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G = C24.177D4order 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⋊C43C23, (C2×C8)⋊1C23, D4.40(C2×D4), (D4×C23)⋊8C2, (C2×D4)⋊2C23, C22⋊D811C2, (C2×Q8)⋊2C23, C4.81C22≀C2, C22⋊SD161C2, (C2×D8)⋊13C22, C22⋊C83C22, C4.40(C22×D4), D4⋊C48C22, C4⋊D450C22, C224(C8⋊C22), C24.4C42C2, (C2×C4).222C24, (C2×SD16)⋊1C22, (C22×C4).419D4, C23.646(C2×D4), C22⋊Q862C22, C42⋊C26C22, C22.19C243C2, C22.57C22≀C2, C23.37D42C2, (C2×M4(2))⋊1C22, (C23×C4).542C22, (C22×C4).960C23, 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

C1C2×C4 — C24.177D4
C1C2C22C2×C4C22×C4C23×C4D4×C23 — C24.177D4
C1C2C2×C4 — C24.177D4
C1C22C23×C4 — C24.177D4
C1C2C2C2×C4 — C24.177D4

Generators and relations for C24.177D4
 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 >

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

Permutation representations of C24.177D4
On 16 points - transitive group 16T255
Generators in S16
(1 5)(2 11)(3 7)(4 13)(6 15)(8 9)(10 14)(12 16)
(1 10)(2 15)(3 12)(4 9)(5 14)(6 11)(7 16)(8 13)
(1 14)(2 15)(3 16)(4 9)(5 10)(6 11)(7 12)(8 13)
(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 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(8 16)

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

G:=Group( (1,5)(2,11)(3,7)(4,13)(6,15)(8,9)(10,14)(12,16), (1,10)(2,15)(3,12)(4,9)(5,14)(6,11)(7,16)(8,13), (1,14)(2,15)(3,16)(4,9)(5,10)(6,11)(7,12)(8,13), (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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(8,16) );

G=PermutationGroup([(1,5),(2,11),(3,7),(4,13),(6,15),(8,9),(10,14),(12,16)], [(1,10),(2,15),(3,12),(4,9),(5,14),(6,11),(7,16),(8,13)], [(1,14),(2,15),(3,16),(4,9),(5,10),(6,11),(7,12),(8,13)], [(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)], [(1,15),(2,14),(3,13),(4,12),(5,11),(6,10),(7,9),(8,16)])

G:=TransitiveGroup(16,255);

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

Matrix representation of C24.177D4 in 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] >;

C24.177D4 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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