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G = C24.177D4  order 128 = 27

32nd non-split extension by C24 of D4 acting via D4/C22=C2

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

Aliases: C24.177D4, (C2×D4)⋊49D4, C4⋊C4⋊3C23, (C2×C8)⋊1C23, D4.40(C2×D4), (C2×D4)⋊2C23, (D4×C23)⋊8C2, C22⋊D8⋊11C2, (C2×Q8)⋊2C23, C4.81C22≀C2, C22⋊SD16⋊1C2, (C2×D8)⋊13C22, C22⋊C8⋊3C22, C4.40(C22×D4), D4⋊C4⋊8C22, C4⋊D4⋊50C22, C22⋊4(C8⋊C22), C24.4C4⋊2C2, (C2×C4).222C24, (C2×SD16)⋊1C22, C23.646(C2×D4), (C22×C4).419D4, C22⋊Q8⋊62C22, C22.19C24⋊3C2, C42⋊C2⋊6C22, C22.57C22≀C2, C23.37D4⋊2C2, (C2×M4(2))⋊1C22, (C22×C4).960C23, (C23×C4).542C22, C22.482(C22×D4), (C22×D4).563C22, (C2×C8⋊C22)⋊5C2, (C2×C4).450(C2×D4), (C2×C4○D4)⋊2C22, C2.10(C2×C8⋊C22), C2.40(C2×C22≀C2), SmallGroup(128,1735)

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

Derived series
C1 — C2×C4 — C24.177D4
Chief series
C1 — C2 — C22 — C2×C4 — C22×C4 — C23×C4 — D4×C23 — C24.177D4
Lower central
C1 — C2 — C2×C4 — C24.177D4
Upper central
C1 — C22 — C23×C4 — C24.177D4
Jennings
C1 — C2 — C2 — C2×C4 — C24.177D4

Subgroups: 1172 in 497 conjugacy classes, 112 normal (18 characteristic)
C1, C2, C2 [×2], C2 [×15], C4 [×4], C4 [×5], C22, C22 [×6], C22 [×81], C8 [×4], C2×C4 [×2], C2×C4 [×6], C2×C4 [×15], D4 [×8], D4 [×36], Q8 [×2], C23, C23 [×2], C23 [×91], C42, C22⋊C4 [×5], C4⋊C4 [×2], C4⋊C4 [×2], C2×C8 [×4], M4(2) [×4], D8 [×8], SD16 [×8], C22×C4 [×2], C22×C4 [×4], C22×C4 [×5], C2×D4, C2×D4 [×12], C2×D4 [×53], C2×Q8, C4○D4 [×4], C24, C24 [×22], C22⋊C8 [×4], D4⋊C4 [×8], C42⋊C2, C4×D4 [×2], C22≀C2, C4⋊D4 [×2], C22⋊Q8 [×2], C22.D4, C2×M4(2) [×2], C2×D8 [×4], C2×SD16 [×4], C8⋊C22 [×8], C23×C4, C22×D4 [×6], C22×D4 [×11], C2×C4○D4, C25, C24.4C4, C23.37D4 [×2], C22⋊D8 [×4], C22⋊SD16 [×4], C22.19C24, C2×C8⋊C22 [×2], D4×C23, C24.177D4

Quotients:
C1, C2 [×15], C22 [×35], D4 [×12], C23 [×15], C2×D4 [×18], C24, C22≀C2 [×4], C8⋊C22 [×4], C22×D4 [×3], C2×C22≀C2, C2×C8⋊C22 [×2], C24.177D4

Generators and relations
 G = < a,b,c,d,e,f | a2=b2=c2=d2=f2=1, e4=d, ab=ba, eae-1=faf=ac=ca, ad=da, bc=cb, ebe-1=fbf=bd=db, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef=de3 >

Permutation representations
►On 16 points - transitive group 16T255
Generators in S16
(1 5)(2 13)(3 7)(4 15)(6 9)(8 11)(10 14)(12 16)

(1 12)(2 9)(3 14)(4 11)(5 16)(6 13)(7 10)(8 15)

(1 16)(2 9)(3 10)(4 11)(5 12)(6 13)(7 14)(8 15)

(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)

(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)

(1 9)(2 16)(3 15)(4 14)(5 13)(6 12)(7 11)(8 10)
 
G:=sub<Sym(16)| (1,5)(2,13)(3,7)(4,15)(6,9)(8,11)(10,14)(12,16), (1,12)(2,9)(3,14)(4,11)(5,16)(6,13)(7,10)(8,15), (1,16)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14)(8,15), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,9)(2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)>;
 
G:=Group( (1,5)(2,13)(3,7)(4,15)(6,9)(8,11)(10,14)(12,16), (1,12)(2,9)(3,14)(4,11)(5,16)(6,13)(7,10)(8,15), (1,16)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14)(8,15), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,9)(2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10) );
 
G=PermutationGroup([(1,5),(2,13),(3,7),(4,15),(6,9),(8,11),(10,14),(12,16)], [(1,12),(2,9),(3,14),(4,11),(5,16),(6,13),(7,10),(8,15)], [(1,16),(2,9),(3,10),(4,11),(5,12),(6,13),(7,14),(8,15)], [(1,5),(2,6),(3,7),(4,8),(9,13),(10,14),(11,15),(12,16)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16)], [(1,9),(2,16),(3,15),(4,14),(5,13),(6,12),(7,11),(8,10)])
 
G:=TransitiveGroup(16,255);
 

Matrix representation ►G ⊆ GL6(ℤ)

100000
0-10000
001000
000100
000010
000001
,
100000
010000
00-1000
000-100
000010
000001
,
-100000
0-10000
001000
000100
000010
000001
,
100000
010000
00-1000
000-100
0000-10
00000-1
,
0-10000
-100000
000001
000010
001000
000-100
,
0-10000
-100000
000010
000001
001000
000100

G:=sub<GL(6,Integers())| [1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1],[0,-1,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,1,0,0,0,0,1,0,0,0],[0,-1,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0] >;
 

32 conjugacy classes

class 1 2A2B2C2D···2I2J···2Q2R4A4B4C4D4E4F4G4H4I8A8B8C8D
order12222···22···224444444448888
size11112···24···482222448888888

32 irreducible representations

dim111111112224
type++++++++++++
imageC1C2C2C2C2C2C2C2D4D4D4C8⋊C22
kernelC24.177D4C24.4C4C23.37D4C22⋊D8C22⋊SD16C22.19C24C2×C8⋊C22D4×C23C22×C4C2×D4C24C22
# reps112441213814

In GAP, Magma, Sage, TeX 

C_2^4._{177}D_4
 
% in TeX
 
G:=Group("C2^4.177D4");
 
// GroupNames label
 
G:=SmallGroup(128,1735);
 
// by ID
 
G=gap.SmallGroup(128,1735);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,758,2019,2804,1411,172]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=f^2=1,e^4=d,a*b=b*a,e*a*e^-1=f*a*f=a*c=c*a,a*d=d*a,b*c=c*b,e*b*e^-1=f*b*f=b*d=d*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f=d*e^3>;
 
// generators/relations
 

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