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G = C42.57D4  order 128 = 27

39th non-split extension by C42 of D4 acting via D4/C2=C22

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

Aliases: C42.57D4, (C4×D4)⋊4C4, (C4×Q8)⋊4C4, C4.34C4≀C2, C4.4D4⋊8C4, C42.C2⋊4C4, C42.74(C2×C4), (C22×C4).662D4, C23.499(C2×D4), C42.6C4⋊30C2, C22.SD16.5C2, C23.31D4⋊19C2, C4⋊D4.135C22, C22⋊C8.131C22, C22.15(C8⋊C22), (C22×C4).631C23, (C2×C42).177C22, C22⋊Q8.140C22, C2.9(C23.36D4), C22.11(C8.C22), C23.36C23.6C2, C2.C42.507C22, C2.17(C23.C23), (C4×C4⋊C4)⋊2C2, C4⋊C4.9(C2×C4), C2.26(C2×C4≀C2), (C2×D4).8(C2×C4), (C2×Q8).8(C2×C4), (C2×C4).1155(C2×D4), (C2×C4).121(C22×C4), (C2×C4).173(C22⋊C4), C22.185(C2×C22⋊C4), SmallGroup(128,241)

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

Derived series
C1 — C2×C4 — C42.57D4
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C23.36C23 — C42.57D4
Lower central
C1 — C22 — C2×C4 — C42.57D4
Upper central
C1 — C22 — C2×C42 — C42.57D4
Jennings
C1 — C2 — C22 — C22×C4 — C42.57D4

Generators and relations for C42.57D4
 G = < a,b,c,d | a4=b4=c4=1, d2=b-1, ab=ba, ac=ca, dad-1=a-1b2, cbc-1=a2b-1, bd=db, dcd-1=b-1c-1 >

Subgroups: 252 in 121 conjugacy classes, 46 normal (36 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2.C42, C8⋊C4, C22⋊C8, C4⋊C8, C2×C42, C2×C42, C2×C4⋊C4, C42⋊C2, C4×D4, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C42.C2, C42⋊2C2, C22.SD16, C23.31D4, C4×C4⋊C4, C42.6C4, C23.36C23, C42.57D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C4≀C2, C2×C22⋊C4, C8⋊C22, C8.C22, C23.C23, C23.36D4, C2×C4≀C2, C42.57D4

Smallest permutation representation of C42.57D4
►On 32 points
Generators in S32
(1 12 31 21)(2 18 32 9)(3 14 25 23)(4 20 26 11)(5 16 27 17)(6 22 28 13)(7 10 29 19)(8 24 30 15)

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

(1 17 31 16)(2 24)(3 14 25 23)(4 13)(5 21 27 12)(6 20)(7 10 29 19)(8 9)(11 28)(15 32)(18 30)(22 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)| (1,12,31,21)(2,18,32,9)(3,14,25,23)(4,20,26,11)(5,16,27,17)(6,22,28,13)(7,10,29,19)(8,24,30,15), (1,7,5,3)(2,8,6,4)(9,15,13,11)(10,16,14,12)(17,23,21,19)(18,24,22,20)(25,31,29,27)(26,32,30,28), (1,17,31,16)(2,24)(3,14,25,23)(4,13)(5,21,27,12)(6,20)(7,10,29,19)(8,9)(11,28)(15,32)(18,30)(22,26), (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( (1,12,31,21)(2,18,32,9)(3,14,25,23)(4,20,26,11)(5,16,27,17)(6,22,28,13)(7,10,29,19)(8,24,30,15), (1,7,5,3)(2,8,6,4)(9,15,13,11)(10,16,14,12)(17,23,21,19)(18,24,22,20)(25,31,29,27)(26,32,30,28), (1,17,31,16)(2,24)(3,14,25,23)(4,13)(5,21,27,12)(6,20)(7,10,29,19)(8,9)(11,28)(15,32)(18,30)(22,26), (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([[(1,12,31,21),(2,18,32,9),(3,14,25,23),(4,20,26,11),(5,16,27,17),(6,22,28,13),(7,10,29,19),(8,24,30,15)], [(1,7,5,3),(2,8,6,4),(9,15,13,11),(10,16,14,12),(17,23,21,19),(18,24,22,20),(25,31,29,27),(26,32,30,28)], [(1,17,31,16),(2,24),(3,14,25,23),(4,13),(5,21,27,12),(6,20),(7,10,29,19),(8,9),(11,28),(15,32),(18,30),(22,26)], [(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 2A2B2C2D2E2F4A···4H4I···4R4S4T4U8A8B8C8D
order12222224···44···44448888
size11112282···24···48888888

32 irreducible representations

dim1111111111222444
type+++++++++-
imageC1C2C2C2C2C2C4C4C4C4D4D4C4≀C2C8⋊C22C8.C22C23.C23
kernelC42.57D4C22.SD16C23.31D4C4×C4⋊C4C42.6C4C23.36C23C4×D4C4×Q8C4.4D4C42.C2C42C22×C4C4C22C22C2
# reps1221112222228112

Matrix representation of C42.57D4 ►in GL6(𝔽17)

1300000
0130000
004000
000400
0000130
0000013
,
1300000
0130000
0041600
00151300
0000131
000024
,
1300000
0160000
0013100
000400
0000160
000091
,
0160000
1300000
0000160
000091
0013100
000400

G:=sub<GL(6,GF(17))| [13,0,0,0,0,0,0,13,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,13,0,0,0,0,0,0,13],[13,0,0,0,0,0,0,13,0,0,0,0,0,0,4,15,0,0,0,0,16,13,0,0,0,0,0,0,13,2,0,0,0,0,1,4],[13,0,0,0,0,0,0,16,0,0,0,0,0,0,13,0,0,0,0,0,1,4,0,0,0,0,0,0,16,9,0,0,0,0,0,1],[0,13,0,0,0,0,16,0,0,0,0,0,0,0,0,0,13,0,0,0,0,0,1,4,0,0,16,9,0,0,0,0,0,1,0,0] >;
 

C42.57D4 in GAP, Magma, Sage, TeX 

C_4^2._{57}D_4
 
% in TeX
 
G:=Group("C4^2.57D4");
 
// GroupNames label
 
G:=SmallGroup(128,241);
 
// by ID
 
G=gap.SmallGroup(128,241);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,387,520,1123,1018,248,1971]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^4=b^4=c^4=1,d^2=b^-1,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1*b^2,c*b*c^-1=a^2*b^-1,b*d=d*b,d*c*d^-1=b^-1*c^-1>;
 
// generators/relations
 

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