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G = M4(2).7D4order 128 = 27

7th non-split extension by M4(2) of D4 acting via D4/C2=C22

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

Aliases: M4(2).7D4, C4⋊C4.93D4, (C2×Q8).93D4, (C2×D4).102D4, (C22×C4).81D4, C426C433C2, C4.21(C4⋊D4), C23.590(C2×D4), M4(2)⋊C46C2, C4.41(C4.4D4), C22.220C22≀C2, C2.29(D4.8D4), C22.67(C4⋊D4), (C22×C4).723C23, (C2×C42).361C22, C2.27(D4.10D4), (C22×Q8).65C22, C42⋊C2.57C22, C23.67C239C2, C23.36D4.10C2, C4.70(C22.D4), C2.11(C23.10D4), (C2×M4(2)).224C22, C23.38C23.5C2, C22.33(C22.D4), (C2×C4≀C2).4C2, (C2×C4).258(C2×D4), (C2×C4).79(C4○D4), (C2×C4.10D4)⋊5C2, (C2×C4⋊C4).123C22, (C2×C4○D4).58C22, SmallGroup(128,770)

Series: Derived Chief Lower central Upper central Jennings

C1C22×C4 — M4(2).7D4
C1C2C22C2×C4C22×C4C2×C4⋊C4C23.67C23 — M4(2).7D4
C1C2C22×C4 — M4(2).7D4
C1C22C22×C4 — M4(2).7D4
C1C2C2C22×C4 — M4(2).7D4

Generators and relations for M4(2).7D4
 G = < a,b,c,d | a8=b2=c4=d2=1, bab=a5, cac-1=a3, dad=a-1b, cbc-1=dbd=a4b, dcd=a4c-1 >

Subgroups: 296 in 139 conjugacy classes, 42 normal (38 characteristic)
C1, C2 [×3], C2 [×3], C4 [×4], C4 [×8], C22 [×3], C22 [×5], C8 [×3], C2×C4 [×6], C2×C4 [×18], D4 [×4], Q8 [×6], C23, C23, C42 [×3], C22⋊C4 [×5], C4⋊C4 [×2], C4⋊C4 [×6], C2×C8 [×2], M4(2) [×2], M4(2) [×3], C22×C4, C22×C4 [×2], C22×C4 [×3], C2×D4, C2×D4, C2×Q8, C2×Q8 [×6], C4○D4 [×4], C2.C42 [×2], C4.10D4 [×2], D4⋊C4, Q8⋊C4, C4≀C2 [×2], C4.Q8, C2.D8, C2×C42, C2×C4⋊C4, C42⋊C2, C22⋊Q8 [×2], C22.D4 [×2], C4.4D4, C4⋊Q8, C2×M4(2) [×2], C22×Q8, C2×C4○D4, C426C4, C23.67C23, C2×C4.10D4, C23.36D4, C2×C4≀C2, M4(2)⋊C4, C23.38C23, M4(2).7D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×8], C23, C2×D4 [×4], C4○D4 [×3], C22≀C2, C4⋊D4 [×3], C22.D4 [×2], C4.4D4, C23.10D4, D4.8D4, D4.10D4, M4(2).7D4

Character table of M4(2).7D4

 class 12A2B2C2D2E2F4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O8A8B8C8D
 size 11112282222444488888888888
ρ111111111111111111111111111    trivial
ρ2111111-11111-1-1-1-1-1-1-11-1-111111    linear of order 2
ρ3111111-111111111-111-1-1-1-111-1-1    linear of order 2
ρ411111111111-1-1-1-11-1-1-111-111-1-1    linear of order 2
ρ511111111111-1-1-1-11111-1-11-1-1-1-1    linear of order 2
ρ6111111-111111111-1-1-11111-1-1-1-1    linear of order 2
ρ7111111-11111-1-1-1-1-111-111-1-1-111    linear of order 2
ρ81111111111111111-1-1-1-1-1-1-1-111    linear of order 2
ρ92222-2-2-22-2-22000020000000000    orthogonal lifted from D4
ρ102222220-2-2-2-20000000200-20000    orthogonal lifted from D4
ρ112222-2-20-222-200000-2200000000    orthogonal lifted from D4
ρ122222220-2-2-2-20000000-20020000    orthogonal lifted from D4
ρ132222-2-20-222-2000002-200000000    orthogonal lifted from D4
ρ142222-2-222-2-220000-20000000000    orthogonal lifted from D4
ρ152-2-222-202-22-20000000000000-22    orthogonal lifted from D4
ρ162-2-222-202-22-200000000000002-2    orthogonal lifted from D4
ρ172-2-22-220-2-222000000002i-2i00000    complex lifted from C4○D4
ρ182-2-22-22022-2-2-2i-2i2i2i00000000000    complex lifted from C4○D4
ρ192-2-222-20-22-2200000000000-2i2i00    complex lifted from C4○D4
ρ202-2-22-220-2-22200000000-2i2i00000    complex lifted from C4○D4
ρ212-2-22-22022-2-22i2i-2i-2i00000000000    complex lifted from C4○D4
ρ222-2-222-20-22-22000000000002i-2i00    complex lifted from C4○D4
ρ2344-4-400000002-22-200000000000    symplectic lifted from D4.10D4, Schur index 2
ρ2444-4-40000000-22-2200000000000    symplectic lifted from D4.10D4, Schur index 2
ρ254-44-400000002i-2i-2i2i00000000000    complex lifted from D4.8D4
ρ264-44-40000000-2i2i2i-2i00000000000    complex lifted from D4.8D4

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

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

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

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

Matrix representation of M4(2).7D4 in GL6(𝔽17)

110000
0160000
000010
0011162
004000
00106016
,
1600000
0160000
0016000
0001600
000010
001101
,
13130000
040000
0011162
000010
0001600
001616016
,
100000
15160000
000010
001616115
001000
0010161

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,1,16,0,0,0,0,0,0,0,1,4,10,0,0,0,1,0,6,0,0,1,16,0,0,0,0,0,2,0,16],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,1,0,0,0,16,0,1,0,0,0,0,1,0,0,0,0,0,0,1],[13,0,0,0,0,0,13,4,0,0,0,0,0,0,1,0,0,16,0,0,1,0,16,16,0,0,16,1,0,0,0,0,2,0,0,16],[1,15,0,0,0,0,0,16,0,0,0,0,0,0,0,16,1,1,0,0,0,16,0,0,0,0,1,1,0,16,0,0,0,15,0,1] >;

M4(2).7D4 in GAP, Magma, Sage, TeX

M_4(2)._7D_4
% in TeX

G:=Group("M4(2).7D4");
// GroupNames label

G:=SmallGroup(128,770);
// by ID

G=gap.SmallGroup(128,770);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,141,456,422,387,58,2804,1411,718,172,4037]);
// 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=a^3,d*a*d=a^-1*b,c*b*c^-1=d*b*d=a^4*b,d*c*d=a^4*c^-1>;
// generators/relations

Export

Character table of M4(2).7D4 in TeX

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