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G = D4:7M4(2)  order 128 = 27

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

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

Aliases: D4:7M4(2), C42.692C23, C4.1682+ 1+4, (C8xD4):45C2, C8:6D4:39C2, C8:9D4:40C2, C4:C8:90C22, (C4xC8):59C22, D4o3(C22:C8), (C4xD4).34C4, C24.85(C2xC4), C8:C4:30C22, C22:C8:79C22, C42.222(C2xC4), (C2xC8).432C23, (C2xC4).671C24, (C22xC8):55C22, (C22xD4).43C4, C4.35(C2xM4(2)), C24.4C4:35C2, C42.6C4:51C2, (C4xD4).363C22, C22.16(C8oD4), C42.12C4:52C2, C22.7(C2xM4(2)), (C2xM4(2)):45C22, C23.229(C22xC4), (C2xC42).781C22, (C23xC4).530C22, (C22xC4).939C23, C22.195(C23xC4), C2.19(C22xM4(2)), C2.45(C22.11C24), (C2xC4xD4).77C2, (C2xC4:C4).77C4, C2.27(C2xC8oD4), C4:C4.229(C2xC4), (C2xC22:C8):46C2, (C2xD4).235(C2xC4), (C2xC22:C4).51C4, C22:C4.77(C2xC4), (C2xC4).276(C22xC4), (C22xC4).352(C2xC4), SmallGroup(128,1706)

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

Derived series
C1 — C22 — D4:7M4(2)
Chief series
C1 — C2 — C4 — C2xC4 — C22xC4 — C23xC4 — C2xC4xD4 — D4:7M4(2)
Lower central
C1 — C22 — D4:7M4(2)
Upper central
C1 — C2xC4 — D4:7M4(2)
Jennings
C1 — C2 — C2 — C2xC4 — D4:7M4(2)

Generators and relations for D4:7M4(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, C2, C4, C4, C22, C22, C22, C8, C2xC4, C2xC4, C2xC4, D4, D4, C23, C23, C23, C42, C22:C4, C4:C4, C4:C4, C2xC8, C2xC8, M4(2), C22xC4, C22xC4, C22xC4, C2xD4, C2xD4, C2xD4, C24, C4xC8, C8:C4, C22:C8, C22:C8, C4:C8, C2xC42, C2xC22:C4, C2xC4:C4, C4xD4, C4xD4, C22xC8, C2xM4(2), C23xC4, C22xD4, C2xC22:C8, C24.4C4, C42.12C4, C42.6C4, C8xD4, C8:9D4, C8:6D4, C2xC4xD4, D4:7M4(2)
Quotients: C1, C2, C4, C22, C2xC4, C23, M4(2), C22xC4, C24, C2xM4(2), C8oD4, C23xC4, 2+ 1+4, C22.11C24, C22xM4(2), C2xC8oD4, D4:7M4(2)

Smallest permutation representation of D4:7M4(2)
►On 32 points
Generators in S32
(1 19 25 12)(2 20 26 13)(3 21 27 14)(4 22 28 15)(5 23 29 16)(6 24 30 9)(7 17 31 10)(8 18 32 11)

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

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

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)C8oD42+ 1+4
kernelD4:7M4(2)C2xC22:C8C24.4C4C42.12C4C42.6C4C8xD4C8:9D4C8:6D4C2xC4xD4C2xC22:C4C2xC4:C4C4xD4C22xD4D4C22C4
# reps1221124214282882

Matrix representation of D4:7M4(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] >;
 

D4:7M4(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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