, C23, C4⋊C4 [×2], C2×C8 [×2], C2×C8 [×9], M4(2) [×4], M4(2) [×8], SD16 [×4], Q16 [×4], C22×C4, C22×C4 [×3], C2×D4, C2×D4, C2×Q8, C2×Q8 [×2], C2×Q8 [×5], C4○D4 [×4], C4○D4 [×2], C4.10D4 [×2], D4⋊C4, Q8⋊C4 [×3], C2×C4⋊C4, C22×C8, C22×C8, C2×M4(2) [×3], C2×M4(2), C8○D4 [×4], C2×SD16, C2×Q16, C8.C22 [×4], C8.C22 [×2], C22×Q8, C2×C4○D4, C4.C42, C22.C42, C2×C4.10D4, C2×Q8⋊C4, C23.36D4, C2×C8○D4, C2×C8.C22, M4(2).49D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×8], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×4], C4○D4 [×2], C2×C22⋊C4, C4×D4 [×2], C22≀C2, C4⋊D4 [×2], C22.D4, C23.23D4, D4.3D4, D4.5D4, M4(2).49D4

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

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

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

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

32 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E4F4G4H4I4J8A8B8C8D8E···8J8K8L8M8N
order12222222444444444488888···88888
size11112244222244888822224···48888

32 irreducible representations

dim111111111222222244
type+++++++++++++-
imageC1C2C2C2C2C2C2C2C4D4D4D4D4D4C4○D4C4○D4D4.3D4D4.5D4
kernelM4(2).49D4C4.C42C22.C42C2×C4.10D4C2×Q8⋊C4C23.36D4C2×C8○D4C2×C8.C22C8.C22C2×C8M4(2)C2×D4C2×Q8C4○D4C2×C4C23C2C2
# reps111111118221122222

Matrix representation of M4(2).49D4 in GL6(𝔽17)

100000
0160000
001212120
0000012
00512155
00010147
,
1600000
0160000
0016000
001010
001100
0001521
,
0130000
1300000
0001610
001611516
00011516
00161611
,
040000
400000
001021
001611516
00160160
00111616

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,16,0,0,0,0,0,0,12,0,5,0,0,0,12,0,12,10,0,0,12,0,15,14,0,0,0,12,5,7],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,1,1,0,0,0,0,0,1,15,0,0,0,1,0,2,0,0,0,0,0,1],[0,13,0,0,0,0,13,0,0,0,0,0,0,0,0,16,0,16,0,0,16,1,1,16,0,0,1,15,15,1,0,0,0,16,16,1],[0,4,0,0,0,0,4,0,0,0,0,0,0,0,1,16,16,1,0,0,0,1,0,1,0,0,2,15,16,16,0,0,1,16,0,16] >;

M4(2).49D4 in GAP, Magma, Sage, TeX

M_4(2)._{49}D_4
% in TeX

G:=Group("M4(2).49D4");
// GroupNames label

G:=SmallGroup(128,640);
// by ID

G=gap.SmallGroup(128,640);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,736,422,2019,1018,521,1411,718,172,2028,1027,124]);
// Polycyclic

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

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