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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), (C8×D4)⋊45C2, C8⋊9D4⋊40C2, C8⋊6D4⋊39C2, C4⋊C8⋊90C22, (C4×C8)⋊59C22, D4○3(C22⋊C8), (C4×D4).34C4, C24.85(C2×C4), C8⋊C4⋊30C22, C22⋊C8⋊79C22, C42.222(C2×C4), (C2×C4).671C24, (C2×C8).432C23, (C22×C8)⋊55C22, (C22×D4).43C4, C4.35(C2×M4(2)), C24.4C4⋊35C2, C42.6C4⋊51C2, (C4×D4).363C22, C22.16(C8○D4), C42.12C4⋊52C2, C22.7(C2×M4(2)), (C2×M4(2))⋊45C22, C22.195(C23×C4), (C22×C4).939C23, (C23×C4).530C22, (C2×C42).781C22, C23.229(C22×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

Derived series
C1 — C22 — D4⋊7M4(2)
Chief series
C1 — C2 — C4 — C2×C4 — C22×C4 — C23×C4 — C2×C4×D4 — D4⋊7M4(2)
Lower central
C1 — C22 — D4⋊7M4(2)
Upper central
C1 — C2×C4 — D4⋊7M4(2)
Jennings
C1 — C2 — C2 — C2×C4 — D4⋊7M4(2)

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], C8⋊9D4 [×4], C8⋊6D4 [×2], C2×C4×D4, D4⋊7M4(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, D4⋊7M4(2)

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

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

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

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

Matrix representation ►G ⊆ 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] >;
 

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)
kernelD4⋊7M4(2)C2×C22⋊C8C24.4C4C42.12C4C42.6C4C8×D4C8⋊9D4C8⋊6D4C2×C4×D4C2×C22⋊C4C2×C4⋊C4C4×D4C22×D4D4C22C4
# reps1221124214282882

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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