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G = C24.180C23order 128 = 27

20th non-split extension by C24 of C23 acting via C23/C2=C22

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

Aliases: C24.180C23, (C22×C4)⋊3Q8, (C22×C4).80D4, C23.53(C2×Q8), C23.588(C2×D4), C232Q8.4C2, (C23×C4).27C22, C23.9D4.7C2, C22.214C22≀C2, C23.129(C4○D4), C2.11(C23⋊Q8), C23.34D4.6C2, C22.34(C22⋊Q8), C22.26(C4.4D4), C2.17(C23.7D4), (C2×C22⋊C4).18C22, SmallGroup(128,762)

Series: Derived Chief Lower central Upper central Jennings

C1C24 — C24.180C23
C1C2C22C23C24C23×C4C23.34D4 — C24.180C23
C1C2C24 — C24.180C23
C1C22C24 — C24.180C23
C1C2C24 — C24.180C23

Generators and relations for C24.180C23
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=1, e2=f2=a, g2=d, ab=ba, ac=ca, ad=da, fef-1=ae=ea, af=fa, ag=ga, bc=cb, fbf-1=bd=db, geg-1=be=eb, bg=gb, ece-1=cd=dc, gfg-1=cf=fc, cg=gc, de=ed, df=fd, dg=gd >

Subgroups: 344 in 145 conjugacy classes, 42 normal (8 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×11], C22, C22 [×6], C22 [×10], C2×C4 [×31], Q8 [×2], C23, C23 [×6], C23 [×2], C22⋊C4 [×12], C4⋊C4 [×6], C22×C4 [×8], C22×C4 [×9], C2×Q8 [×2], C24, C2.C42 [×6], C2×C22⋊C4 [×6], C22⋊Q8 [×6], C23×C4, C23.9D4 [×3], C23.34D4 [×3], C232Q8, C24.180C23
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], Q8 [×2], C23, C2×D4 [×3], C2×Q8, C4○D4 [×3], C22≀C2, C22⋊Q8 [×3], C4.4D4 [×3], C23⋊Q8, C23.7D4 [×2], C24.180C23

Character table of C24.180C23

 class 12A2B2C2D2E2F2G2H2I4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O4P
 size 11112222224444888888888888
ρ111111111111111111111111111    trivial
ρ21111111111111111-1-1-11-1-11-1-1-1    linear of order 2
ρ31111111111-1-1-1-1-1-11-1-11-111-111    linear of order 2
ρ41111111111-1-1-1-1-1-1-11111-111-1-1    linear of order 2
ρ51111111111-1-1-1-1111-1-1-11-1-111-1    linear of order 2
ρ61111111111-1-1-1-111-111-1-11-1-1-11    linear of order 2
ρ711111111111111-1-1111-1-1-1-1-11-1    linear of order 2
ρ811111111111111-1-1-1-1-1-111-11-11    linear of order 2
ρ922222-22-2-2-200002-20000000000    orthogonal lifted from D4
ρ1022222-22-2-2-20000-220000000000    orthogonal lifted from D4
ρ112222-22-2-2-22000000-2000000020    orthogonal lifted from D4
ρ122222-22-2-2-2200000020000000-20    orthogonal lifted from D4
ρ132222-2-2-222-20000000000200-200    orthogonal lifted from D4
ρ142222-2-2-222-20000000000-200200    orthogonal lifted from D4
ρ152-2-22-2-22-222-2-222000000000000    symplectic lifted from Q8, Schur index 2
ρ162-2-22-2-22-22222-2-2000000000000    symplectic lifted from Q8, Schur index 2
ρ172-2-2222-2-22-200000000000-2i0002i    complex lifted from C4○D4
ρ182-2-22-2222-2-2000000000-2i002i000    complex lifted from C4○D4
ρ192-2-22-2222-2-20000000002i00-2i000    complex lifted from C4○D4
ρ202-2-222-2-22-2200000002i-2i0000000    complex lifted from C4○D4
ρ212-2-2222-2-22-2000000000002i000-2i    complex lifted from C4○D4
ρ222-2-222-2-22-220000000-2i2i0000000    complex lifted from C4○D4
ρ2344-4-40000002i-2i-2i2i000000000000    complex lifted from C23.7D4
ρ244-44-40000002i-2i2i-2i000000000000    complex lifted from C23.7D4
ρ2544-4-4000000-2i2i2i-2i000000000000    complex lifted from C23.7D4
ρ264-44-4000000-2i2i-2i2i000000000000    complex lifted from C23.7D4

Smallest permutation representation of C24.180C23
On 32 points
Generators in S32
(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(13 15)(14 16)(17 19)(18 20)(21 23)(22 24)(25 27)(26 28)(29 31)(30 32)
(1 15)(2 16)(3 13)(4 14)(5 21)(6 22)(7 23)(8 24)(9 25)(10 26)(11 27)(12 28)(17 29)(18 30)(19 31)(20 32)
(1 5)(2 4)(3 7)(6 8)(9 31)(10 12)(11 29)(13 23)(14 16)(15 21)(17 27)(18 20)(19 25)(22 24)(26 28)(30 32)
(1 7)(2 8)(3 5)(4 6)(9 29)(10 30)(11 31)(12 32)(13 21)(14 22)(15 23)(16 24)(17 25)(18 26)(19 27)(20 28)
(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 31 3 29)(2 30 4 32)(5 9 7 11)(6 12 8 10)(13 25 15 27)(14 28 16 26)(17 23 19 21)(18 22 20 24)
(1 11 7 31)(2 28 8 20)(3 9 5 29)(4 26 6 18)(10 22 30 14)(12 24 32 16)(13 25 21 17)(15 27 23 19)

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

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

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

Matrix representation of C24.180C23 in GL6(𝔽5)

400000
040000
001000
000100
000010
000001
,
100000
010000
001300
000400
002143
000201
,
100000
010000
004000
000400
000010
003401
,
100000
010000
004000
000400
000040
000004
,
010000
400000
000010
004211
001000
001243
,
200000
030000
003400
003200
001324
004133
,
400000
040000
003400
000200
000030
000103

G:=sub<GL(6,GF(5))| [4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,2,0,0,0,3,4,1,2,0,0,0,0,4,0,0,0,0,0,3,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,3,0,0,0,4,0,4,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[0,4,0,0,0,0,1,0,0,0,0,0,0,0,0,4,1,1,0,0,0,2,0,2,0,0,1,1,0,4,0,0,0,1,0,3],[2,0,0,0,0,0,0,3,0,0,0,0,0,0,3,3,1,4,0,0,4,2,3,1,0,0,0,0,2,3,0,0,0,0,4,3],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,3,0,0,0,0,0,4,2,0,1,0,0,0,0,3,0,0,0,0,0,0,3] >;

C24.180C23 in GAP, Magma, Sage, TeX

C_2^4._{180}C_2^3
% in TeX

G:=Group("C2^4.180C2^3");
// GroupNames label

G:=SmallGroup(128,762);
// by ID

G=gap.SmallGroup(128,762);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,56,141,64,422,387,352,1411,4037]);
// Polycyclic

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

Export

Character table of C24.180C23 in TeX

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