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G = C4.2+ 1+4order 128 = 27

13rd 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: C4.132+ 1+4, C88D43C2, C87D418C2, C8⋊D416C2, C82D412C2, C22⋊D819C2, D4⋊D423C2, C2.D88C22, (C2×D4).153D4, C22⋊SD169C2, C2.27(D4○D8), C4⋊D45C22, (C2×Q8).129D4, C4.Q820C22, C22⋊Q85C22, C4⋊C4.137C23, C22⋊C816C22, (C2×C8).154C23, (C2×C4).396C24, (C22×C8)⋊17C22, Q8⋊C44C22, (C2×D8).68C22, C23.280(C2×D4), D4⋊C429C22, C2.41(D4○SD16), (C2×D4).147C23, C22.D821C2, (C2×Q8).135C23, C22.29C2414C2, C23.20D424C2, C23.19D424C2, C23.46D410C2, C2.77(C233D4), (C2×M4(2))⋊16C22, (C22×C4).299C23, (C2×SD16).80C22, C22.656(C22×D4), C22.31C248C2, (C22×D4).384C22, C42⋊C2.154C22, (C2×C4⋊C4)⋊54C22, (C2×C4).150(C2×D4), (C2×C4○D4)⋊10C22, (C22×C8)⋊C212C2, SmallGroup(128,1930)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C4.2+ 1+4
C1C2C4C2×C4C22×C4C22×D4C22.29C24 — C4.2+ 1+4
C1C2C2×C4 — C4.2+ 1+4
C1C22C2×C4○D4 — C4.2+ 1+4
C1C2C2C2×C4 — C4.2+ 1+4

Generators and relations for C4.2+ 1+4
 G = < a,b,c,d,e | a4=c2=e2=1, b4=a2, d2=ab2, dbd-1=ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc=a-1b3, be=eb, dcd-1=ece=a2c, ede=ab2d >

Subgroups: 516 in 211 conjugacy classes, 84 normal (all characteristic)
C1, C2 [×3], C2 [×6], C4 [×2], C4 [×8], C22, C22 [×22], C8 [×4], C2×C4 [×4], C2×C4 [×14], D4 [×19], Q8 [×3], C23 [×3], C23 [×7], C42, C22⋊C4 [×9], C4⋊C4 [×4], C4⋊C4 [×3], C2×C8 [×4], C2×C8, M4(2), D8 [×2], SD16 [×2], C22×C4 [×3], C22×C4 [×2], C2×D4 [×6], C2×D4 [×10], C2×Q8 [×2], C4○D4 [×4], C24, C22⋊C8 [×4], D4⋊C4 [×6], Q8⋊C4 [×2], C4.Q8 [×2], C2.D8 [×2], C2×C4⋊C4, C42⋊C2, C22≀C2 [×2], C4⋊D4 [×6], C4⋊D4 [×3], C22⋊Q8 [×2], C22⋊Q8, C4.4D4, C41D4, C22×C8, C2×M4(2), C2×D8 [×2], C2×SD16 [×2], C22×D4, C2×C4○D4 [×2], (C22×C8)⋊C2, C22⋊D8, D4⋊D4 [×2], C22⋊SD16, C88D4, C87D4, C8⋊D4, C82D4, C22.D8, C23.46D4, C23.19D4, C23.20D4, C22.29C24, C22.31C24, C4.2+ 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, D4○SD16, C4.2+ 1+4

Character table of C4.2+ 1+4

 class 12A2B2C2D2E2F2G2H2I4A4B4C4D4E4F4G4H4I4J8A8B8C8D8E8F
 size 11114448882244488888444488
ρ111111111111111111111111111    trivial
ρ21111111-1-1-111111-1-1-1-1-1111111    linear of order 2
ρ31111-11-111-111-11-1-11-1-111-11-1-11    linear of order 2
ρ41111-11-1-1-1111-11-11-111-11-11-1-11    linear of order 2
ρ5111111111-111111-1-111-1-1-1-1-1-1-1    linear of order 2
ρ61111111-1-111111111-1-11-1-1-1-1-1-1    linear of order 2
ρ71111-11-111111-11-11-1-1-1-1-11-111-1    linear of order 2
ρ81111-11-1-1-1-111-11-1-11111-11-111-1    linear of order 2
ρ91111-1-111-11111-1-1-1-11-111-11-11-1    linear of order 2
ρ101111-1-11-11-1111-1-111-11-11-11-11-1    linear of order 2
ρ1111111-1-11-1-111-1-111-1-1111111-1-1    linear of order 2
ρ1211111-1-1-11111-1-11-111-1-11111-1-1    linear of order 2
ρ131111-1-111-1-1111-1-1111-1-1-11-11-11    linear of order 2
ρ141111-1-11-111111-1-1-1-1-111-11-11-11    linear of order 2
ρ1511111-1-11-1111-1-11-11-11-1-1-1-1-111    linear of order 2
ρ1611111-1-1-11-111-1-111-11-11-1-1-1-111    linear of order 2
ρ1722222-22000-2-2-22-200000000000    orthogonal lifted from D4
ρ182222-2-2-2000-2-222200000000000    orthogonal lifted from D4
ρ192222-222000-2-2-2-2200000000000    orthogonal lifted from D4
ρ20222222-2000-2-22-2-200000000000    orthogonal lifted from D4
ρ214-44-40000004-400000000000000    orthogonal lifted from 2+ 1+4
ρ224-44-4000000-4400000000000000    orthogonal lifted from 2+ 1+4
ρ234-4-4400000000000000000-2202200    orthogonal lifted from D4○D8
ρ244-4-4400000000000000000220-2200    orthogonal lifted from D4○D8
ρ2544-4-400000000000000002-20-2-2000    complex lifted from D4○SD16
ρ2644-4-40000000000000000-2-202-2000    complex lifted from D4○SD16

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

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

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

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

Matrix representation of C4.2+ 1+4 in GL8(𝔽17)

12000000
1616000000
010160000
1616100000
00001000
00000100
00000010
00000001
,
010000000
127000000
0512120000
12125120000
0000501313
0000701315
000011988
00004844
,
10000000
01000000
101600000
1600160000
00001000
00000100
0000130160
0000150016
,
1600150000
00110000
16160160000
10010000
000016200
000016100
000016131616
000061521
,
101500000
00110000
001600000
01100000
000011500
000001600
00001241616
00000201

G:=sub<GL(8,GF(17))| [1,16,0,16,0,0,0,0,2,16,1,16,0,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,12,0,12,0,0,0,0,10,7,5,12,0,0,0,0,0,0,12,5,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,5,7,11,4,0,0,0,0,0,0,9,8,0,0,0,0,13,13,8,4,0,0,0,0,13,15,8,4],[1,0,1,16,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,13,15,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16],[16,0,16,1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,15,1,16,1,0,0,0,0,0,0,0,0,16,16,16,6,0,0,0,0,2,1,13,15,0,0,0,0,0,0,16,2,0,0,0,0,0,0,16,1],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,15,1,16,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,12,0,0,0,0,0,15,16,4,2,0,0,0,0,0,0,16,0,0,0,0,0,0,0,16,1] >;

C4.2+ 1+4 in GAP, Magma, Sage, TeX

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

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

G:=SmallGroup(128,1930);
// by ID

G=gap.SmallGroup(128,1930);
# by ID

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

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

Export

Character table of C4.2+ 1+4 in TeX

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