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G = C4.142+ 1+4order 128 = 27

14th non-split extension by C4 of 2+ 1+4 acting via 2+ 1+4/C2×D4=C2

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

Aliases: C4.142+ 1+4, C87D45C2, C82D413C2, C22⋊D820C2, C2.D89C22, (C2×D4).154D4, C2.28(D4○D8), C4⋊D46C22, (C2×Q8).130D4, C23.89(C2×D4), C4.Q821C22, C4⋊C4.138C23, C22⋊C817C22, (C2×C4).397C24, (C2×C8).155C23, (C22×C8)⋊10C22, (C2×D8).21C22, D4⋊C430C22, (C2×D4).148C23, (C22×D4)⋊25C22, C42⋊C218C22, C22.29C2415C2, C23.19D425C2, C2.78(C233D4), (C2×M4(2))⋊17C22, (C22×C4).300C23, C22.657(C22×D4), (C2×C4).534(C2×D4), (C22×C8)⋊C213C2, (C2×C4○D4).165C22, SmallGroup(128,1931)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C4.142+ 1+4
C1C2C4C2×C4C22×C4C22×D4C22.29C24 — C4.142+ 1+4
C1C2C2×C4 — C4.142+ 1+4
C1C22C2×C4○D4 — C4.142+ 1+4
C1C2C2C2×C4 — C4.142+ 1+4

Generators and relations for C4.142+ 1+4
 G = < a,b,c,d,e | a4=c2=1, b4=e2=a2, d2=ab2, dbd-1=ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc=ab3, be=eb, dcd-1=ece-1=a2c, ede-1=a-1b2d >

Subgroups: 588 in 221 conjugacy classes, 84 normal (16 characteristic)
C1, C2, C2 [×2], C2 [×7], C4 [×2], C4 [×7], C22, C22 [×29], C8 [×4], C2×C4 [×2], C2×C4 [×2], C2×C4 [×10], D4 [×23], Q8, C23, C23 [×2], C23 [×12], C42 [×2], C22⋊C4 [×10], C4⋊C4 [×4], C2×C8 [×4], C2×C8, M4(2), D8 [×4], C22×C4, C22×C4 [×2], C2×D4, C2×D4 [×6], C2×D4 [×14], C2×Q8, C4○D4 [×2], C24 [×2], C22⋊C8 [×4], D4⋊C4 [×8], C4.Q8 [×2], C2.D8 [×2], C42⋊C2 [×2], C22≀C2 [×4], C4⋊D4 [×8], C4.4D4 [×2], C41D4 [×2], C22×C8, C2×M4(2), C2×D8 [×4], C22×D4 [×2], C2×C4○D4, (C22×C8)⋊C2, C22⋊D8 [×4], C87D4 [×2], C82D4 [×2], C23.19D4 [×4], C22.29C24 [×2], C4.142+ 1+4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C24, C22×D4, 2+ 1+4 [×2], C233D4, D4○D8 [×2], C4.142+ 1+4

Character table of C4.142+ 1+4

 class 12A2B2C2D2E2F2G2H2I2J4A4B4C4D4E4F4G4H4I8A8B8C8D8E8F
 size 11114448888224448888444488
ρ111111111111111111111111111    trivial
ρ21111111-1-1-1-111111-1-1-1-1111111    linear of order 2
ρ31111111-111-111111-111-1-1-1-1-1-1-1    linear of order 2
ρ411111111-1-11111111-1-11-1-1-1-1-1-1    linear of order 2
ρ51111-11-1-111111-1-11-1-1-111-11-1-11    linear of order 2
ρ61111-11-11-1-1-111-1-11111-11-11-1-11    linear of order 2
ρ71111-11-1111-111-1-111-1-1-1-11-111-1    linear of order 2
ρ81111-11-1-1-1-1111-1-11-1111-11-111-1    linear of order 2
ρ911111-1-1-1-111111-1-111-1-1-1-1-1-111    linear of order 2
ρ1011111-1-111-1-1111-1-1-1-111-1-1-1-111    linear of order 2
ρ1111111-1-11-11-1111-1-1-11-111111-1-1    linear of order 2
ρ1211111-1-1-11-11111-1-11-11-11111-1-1    linear of order 2
ρ131111-1-111-11111-11-1-1-11-1-11-11-11    linear of order 2
ρ141111-1-11-11-1-111-11-111-11-11-11-11    linear of order 2
ρ151111-1-11-1-11-111-11-11-1111-11-11-1    linear of order 2
ρ161111-1-1111-1111-11-1-11-1-11-11-11-1    linear of order 2
ρ17222222-20000-2-2-22-20000000000    orthogonal lifted from D4
ρ182222-2-2-20000-2-22220000000000    orthogonal lifted from D4
ρ1922222-220000-2-2-2-220000000000    orthogonal lifted from D4
ρ202222-2220000-2-22-2-20000000000    orthogonal lifted from D4
ρ214-44-40000000-440000000000000    orthogonal lifted from 2+ 1+4
ρ224-44-400000004-40000000000000    orthogonal lifted from 2+ 1+4
ρ2344-4-400000000000000000220-2200    orthogonal lifted from D4○D8
ρ2444-4-400000000000000000-2202200    orthogonal lifted from D4○D8
ρ254-4-440000000000000000-22022000    orthogonal lifted from D4○D8
ρ264-4-440000000000000000220-22000    orthogonal lifted from D4○D8

Smallest permutation representation of C4.142+ 1+4
On 32 points
Generators in S32
(1 25 5 29)(2 26 6 30)(3 27 7 31)(4 28 8 32)(9 21 13 17)(10 22 14 18)(11 23 15 19)(12 24 16 20)
(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)
(2 28)(4 30)(6 32)(8 26)(9 13)(10 20)(11 15)(12 22)(14 24)(16 18)(17 21)(19 23)
(1 19 27 13)(2 12 28 18)(3 17 29 11)(4 10 30 24)(5 23 31 9)(6 16 32 22)(7 21 25 15)(8 14 26 20)
(1 11 5 15)(2 12 6 16)(3 13 7 9)(4 14 8 10)(17 31 21 27)(18 32 22 28)(19 25 23 29)(20 26 24 30)

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

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

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

Matrix representation of C4.142+ 1+4 in GL8(𝔽17)

10000000
01000000
00100000
00010000
0000161500
00001100
0000001615
00000011
,
0016150000
00110000
1615000000
11000000
00000066
000000140
0000111100
00003000
,
10000000
01000000
001600000
000160000
00001000
00000100
000000160
000000016
,
12000000
1616000000
00120000
0016160000
0000001615
00000001
00001200
000001600
,
1615000000
01000000
00120000
000160000
00000010
00000001
000016000
000001600

G:=sub<GL(8,GF(17))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,1,0,0,0,0,0,0,15,1,0,0,0,0,0,0,0,0,16,1,0,0,0,0,0,0,15,1],[0,0,16,1,0,0,0,0,0,0,15,1,0,0,0,0,16,1,0,0,0,0,0,0,15,1,0,0,0,0,0,0,0,0,0,0,0,0,11,3,0,0,0,0,0,0,11,0,0,0,0,0,6,14,0,0,0,0,0,0,6,0,0,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16],[1,16,0,0,0,0,0,0,2,16,0,0,0,0,0,0,0,0,1,16,0,0,0,0,0,0,2,16,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,2,16,0,0,0,0,16,0,0,0,0,0,0,0,15,1,0,0],[16,0,0,0,0,0,0,0,15,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,2,16,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0] >;

C4.142+ 1+4 in GAP, Magma, Sage, TeX

C_4._{14}2_+^{1+4}
% in TeX

G:=Group("C4.14ES+(2,2)");
// GroupNames label

G:=SmallGroup(128,1931);
// by ID

G=gap.SmallGroup(128,1931);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,456,758,219,675,1018,4037,1027,124]);
// Polycyclic

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

Export

Character table of C4.142+ 1+4 in TeX

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