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G = C42.130D4order 128 = 27

112nd non-split extension by C42 of D4 acting via D4/C2=C22

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

Aliases: C42.130D4, M4(2).2D4, C4⋊C4.81D4, C4.4C22≀C2, (C2×D4).88D4, (C2×Q8).79D4, C426C49C2, (C22×C4).74D4, C4.42(C4⋊D4), C4.32(C41D4), C23.580(C2×D4), C22.198C22≀C2, C23.38D428C2, C22.58(C4⋊D4), C2.10(C232D4), (C22×C4).711C23, (C2×C42).344C22, C2.24(D4.10D4), (C22×Q8).50C22, C42⋊C2.49C22, (C2×M4(2)).14C22, C23.38C23.4C2, (C2×C4⋊Q8)⋊1C2, (C2×C4≀C2).14C2, (C2×C4).75(C4○D4), (C2×C4.10D4)⋊2C2, (C2×C4).1026(C2×D4), (C2×C8.C22).5C2, (C2×C4○D4).46C22, SmallGroup(128,737)

Series: Derived Chief Lower central Upper central Jennings

C1C22×C4 — C42.130D4
C1C2C22C23C22×C4C22×Q8C2×C4⋊Q8 — C42.130D4
C1C2C22×C4 — C42.130D4
C1C22C22×C4 — C42.130D4
C1C2C2C22×C4 — C42.130D4

Generators and relations for C42.130D4
 G = < a,b,c,d | a4=b4=d2=1, c4=b2, ab=ba, cac-1=a-1b, dad=ab-1, bc=cb, dbd=b-1, dcd=c3 >

Subgroups: 368 in 181 conjugacy classes, 46 normal (34 characteristic)
C1, C2, C2 [×2], C2 [×3], C4 [×4], C4 [×11], C22 [×3], C22 [×5], C8 [×3], C2×C4 [×6], C2×C4 [×21], D4 [×4], Q8 [×12], C23, C23, C42 [×2], C42 [×2], C22⋊C4 [×5], C4⋊C4 [×2], C4⋊C4 [×12], C2×C8 [×2], M4(2) [×2], M4(2) [×3], SD16 [×4], Q16 [×4], C22×C4, C22×C4 [×2], C22×C4 [×3], C2×D4, C2×D4, C2×Q8, C2×Q8 [×2], C2×Q8 [×11], C4○D4 [×4], C4.10D4 [×2], Q8⋊C4 [×2], C4≀C2 [×2], C2×C42, C2×C4⋊C4 [×2], C42⋊C2, C22⋊Q8 [×2], C22.D4 [×2], C4.4D4, C4⋊Q8 [×5], C2×M4(2) [×2], C2×SD16, C2×Q16, C8.C22 [×4], C22×Q8 [×2], C2×C4○D4, C426C4, C2×C4.10D4, C23.38D4, C2×C4≀C2, C2×C4⋊Q8, C23.38C23, C2×C8.C22, C42.130D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×12], C23, C2×D4 [×6], C4○D4, C22≀C2 [×3], C4⋊D4 [×3], C41D4, C232D4, D4.10D4 [×2], C42.130D4

Character table of C42.130D4

 class 12A2B2C2D2E2F4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O8A8B8C8D
 size 11112282222444488888888888
ρ111111111111111111111111111    trivial
ρ21111111111111111-1-1-1-1-1-1-11-11    linear of order 2
ρ3111111-11111-1-1-1-1-1-1-11-11-11111    linear of order 2
ρ4111111-11111-1-1-1-1-111-11-11-11-11    linear of order 2
ρ511111111111-1-1-1-11-1-1-11-111-11-1    linear of order 2
ρ611111111111-1-1-1-11111-11-1-1-1-1-1    linear of order 2
ρ7111111-111111111-111-1-1-1-11-11-1    linear of order 2
ρ8111111-111111111-1-1-11111-1-1-1-1    linear of order 2
ρ92222-2-2-22-2-22000020000000000    orthogonal lifted from D4
ρ102-2-22-220-2-2220000000020-20000    orthogonal lifted from D4
ρ112222220-2-2-2-2000000020-200000    orthogonal lifted from D4
ρ122222-2-20-222-200000-2200000000    orthogonal lifted from D4
ρ132222-2-20-222-2000002-200000000    orthogonal lifted from D4
ρ142-2-22-22022-2-2-2-22200000000000    orthogonal lifted from D4
ρ152222220-2-2-2-20000000-20200000    orthogonal lifted from D4
ρ162222-2-222-2-220000-20000000000    orthogonal lifted from D4
ρ172-2-222-20-22-2200000000000-2020    orthogonal lifted from D4
ρ182-2-22-220-2-22200000000-2020000    orthogonal lifted from D4
ρ192-2-22-22022-2-222-2-200000000000    orthogonal lifted from D4
ρ202-2-222-20-22-220000000000020-20    orthogonal lifted from D4
ρ212-2-222-202-22-2000000000000-2i02i    complex lifted from C4○D4
ρ222-2-222-202-22-20000000000002i0-2i    complex lifted from C4○D4
ρ234-44-400000002-2-2200000000000    symplectic lifted from D4.10D4, Schur index 2
ρ244-44-40000000-222-200000000000    symplectic lifted from D4.10D4, Schur index 2
ρ2544-4-400000002-22-200000000000    symplectic lifted from D4.10D4, Schur index 2
ρ2644-4-40000000-22-2200000000000    symplectic lifted from D4.10D4, Schur index 2

Smallest permutation representation of C42.130D4
On 32 points
Generators in S32
(1 26 5 30)(2 6)(3 28 7 32)(4 8)(9 21 13 17)(10 14)(11 23 15 19)(12 16)(18 22)(20 24)(25 29)(27 31)
(1 30 5 26)(2 31 6 27)(3 32 7 28)(4 25 8 29)(9 17 13 21)(10 18 14 22)(11 19 15 23)(12 20 16 24)
(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 12)(2 15)(3 10)(4 13)(5 16)(6 11)(7 14)(8 9)(17 25)(18 28)(19 31)(20 26)(21 29)(22 32)(23 27)(24 30)

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

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

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

Matrix representation of C42.130D4 in GL6(𝔽17)

1610000
1510000
0016200
0016100
0000160
0000016
,
100000
010000
0011500
0011600
0000162
0000161
,
1300000
940000
0000914
0000168
006200
0071100
,
4130000
8130000
000010
000001
001000
000100

G:=sub<GL(6,GF(17))| [16,15,0,0,0,0,1,1,0,0,0,0,0,0,16,16,0,0,0,0,2,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,15,16,0,0,0,0,0,0,16,16,0,0,0,0,2,1],[13,9,0,0,0,0,0,4,0,0,0,0,0,0,0,0,6,7,0,0,0,0,2,11,0,0,9,16,0,0,0,0,14,8,0,0],[4,8,0,0,0,0,13,13,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0] >;

C42.130D4 in GAP, Magma, Sage, TeX

C_4^2._{130}D_4
% in TeX

G:=Group("C4^2.130D4");
// GroupNames label

G:=SmallGroup(128,737);
// by ID

G=gap.SmallGroup(128,737);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,141,456,422,387,352,2019,1018,521,248,4037]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^4=d^2=1,c^4=b^2,a*b=b*a,c*a*c^-1=a^-1*b,d*a*d=a*b^-1,b*c=c*b,d*b*d=b^-1,d*c*d=c^3>;
// generators/relations

Export

Character table of C42.130D4 in TeX

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