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G = D47M4(2)  order 128 = 27

2nd semidirect product of D4 and M4(2) acting through Inn(D4)

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

Aliases: D47M4(2), C42.692C23, C4.1682+ 1+4, (C8×D4)⋊45C2, C86D439C2, C89D440C2, C4⋊C890C22, (C4×C8)⋊59C22, D43(C22⋊C8), (C4×D4).34C4, C24.85(C2×C4), C8⋊C430C22, C22⋊C879C22, C42.222(C2×C4), (C2×C8).432C23, (C2×C4).671C24, (C22×C8)⋊55C22, (C22×D4).43C4, C4.35(C2×M4(2)), C24.4C435C2, C42.6C451C2, (C4×D4).363C22, C22.16(C8○D4), C42.12C452C2, C22.7(C2×M4(2)), (C2×M4(2))⋊45C22, C23.229(C22×C4), (C2×C42).781C22, (C23×C4).530C22, (C22×C4).939C23, C22.195(C23×C4), C2.19(C22×M4(2)), C2.45(C22.11C24), (C2×C4×D4).77C2, (C2×C4⋊C4).77C4, C2.27(C2×C8○D4), C4⋊C4.229(C2×C4), (C2×C22⋊C8)⋊46C2, (C2×D4).235(C2×C4), (C2×C22⋊C4).51C4, C22⋊C4.77(C2×C4), (C2×C4).276(C22×C4), (C22×C4).352(C2×C4), SmallGroup(128,1706)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — D47M4(2)
C1C2C4C2×C4C22×C4C23×C4C2×C4×D4 — D47M4(2)
C1C22 — D47M4(2)
C1C2×C4 — D47M4(2)
C1C2C2C2×C4 — D47M4(2)

Generators and relations for D47M4(2)
 G = < a,b,c,d | a4=b2=c8=d2=1, bab=dad=a-1, ac=ca, cbc-1=a2b, bd=db, dcd=c5 >

Subgroups: 388 in 232 conjugacy classes, 136 normal (36 characteristic)
C1, C2 [×3], C2 [×8], C4 [×4], C4 [×8], C22, C22 [×6], C22 [×20], C8 [×8], C2×C4 [×6], C2×C4 [×4], C2×C4 [×20], D4 [×4], D4 [×6], C23, C23 [×4], C23 [×10], C42 [×4], C22⋊C4 [×8], C4⋊C4 [×2], C4⋊C4 [×2], C2×C8 [×8], C2×C8 [×4], M4(2) [×4], C22×C4 [×3], C22×C4 [×10], C22×C4 [×4], C2×D4 [×2], C2×D4 [×2], C2×D4 [×4], C24 [×2], C4×C8 [×2], C8⋊C4 [×2], C22⋊C8 [×2], C22⋊C8 [×10], C4⋊C8 [×4], C2×C42, C2×C22⋊C4 [×2], C2×C4⋊C4, C4×D4 [×4], C4×D4 [×4], C22×C8 [×4], C2×M4(2) [×4], C23×C4 [×2], C22×D4, C2×C22⋊C8 [×2], C24.4C4 [×2], C42.12C4, C42.6C4, C8×D4 [×2], C89D4 [×4], C86D4 [×2], C2×C4×D4, D47M4(2)
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], M4(2) [×4], C22×C4 [×14], C24, C2×M4(2) [×6], C8○D4 [×2], C23×C4, 2+ 1+4 [×2], C22.11C24, C22×M4(2), C2×C8○D4, D47M4(2)

Smallest permutation representation of D47M4(2)
On 32 points
Generators in S32
(1 19 31 10)(2 20 32 11)(3 21 25 12)(4 22 26 13)(5 23 27 14)(6 24 28 15)(7 17 29 16)(8 18 30 9)
(1 14)(2 24)(3 16)(4 18)(5 10)(6 20)(7 12)(8 22)(9 26)(11 28)(13 30)(15 32)(17 25)(19 27)(21 29)(23 31)
(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)
(1 10)(2 15)(3 12)(4 9)(5 14)(6 11)(7 16)(8 13)(17 29)(18 26)(19 31)(20 28)(21 25)(22 30)(23 27)(24 32)

G:=sub<Sym(32)| (1,19,31,10)(2,20,32,11)(3,21,25,12)(4,22,26,13)(5,23,27,14)(6,24,28,15)(7,17,29,16)(8,18,30,9), (1,14)(2,24)(3,16)(4,18)(5,10)(6,20)(7,12)(8,22)(9,26)(11,28)(13,30)(15,32)(17,25)(19,27)(21,29)(23,31), (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), (1,10)(2,15)(3,12)(4,9)(5,14)(6,11)(7,16)(8,13)(17,29)(18,26)(19,31)(20,28)(21,25)(22,30)(23,27)(24,32)>;

G:=Group( (1,19,31,10)(2,20,32,11)(3,21,25,12)(4,22,26,13)(5,23,27,14)(6,24,28,15)(7,17,29,16)(8,18,30,9), (1,14)(2,24)(3,16)(4,18)(5,10)(6,20)(7,12)(8,22)(9,26)(11,28)(13,30)(15,32)(17,25)(19,27)(21,29)(23,31), (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), (1,10)(2,15)(3,12)(4,9)(5,14)(6,11)(7,16)(8,13)(17,29)(18,26)(19,31)(20,28)(21,25)(22,30)(23,27)(24,32) );

G=PermutationGroup([(1,19,31,10),(2,20,32,11),(3,21,25,12),(4,22,26,13),(5,23,27,14),(6,24,28,15),(7,17,29,16),(8,18,30,9)], [(1,14),(2,24),(3,16),(4,18),(5,10),(6,20),(7,12),(8,22),(9,26),(11,28),(13,30),(15,32),(17,25),(19,27),(21,29),(23,31)], [(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)], [(1,10),(2,15),(3,12),(4,9),(5,14),(6,11),(7,16),(8,13),(17,29),(18,26),(19,31),(20,28),(21,25),(22,30),(23,27),(24,32)])

50 conjugacy classes

class 1 2A2B2C2D···2I2J2K4A4B4C4D4E···4N4O4P4Q4R8A···8H8I···8T
order12222···22244444···444448···88···8
size11112···24411112···244442···24···4

50 irreducible representations

dim1111111111111224
type++++++++++
imageC1C2C2C2C2C2C2C2C2C4C4C4C4M4(2)C8○D42+ 1+4
kernelD47M4(2)C2×C22⋊C8C24.4C4C42.12C4C42.6C4C8×D4C89D4C86D4C2×C4×D4C2×C22⋊C4C2×C4⋊C4C4×D4C22×D4D4C22C4
# reps1221124214282882

Matrix representation of D47M4(2) in GL4(𝔽17) generated by

16000
01600
00161
00151
,
1000
0100
00116
00016
,
01500
2000
00152
00132
,
16000
0100
00161
0001
G:=sub<GL(4,GF(17))| [16,0,0,0,0,16,0,0,0,0,16,15,0,0,1,1],[1,0,0,0,0,1,0,0,0,0,1,0,0,0,16,16],[0,2,0,0,15,0,0,0,0,0,15,13,0,0,2,2],[16,0,0,0,0,1,0,0,0,0,16,0,0,0,1,1] >;

D47M4(2) in GAP, Magma, Sage, TeX

D_4\rtimes_7M_4(2)
% in TeX

G:=Group("D4:7M4(2)");
// GroupNames label

G:=SmallGroup(128,1706);
// by ID

G=gap.SmallGroup(128,1706);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,891,675,124]);
// Polycyclic

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

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