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G = C24.182C23order 128 = 27

22nd non-split extension by C24 of C23 acting via C23/C2=C22

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

Aliases: C24.182C23, C22⋊C41Q8, C2.19C2≀C22, C23.7(C2×Q8), (C22×C4).87D4, C23.595(C2×D4), C22.15(C4⋊Q8), C232Q8.5C2, C23.12(C4○D4), (C23×C4).29C22, C23.9D4.8C2, C22.232C22≀C2, C23.8Q8.2C2, C22.37(C22⋊Q8), C2.19(C23.7D4), C2.9(C23.78C23), (C2×C22⋊C4).104C22, SmallGroup(128,794)

Series: Derived Chief Lower central Upper central Jennings

C1C24 — C24.182C23
C1C2C22C23C24C23×C4C23.8Q8 — C24.182C23
C1C2C24 — C24.182C23
C1C22C24 — C24.182C23
C1C2C24 — C24.182C23

Generators and relations for C24.182C23
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=1, e2=f2=g2=a, 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: 368 in 157 conjugacy classes, 46 normal (10 characteristic)
C1, C2 [×3], C2 [×6], C4 [×14], C22, C22 [×6], C22 [×10], C2×C4 [×34], Q8 [×2], C23, C23 [×6], C23 [×2], C22⋊C4 [×6], C22⋊C4 [×9], C4⋊C4 [×12], C22×C4 [×6], C22×C4 [×8], C2×Q8 [×2], C24, C2.C42 [×3], C2×C22⋊C4 [×6], C2×C4⋊C4 [×3], C22⋊Q8 [×6], C23×C4, C23.9D4 [×3], C23.8Q8 [×3], C232Q8, C24.182C23
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], Q8 [×6], C23, C2×D4 [×3], C2×Q8 [×3], C4○D4, C22≀C2, C22⋊Q8 [×3], C4⋊Q8 [×3], C23.78C23, C2≀C22, C23.7D4, C24.182C23

Character table of C24.182C23

 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
ρ922222-22-2-2-20000-220000000000    orthogonal lifted from D4
ρ102222-2-2-222-20000000000200-200    orthogonal lifted from D4
ρ112222-22-2-2-22000000-2000000020    orthogonal lifted from D4
ρ122222-22-2-2-2200000020000000-20    orthogonal lifted from D4
ρ1322222-22-2-2-200002-20000000000    orthogonal lifted from D4
ρ142222-2-2-222-20000000000-200200    orthogonal lifted from D4
ρ152-2-2222-2-22-200000000000-20002    symplectic lifted from Q8, Schur index 2
ρ162-2-22-2222-2-2000000000-2002000    symplectic lifted from Q8, Schur index 2
ρ172-2-222-2-22-2200000002-20000000    symplectic lifted from Q8, Schur index 2
ρ182-2-2222-2-22-2000000000002000-2    symplectic lifted from Q8, Schur index 2
ρ192-2-222-2-22-220000000-220000000    symplectic lifted from Q8, Schur index 2
ρ202-2-22-2222-2-2000000000200-2000    symplectic lifted from Q8, Schur index 2
ρ212-2-22-2-22-222-2i-2i2i2i000000000000    complex lifted from C4○D4
ρ222-2-22-2-22-2222i2i-2i-2i000000000000    complex lifted from C4○D4
ρ2344-4-40000002-2-22000000000000    orthogonal lifted from C2≀C22
ρ2444-4-4000000-222-2000000000000    orthogonal lifted from C2≀C22
ρ254-44-40000002i-2i2i-2i000000000000    complex lifted from C23.7D4
ρ264-44-4000000-2i2i-2i2i000000000000    complex lifted from C23.7D4

Smallest permutation representation of C24.182C23
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 25)(2 26)(3 27)(4 28)(5 12)(6 9)(7 10)(8 11)(13 18)(14 19)(15 20)(16 17)(21 32)(22 29)(23 30)(24 31)
(2 12)(4 10)(5 26)(7 28)(14 21)(16 23)(17 30)(19 32)
(1 11)(2 12)(3 9)(4 10)(5 26)(6 27)(7 28)(8 25)(13 24)(14 21)(15 22)(16 23)(17 30)(18 31)(19 32)(20 29)
(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 20 3 18)(2 19 4 17)(5 14 7 16)(6 13 8 15)(9 31 11 29)(10 30 12 32)(21 28 23 26)(22 27 24 25)
(1 20 3 18)(2 16 4 14)(5 30 7 32)(6 24 8 22)(9 31 11 29)(10 21 12 23)(13 25 15 27)(17 28 19 26)

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

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

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

Matrix representation of C24.182C23 in GL6(𝔽5)

400000
040000
004000
000400
000040
000004
,
400000
040000
001300
000400
000101
000110
,
400000
040000
001000
000100
004040
004004
,
100000
010000
004000
000400
000040
000004
,
010000
400000
004030
000041
001010
001410
,
030000
300000
002000
002300
003030
000002
,
200000
030000
002000
000200
000002
000020

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

C24.182C23 in GAP, Magma, Sage, TeX

C_2^4._{182}C_2^3
% in TeX

G:=Group("C2^4.182C2^3");
// GroupNames label

G:=SmallGroup(128,794);
// by ID

G=gap.SmallGroup(128,794);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,56,141,64,422,387,58,1411,4037]);
// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=1,e^2=f^2=g^2=a,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 C24.182C23 in TeX

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