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G = M4(2).41D4order 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≀C23C4, (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), C426C418C2, C4.118C22≀C2, C42.146(C2×C4), Q8○M4(2).2C2, M4(2).3(C2×C4), C4.97(C22⋊Q8), M4(2).C42C2, M4(2)⋊4C41C2, 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

C1C2×C4 — M4(2).41D4
C1C2C4C2×C4C22×C4C2×C4○D4Q8○M4(2) — M4(2).41D4
C1C2C2×C4 — M4(2).41D4
C1C4C22×C4 — M4(2).41D4
C1C2C2C22×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 [×5], C4 [×4], C4 [×6], C22 [×3], C22 [×4], C8 [×7], C2×C4 [×6], C2×C4 [×10], D4 [×2], D4 [×5], Q8 [×2], Q8, C23, C23, C42 [×2], C42 [×2], C22⋊C4, C4⋊C4, C2×C8 [×2], C2×C8 [×7], M4(2) [×4], M4(2) [×10], C22×C4, C22×C4 [×2], C2×D4, C2×D4, C2×Q8, C4○D4 [×4], C4○D4 [×2], C4≀C2 [×4], C4≀C2 [×2], C4⋊C8 [×2], C8.C4 [×2], C2×C42, C42⋊C2, C2×M4(2) [×4], C2×M4(2) [×2], C8○D4 [×4], C2×C4○D4, C426C4, M4(2)⋊4C4, C2×C4≀C2, C42⋊C22, C4⋊M4(2), M4(2).C4, Q8○M4(2), M4(2).41D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×6], Q8 [×2], C23, C4⋊C4 [×4], C22×C4, C2×D4 [×3], C2×Q8, C4○D4 [×2], C2×C4⋊C4, C4×D4 [×2], C22≀C2, C22⋊Q8 [×2], 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).41D4C426C4M4(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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