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G = C42.427D4  order 128 = 27

60th non-split extension by C42 of D4 acting via D4/C22=C2

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

Aliases: C42.427D4, M4(2).25D4, C4.121(C4×D4), C4.D4⋊4C4, C42⋊6C4⋊5C2, C4.10D4⋊4C4, (C4×M4(2))⋊23C2, M4(2).9(C2×C4), M4(2).C4⋊3C2, C4.194(C4⋊D4), M4(2)⋊4C4⋊18C2, C4.17(C42⋊2C2), C23.125(C4○D4), C42⋊C22.7C2, (C2×C42).301C22, (C22×C4).695C23, C22.7(C4.4D4), C42⋊C2.37C22, C22.11(C42⋊C2), (C2×M4(2)).322C22, C2.17(C24.C22), C22.29(C22.D4), M4(2).8C22.2C2, (C2×C4≀C2).11C2, (C2×D4).98(C2×C4), (C2×Q8).83(C2×C4), (C2×C4).1340(C2×D4), (C2×C4).17(C22×C4), (C2×C4).336(C4○D4), (C2×C4○D4).34C22, SmallGroup(128,664)

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

Derived series
C1 — C2×C4 — C42.427D4
Chief series
C1 — C2 — C4 — C2×C4 — C22×C4 — C2×M4(2) — C4×M4(2) — C42.427D4
Lower central
C1 — C2 — C2×C4 — C42.427D4
Upper central
C1 — C4 — C22×C4 — C42.427D4
Jennings
C1 — C2 — C2 — C22×C4 — C42.427D4

Generators and relations for C42.427D4
 G = < a,b,c,d | a4=b4=c4=1, d2=b, ab=ba, cac-1=dad-1=a-1b-1, bc=cb, bd=db, dcd-1=a2c-1 >

Subgroups: 212 in 108 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, C2×C8, M4(2), M4(2), C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C4×C8, C8⋊C4, C4.D4, C4.10D4, C4≀C2, C8.C4, C2×C42, C42⋊C2, C2×M4(2), C2×C4○D4, C42⋊6C4, M4(2)⋊4C4, C4×M4(2), M4(2).8C22, C2×C4≀C2, C42⋊C22, M4(2).C4, C42.427D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22×C4, C2×D4, C4○D4, C42⋊C2, C4×D4, C4⋊D4, C22.D4, C4.4D4, C42⋊2C2, C24.C22, C42.427D4

Permutation representations of C42.427D4
►On 16 points - transitive group 16T321
Generators in S16
(2 8 6 4)(10 16 14 12)

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

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

(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
 
G:=sub<Sym(16)| (2,8,6,4)(10,16,14,12), (1,3,5,7)(2,4,6,8)(9,11,13,15)(10,12,14,16), (1,14,5,10)(2,11)(3,16,7,12)(4,13)(6,15)(8,9), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)>;
 
G:=Group( (2,8,6,4)(10,16,14,12), (1,3,5,7)(2,4,6,8)(9,11,13,15)(10,12,14,16), (1,14,5,10)(2,11)(3,16,7,12)(4,13)(6,15)(8,9), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16) );
 
G=PermutationGroup([[(2,8,6,4),(10,16,14,12)], [(1,3,5,7),(2,4,6,8),(9,11,13,15),(10,12,14,16)], [(1,14,5,10),(2,11),(3,16,7,12),(4,13),(6,15),(8,9)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16)]])
 
G:=TransitiveGroup(16,321);
 

32 conjugacy classes

class 1 2A2B2C2D2E4A4B4C···4I4J4K4L4M4N8A···8H8I8J8K8L
order122222444···4444448···88888
size112228112···2448884···48888

32 irreducible representations

dim111111111122224
type++++++++++
imageC1C2C2C2C2C2C2C2C4C4D4D4C4○D4C4○D4C42.427D4
kernelC42.427D4C42⋊6C4M4(2)⋊4C4C4×M4(2)M4(2).8C22C2×C4≀C2C42⋊C22M4(2).C4C4.D4C4.10D4C42M4(2)C2×C4C23C1
# reps111111114422624

Matrix representation of C42.427D4 ►in GL4(𝔽5) generated by

0300
1300
0003
0013
,
3000
0300
0030
0003
,
0034
0002
1000
1400
,
0400
2000
0014
0034
G:=sub<GL(4,GF(5))| [0,1,0,0,3,3,0,0,0,0,0,1,0,0,3,3],[3,0,0,0,0,3,0,0,0,0,3,0,0,0,0,3],[0,0,1,1,0,0,0,4,3,0,0,0,4,2,0,0],[0,2,0,0,4,0,0,0,0,0,1,3,0,0,4,4] >;
 

C42.427D4 in GAP, Magma, Sage, TeX 

C_4^2._{427}D_4
 
% in TeX
 
G:=Group("C4^2.427D4");
 
// GroupNames label
 
G:=SmallGroup(128,664);
 
// by ID
 
G=gap.SmallGroup(128,664);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,288,422,58,2019,248,718,124]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^4=b^4=c^4=1,d^2=b,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1*b^-1,b*c=c*b,b*d=d*b,d*c*d^-1=a^2*c^-1>;
 
// generators/relations
 

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