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G = C42.56D4  order 128 = 27

38th non-split extension by C42 of D4 acting via D4/C2=C22

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

Aliases: C42.56D4, C4⋊Q8⋊9C4, (C4×Q8)⋊3C4, C4.14C4≀C2, C42.73(C2×C4), C42⋊4C4.5C2, C23.496(C2×D4), (C22×C4).735D4, C22⋊C8.129C22, C42.6C4.17C2, (C2×C42).176C22, (C22×C4).628C23, C23.31D4.5C2, C22⋊Q8.138C22, C2.7(C23.38D4), C22.23(C8.C22), C23.37C23.6C2, C2.C42.505C22, C2.16(C23.C23), C4⋊C4.6(C2×C4), C2.23(C2×C4≀C2), (C2×Q8).6(C2×C4), (C2×C4).1152(C2×D4), (C2×C4).118(C22×C4), (C2×C4).318(C22⋊C4), C22.182(C2×C22⋊C4), SmallGroup(128,238)

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

Derived series
C1 — C2×C4 — C42.56D4
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C23.37C23 — C42.56D4
Lower central
C1 — C22 — C2×C4 — C42.56D4
Upper central
C1 — C22 — C2×C42 — C42.56D4
Jennings
C1 — C2 — C22 — C22×C4 — C42.56D4

Generators and relations for C42.56D4
 G = < a,b,c,d | a4=b4=c4=1, d2=a2b-1, ab=ba, ac=ca, dad-1=a-1, cbc-1=a2b, bd=db, dcd-1=b-1c-1 >

Subgroups: 220 in 114 conjugacy classes, 46 normal (26 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, Q8, C23, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C2×Q8, C2×Q8, C2.C42, C2.C42, C8⋊C4, C22⋊C8, C4⋊C8, C2×C42, C2×C42, C42⋊C2, C4×Q8, C4×Q8, C22⋊Q8, C22⋊Q8, C42.C2, C4⋊Q8, C23.31D4, C42⋊4C4, C42.6C4, C23.37C23, C42.56D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C4≀C2, C2×C22⋊C4, C8.C22, C23.C23, C23.38D4, C2×C4≀C2, C42.56D4

Smallest permutation representation of C42.56D4
►On 32 points
Generators in S32
(1 12 25 21)(2 22 26 13)(3 14 27 23)(4 24 28 15)(5 16 29 17)(6 18 30 9)(7 10 31 19)(8 20 32 11)

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

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

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

32 conjugacy classes

class 1 2A2B2C2D2E4A···4H4I···4R4S4T4U4V8A8B8C8D
order1222224···44···444448888
size1111222···24···488888888

32 irreducible representations

dim111111122244
type+++++++-
imageC1C2C2C2C2C4C4D4D4C4≀C2C8.C22C23.C23
kernelC42.56D4C23.31D4C42⋊4C4C42.6C4C23.37C23C4×Q8C4⋊Q8C42C22×C4C4C22C2
# reps141114422822

Matrix representation of C42.56D4 ►in GL6(𝔽17)

100000
010000
0013000
0001300
000040
000004
,
400000
040000
0001600
0016000
000001
000010
,
1600000
140000
004000
0001300
0000013
000040
,
580000
7120000
0000013
000040
0013000
000400

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,16,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[16,1,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,13,0,0,0,0,0,0,0,4,0,0,0,0,13,0],[5,7,0,0,0,0,8,12,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,4,0,0,0,4,0,0,0,0,13,0,0,0] >;
 

C42.56D4 in GAP, Magma, Sage, TeX 

C_4^2._{56}D_4
 
% in TeX
 
G:=Group("C4^2.56D4");
 
// GroupNames label
 
G:=SmallGroup(128,238);
 
// by ID
 
G=gap.SmallGroup(128,238);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,232,387,184,1123,1018,248,1971]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^4=b^4=c^4=1,d^2=a^2*b^-1,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,c*b*c^-1=a^2*b,b*d=d*b,d*c*d^-1=b^-1*c^-1>;
 
// generators/relations
 

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