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G = M4(2)⋊20D4order 128 = 27

7th semidirect product of M4(2) and D4 acting via D4/C22=C2

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

Aliases: M4(2)⋊20D4, C24.8(C2×C4), C4.93C22≀C2, C22.51(C4×D4), (C22×D4).13C4, (C22×C4).294D4, C4.133(C4⋊D4), C24.4C429C2, C221(C4.D4), C22.C4217C2, (C22×M4(2))⋊12C2, (C22×C4).688C23, C23.194(C22×C4), (C23×C4).263C22, (C22×D4).28C22, C23.125(C22⋊C4), C4.88(C22.D4), C2.39(C23.23D4), (C2×M4(2)).187C22, C2.26(M4(2).8C22), (C2×C4⋊D4).10C2, (C2×C4.D4)⋊17C2, (C2×C4).1336(C2×D4), (C2×C4⋊C4).64C22, (C2×C22⋊C4).11C4, (C22×C4).20(C2×C4), C2.27(C2×C4.D4), (C2×C4).322(C4○D4), (C2×C4).197(C22⋊C4), C22.275(C2×C22⋊C4), SmallGroup(128,632)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — M4(2)⋊20D4
C1C2C4C2×C4C22×C4C23×C4C22×M4(2) — M4(2)⋊20D4
C1C2C23 — M4(2)⋊20D4
C1C22C23×C4 — M4(2)⋊20D4
C1C2C2C22×C4 — M4(2)⋊20D4

Generators and relations for M4(2)⋊20D4
 G = < a,b,c,d | a8=b2=c4=d2=1, bab=a5, cac-1=dad=ab, bc=cb, bd=db, dcd=c-1 >

Subgroups: 444 in 195 conjugacy classes, 58 normal (28 characteristic)
C1, C2 [×3], C2 [×7], C4 [×2], C4 [×2], C4 [×4], C22, C22 [×4], C22 [×21], C8 [×6], C2×C4 [×4], C2×C4 [×4], C2×C4 [×14], D4 [×12], C23 [×3], C23 [×15], C22⋊C4 [×4], C4⋊C4 [×2], C2×C8 [×8], M4(2) [×4], M4(2) [×10], C22×C4 [×6], C22×C4 [×2], C22×C4 [×4], C2×D4 [×14], C24, C24 [×2], C22⋊C8 [×2], C4.D4 [×4], C2×C22⋊C4 [×2], C2×C4⋊C4, C4⋊D4 [×4], C22×C8, C2×M4(2) [×4], C2×M4(2) [×5], C23×C4, C22×D4, C22×D4 [×2], C22.C42 [×2], C24.4C4, C2×C4.D4 [×2], C2×C4⋊D4, C22×M4(2), M4(2)⋊20D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×8], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×4], C4○D4 [×2], C4.D4 [×2], C2×C22⋊C4, C4×D4 [×2], C22≀C2, C4⋊D4 [×2], C22.D4, C23.23D4, C2×C4.D4, M4(2).8C22, M4(2)⋊20D4

Smallest permutation representation of M4(2)⋊20D4
On 32 points
Generators in S32
(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)
(1 5)(3 7)(9 13)(11 15)(17 21)(19 23)(25 29)(27 31)
(1 9 31 19)(2 10 32 20)(3 15 25 17)(4 16 26 18)(5 13 27 23)(6 14 28 24)(7 11 29 21)(8 12 30 22)
(3 7)(4 8)(9 19)(10 20)(11 17)(12 18)(13 23)(14 24)(15 21)(16 22)(25 29)(26 30)

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

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

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

32 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J4A···4F4G4H4I8A···8H8I8J8K8L
order122222222224···44448···88888
size111122224882···24884···48888

32 irreducible representations

dim1111111122244
type+++++++++
imageC1C2C2C2C2C2C4C4D4D4C4○D4C4.D4M4(2).8C22
kernelM4(2)⋊20D4C22.C42C24.4C4C2×C4.D4C2×C4⋊D4C22×M4(2)C2×C22⋊C4C22×D4M4(2)C22×C4C2×C4C22C2
# reps1212114444422

Matrix representation of M4(2)⋊20D4 in GL6(𝔽17)

0130000
400000
0000160
0000016
0011500
0011600
,
1600000
0160000
0016000
0001600
000010
000001
,
1300000
040000
0013800
000400
000049
0000013
,
0160000
1600000
001000
0011600
0000160
0000161

G:=sub<GL(6,GF(17))| [0,4,0,0,0,0,13,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,15,16,0,0,16,0,0,0,0,0,0,16,0,0],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[13,0,0,0,0,0,0,4,0,0,0,0,0,0,13,0,0,0,0,0,8,4,0,0,0,0,0,0,4,0,0,0,0,0,9,13],[0,16,0,0,0,0,16,0,0,0,0,0,0,0,1,1,0,0,0,0,0,16,0,0,0,0,0,0,16,16,0,0,0,0,0,1] >;

M4(2)⋊20D4 in GAP, Magma, Sage, TeX

M_4(2)\rtimes_{20}D_4
% in TeX

G:=Group("M4(2):20D4");
// GroupNames label

G:=SmallGroup(128,632);
// by ID

G=gap.SmallGroup(128,632);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,288,422,2019,1018,2028,124]);
// Polycyclic

G:=Group<a,b,c,d|a^8=b^2=c^4=d^2=1,b*a*b=a^5,c*a*c^-1=d*a*d=a*b,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

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