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G = M4(2).41D4  order 128 = 27

5th non-split extension by M4(2) of D4 acting via D4/C22=C2

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

Aliases: M4(2).41D4, C4≀C2⋊3C4, (C2×C8).25D4, (C2×D4).7Q8, (C2×Q8).4Q8, D4.6(C4⋊C4), C4○D4.43D4, C4.142(C4×D4), Q8.6(C4⋊C4), C22.32(C4×D4), C42⋊6C4⋊18C2, C4.118C22≀C2, C42.146(C2×C4), Q8○M4(2).2C2, M4(2).3(C2×C4), C4.97(C22⋊Q8), M4(2).C4⋊2C2, M4(2)⋊4C4⋊1C2, C4⋊M4(2)⋊25C2, C22.2(C22⋊Q8), C23.117(C4○D4), (C2×C42).267C22, (C22×C4).677C23, C42⋊C22.3C2, C42⋊C2.12C22, C2.17(C23.8Q8), (C2×M4(2)).175C22, C22.3(C22.D4), C4.11(C2×C4⋊C4), (C2×C4≀C2).3C2, C4○D4.6(C2×C4), (C2×C4).11(C2×Q8), (C2×C4).987(C2×D4), (C2×C4).52(C4○D4), (C2×C4).183(C22×C4), (C2×C4○D4).13C22, SmallGroup(128,593)

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

Derived series
C1 — C2×C4 — M4(2).41D4
Chief series
C1 — C2 — C4 — C2×C4 — C22×C4 — C2×C4○D4 — Q8○M4(2) — M4(2).41D4
Lower central
C1 — C2 — C2×C4 — M4(2).41D4
Upper central
C1 — C4 — C22×C4 — M4(2).41D4
Jennings
C1 — C2 — C2 — C22×C4 — M4(2).41D4

Generators and relations for M4(2).41D4
 G = < a,b,c,d | a8=b2=c4=1, d2=a6, bab=a5, cac-1=dad-1=a-1b, cbc-1=dbd-1=a4b, dcd-1=a6c-1 >

Subgroups: 236 in 131 conjugacy classes, 54 normal (36 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, C42, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), M4(2), C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C4○D4, C4≀C2, C4≀C2, C4⋊C8, C8.C4, C2×C42, C42⋊C2, C2×M4(2), C2×M4(2), C8○D4, C2×C4○D4, C42⋊6C4, M4(2)⋊4C4, C2×C4≀C2, C42⋊C22, C4⋊M4(2), M4(2).C4, Q8○M4(2), M4(2).41D4
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C4⋊C4, C22×C4, C2×D4, C2×Q8, C4○D4, C2×C4⋊C4, C4×D4, C22≀C2, C22⋊Q8, C22.D4, C23.8Q8, M4(2).41D4

Permutation representations of M4(2).41D4
►On 16 points - transitive group 16T215
Generators in S16
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)

(2 6)(4 8)(9 13)(11 15)

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

(1 9 7 15 5 13 3 11)(2 16 8 14 6 12 4 10)
 
G:=sub<Sym(16)| (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (2,6)(4,8)(9,13)(11,15), (1,4,5,8)(2,3,6,7)(9,12)(10,15)(11,14)(13,16), (1,9,7,15,5,13,3,11)(2,16,8,14,6,12,4,10)>;
 
G:=Group( (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (2,6)(4,8)(9,13)(11,15), (1,4,5,8)(2,3,6,7)(9,12)(10,15)(11,14)(13,16), (1,9,7,15,5,13,3,11)(2,16,8,14,6,12,4,10) );
 
G=PermutationGroup([[(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16)], [(2,6),(4,8),(9,13),(11,15)], [(1,4,5,8),(2,3,6,7),(9,12),(10,15),(11,14),(13,16)], [(1,9,7,15,5,13,3,11),(2,16,8,14,6,12,4,10)]])
 
G:=TransitiveGroup(16,215);
 

32 conjugacy classes

class 1 2A2B2C2D2E2F4A4B4C4D4E4F···4K4L4M8A···8H8I8J8K8L
order1222222444444···4448···88888
size1122244112224···4884···48888

32 irreducible representations

dim11111111122222224
type++++++++++--+
imageC1C2C2C2C2C2C2C2C4D4D4Q8Q8D4C4○D4C4○D4M4(2).41D4
kernelM4(2).41D4C42⋊6C4M4(2)⋊4C4C2×C4≀C2C42⋊C22C4⋊M4(2)M4(2).C4Q8○M4(2)C4≀C2C2×C8M4(2)C2×D4C2×Q8C4○D4C2×C4C23C1
# reps11111111822112224

Matrix representation of M4(2).41D4 ►in GL4(𝔽5) generated by

0004
0030
0400
3000
,
2003
0240
0330
4003
,
3002
0430
0010
0002
,
0400
2000
3001
0330
G:=sub<GL(4,GF(5))| [0,0,0,3,0,0,4,0,0,3,0,0,4,0,0,0],[2,0,0,4,0,2,3,0,0,4,3,0,3,0,0,3],[3,0,0,0,0,4,0,0,0,3,1,0,2,0,0,2],[0,2,3,0,4,0,0,3,0,0,0,3,0,0,1,0] >;
 

M4(2).41D4 in GAP, Magma, Sage, TeX 

M_4(2)._{41}D_4
 
% in TeX
 
G:=Group("M4(2).41D4");
 
// GroupNames label
 
G:=SmallGroup(128,593);
 
// by ID
 
G=gap.SmallGroup(128,593);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,232,422,2019,1018,248,2804,718,172,124]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^8=b^2=c^4=1,d^2=a^6,b*a*b=a^5,c*a*c^-1=d*a*d^-1=a^-1*b,c*b*c^-1=d*b*d^-1=a^4*b,d*c*d^-1=a^6*c^-1>;
 
// generators/relations
 

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