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G = C4.142+ (1+4)  order 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), C8⋊7D4⋊5C2, C8⋊2D4⋊13C2, C22⋊D8⋊20C2, C2.D8⋊9C22, (C2×D4).154D4, C2.28(D4○D8), C4⋊D4⋊6C22, (C2×Q8).130D4, C23.89(C2×D4), C4.Q8⋊21C22, C4⋊C4.138C23, C22⋊C8⋊17C22, (C2×C4).397C24, (C2×C8).155C23, (C22×C8)⋊10C22, (C2×D8).21C22, D4⋊C4⋊30C22, (C2×D4).148C23, (C22×D4)⋊25C22, C22.29C24⋊15C2, C42⋊C2⋊18C22, C23.19D4⋊25C2, C2.78(C23⋊3D4), (C2×M4(2))⋊17C22, (C22×C4).300C23, C22.657(C22×D4), (C2×C4).534(C2×D4), (C22×C8)⋊C2⋊13C2, (C2×C4○D4).165C22, SmallGroup(128,1931)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series
C1 — C2×C4 — C4.142+ (1+4)
Chief series
C1 — C2 — C4 — C2×C4 — C22×C4 — C22×D4 — C22.29C24 — C4.142+ (1+4)
Lower central
C1 — C2 — C2×C4 — C4.142+ (1+4)
Upper central
C1 — C22 — C2×C4○D4 — C4.142+ (1+4)
Jennings
C1 — C2 — C2 — C2×C4 — C4.142+ (1+4)

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], C4⋊1D4 [×2], C22×C8, C2×M4(2), C2×D8 [×4], C22×D4 [×2], C2×C4○D4, (C22×C8)⋊C2, C22⋊D8 [×4], C8⋊7D4 [×2], C8⋊2D4 [×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], C23⋊3D4, D4○D8 [×2], C4.142+ (1+4)

Generators and relations
 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 >

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

(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 22)(11 15)(12 24)(14 18)(16 20)(17 21)(19 23)

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

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

Matrix representation ►G ⊆ 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] >;
 

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)
ρ2344-4-4000000000000000002√202√200    orthogonal lifted from D4○D8
ρ2444-4-4000000000000000002√202√200    orthogonal lifted from D4○D8
ρ254-4-4400000000000000002√202√2000    orthogonal lifted from D4○D8
ρ264-4-4400000000000000002√202√2000    orthogonal lifted from D4○D8

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
 

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