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G = C23.8C42order 128 = 27

3rd non-split extension by C23 of C42 acting via C42/C22=C22

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

Aliases: C23.8C42, C22⋊C83C4, C24.39(C2×C4), (C22×C4).27Q8, C23.18(C4⋊C4), (C22×C4).184D4, C24.4C4.9C2, C22.13(C23⋊C4), (C23×C4).196C22, C23.34D4.3C2, C23.146(C22⋊C4), C2.11(C23.9D4), C2.13(M4(2)⋊4C4), C22.56(C2.C42), (C2×C4).24(C4⋊C4), (C2×C22⋊C4).2C4, (C22×C4).160(C2×C4), (C2×C4).305(C22⋊C4), SmallGroup(128,38)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C23.8C42
C1C2C22C2×C4C22×C4C23×C4C24.4C4 — C23.8C42
C1C22C23 — C23.8C42
C1C22C23×C4 — C23.8C42
C1C2C22C23×C4 — C23.8C42

Generators and relations for C23.8C42
 G = < a,b,c,d,e | a2=b2=c2=d4=1, e4=b, dad-1=ab=ba, eae-1=ac=ca, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=abcd >

Subgroups: 232 in 96 conjugacy classes, 32 normal (12 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×6], C22, C22 [×2], C22 [×12], C8 [×4], C2×C4 [×4], C2×C4 [×14], C23 [×3], C23 [×4], C22⋊C4 [×2], C2×C8 [×4], M4(2) [×4], C22×C4 [×2], C22×C4 [×4], C22×C4 [×4], C24, C2.C42 [×2], C22⋊C8 [×4], C22⋊C8 [×2], C2×C22⋊C4 [×2], C2×M4(2) [×2], C23×C4, C23.34D4, C24.4C4 [×2], C23.8C42
Quotients: C1, C2 [×3], C4 [×6], C22, C2×C4 [×3], D4 [×3], Q8, C42, C22⋊C4 [×3], C4⋊C4 [×3], C2.C42, C23⋊C4 [×2], C23.9D4, M4(2)⋊4C4 [×2], C23.8C42

Character table of C23.8C42

 class 12A2B2C2D2E2F2G4A4B4C4D4E4F4G4H4I4J8A8B8C8D8E8F8G8H
 size 11112244222244888888888888
ρ111111111111111111111111111    trivial
ρ211111111111111-1-1-1-1-11-111-11-1    linear of order 2
ρ311111111111111-1-1-1-11-11-1-11-11    linear of order 2
ρ4111111111111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ511111111-1-1-1-1-1-1-11-11i-i-ii-iii-i    linear of order 4
ρ611111111-1-1-1-1-1-11-11-1-i-iii-i-iii    linear of order 4
ρ71111-1-1-11-11-111-1-iii-i-1-i-1ii1-i1    linear of order 4
ρ81111-1-1-111-11-1-11-i-iiii-1-i-11-i1i    linear of order 4
ρ911111111-1-1-1-1-1-1-11-11-iii-ii-i-ii    linear of order 4
ρ1011111111-1-1-1-1-1-11-11-1ii-i-iii-i-i    linear of order 4
ρ111111-1-1-11-11-111-1-iii-i1i1-i-i-1i-1    linear of order 4
ρ121111-1-1-111-11-1-11-i-iii-i1i1-1i-1-i    linear of order 4
ρ131111-1-1-11-11-111-1i-i-ii1-i1ii-1-i-1    linear of order 4
ρ141111-1-1-111-11-1-11ii-i-i-i-1i-11i1-i    linear of order 4
ρ151111-1-1-111-11-1-11ii-i-ii1-i1-1-i-1i    linear of order 4
ρ161111-1-1-11-11-111-1i-i-ii-1i-1-i-i1i1    linear of order 4
ρ172222-2-22-22222-2-2000000000000    orthogonal lifted from D4
ρ18222222-2-2-22-22-22000000000000    orthogonal lifted from D4
ρ19222222-2-22-22-22-2000000000000    orthogonal lifted from D4
ρ202222-2-22-2-2-2-2-222000000000000    symplectic lifted from Q8, Schur index 2
ρ214-44-44-400000000000000000000    orthogonal lifted from C23⋊C4
ρ224-44-4-4400000000000000000000    orthogonal lifted from C23⋊C4
ρ234-4-440000-4i04i000000000000000    complex lifted from M4(2)⋊4C4
ρ2444-4-4000004i0-4i00000000000000    complex lifted from M4(2)⋊4C4
ρ2544-4-400000-4i04i00000000000000    complex lifted from M4(2)⋊4C4
ρ264-4-4400004i0-4i000000000000000    complex lifted from M4(2)⋊4C4

Smallest permutation representation of C23.8C42
On 32 points
Generators in S32
(1 15)(2 23)(3 9)(4 17)(5 11)(6 19)(7 13)(8 21)(10 26)(12 28)(14 30)(16 32)(18 27)(20 29)(22 31)(24 25)
(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)(25 29)(26 30)(27 31)(28 32)
(1 31)(2 32)(3 25)(4 26)(5 27)(6 28)(7 29)(8 30)(9 24)(10 17)(11 18)(12 19)(13 20)(14 21)(15 22)(16 23)
(2 23 6 19)(3 25)(4 10 8 14)(7 29)(9 20)(11 15)(12 32 16 28)(13 24)(17 30 21 26)(18 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)

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

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

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

Matrix representation of C23.8C42 in GL8(𝔽17)

013000000
40000000
00040000
001300000
00001210132
000015130
000000119
000000156
,
160000000
016000000
001600000
000160000
000016000
000001600
000000160
000000016
,
160000000
016000000
001600000
000160000
00001000
00000100
00000010
00000001
,
10000000
016000000
00040000
00400000
0000110137
00000161314
0000001110
000000156
,
00100000
00010000
01000000
160000000
00000116
00000106
000041300
000009016

G:=sub<GL(8,GF(17))| [0,4,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,12,1,0,0,0,0,0,0,10,5,0,0,0,0,0,0,13,13,11,15,0,0,0,0,2,0,9,6],[16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16],[16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,10,16,0,0,0,0,0,0,13,13,11,15,0,0,0,0,7,14,10,6],[0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,1,1,13,9,0,0,0,0,1,0,0,0,0,0,0,0,6,6,0,16] >;

C23.8C42 in GAP, Magma, Sage, TeX

C_2^3._8C_4^2
% in TeX

G:=Group("C2^3.8C4^2");
// GroupNames label

G:=SmallGroup(128,38);
// by ID

G=gap.SmallGroup(128,38);
# by ID

G:=PCGroup([7,-2,2,-2,2,2,-2,2,56,85,120,758,723,570,2804,102]);
// Polycyclic

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

Export

Character table of C23.8C42 in TeX

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