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G = C23.10C24order 128 = 27

10th non-split extension by C23 of C24 acting via C24/C22=C22

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

Aliases: C23.10C24, 2+ 1+4.16C22, C4.48C22≀C2, (C2×D4).149D4, (C2×Q8).125D4, C23.29(C2×D4), C23⋊C410C22, (C2×D4).44C23, C23.7D45C2, (C22×C4).115D4, C22⋊C4.3C23, C2.C25.5C2, (C22×Q8)⋊18C22, C22.44(C22×D4), C42⋊C214C22, (C22×C4).286C23, C22.D42C22, C23.38C237C2, C23.C2320C2, (C2×C4).30(C2×D4), C2.65(C2×C22≀C2), (C2×C4○D4).113C22, SmallGroup(128,1760)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C23.10C24
C1C2C22C23C22×C4C2×C4○D4C2.C25 — C23.10C24
C1C2C23 — C23.10C24
C1C2C22×C4 — C23.10C24
C1C2C23 — C23.10C24

Generators and relations for C23.10C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=e2=f2=1, d2=g2=c, ab=ba, faf=ac=ca, ede=ad=da, ae=ea, ag=ga, ebe=bc=cb, fdf=bd=db, bf=fb, bg=gb, gdg-1=cd=dc, ce=ec, cf=fc, cg=gc, ef=fe, eg=ge, fg=gf >

Subgroups: 684 in 358 conjugacy classes, 106 normal (8 characteristic)
C1, C2, C2 [×9], C4, C4 [×3], C4 [×14], C22 [×3], C22 [×17], C2×C4 [×12], C2×C4 [×36], D4 [×30], Q8 [×14], C23, C23 [×6], C23 [×4], C42 [×3], C22⋊C4 [×6], C22⋊C4 [×12], C4⋊C4 [×15], C22×C4, C22×C4 [×8], C22×C4 [×4], C2×D4 [×9], C2×D4 [×18], C2×Q8 [×3], C2×Q8 [×12], C4○D4 [×40], C23⋊C4 [×12], C42⋊C2 [×3], C22⋊Q8 [×6], C22.D4 [×12], C4.4D4 [×3], C4⋊Q8 [×3], C22×Q8, C2×C4○D4 [×3], C2×C4○D4 [×6], 2+ 1+4 [×4], 2+ 1+4 [×3], 2- 1+4 [×3], C23.C23 [×3], C23.7D4 [×8], C23.38C23 [×3], C2.C25, C23.10C24
Quotients: C1, C2 [×15], C22 [×35], D4 [×12], C23 [×15], C2×D4 [×18], C24, C22≀C2 [×4], C22×D4 [×3], C2×C22≀C2, C23.10C24

Character table of C23.10C24

 class 12A2B2C2D2E2F2G2H2I2J4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O4P4Q4R
 size 11222444444222244444488888888
ρ111111111111111111111111111111    trivial
ρ211111-11-1-1-1111111-1-11-1-111-11-1-1-11    linear of order 2
ρ311111-1-111-11-1-1-1-1-1-111-11111-11-1-1-1    linear of order 2
ρ4111111-1-1-111-1-1-1-1-11-111-111-1-1-111-1    linear of order 2
ρ511111-11-1-1-1111111-1-11-1-1-1-11-1111-1    linear of order 2
ρ6111111111111111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ7111111-1-1-111-1-1-1-1-11-111-1-1-1111-1-11    linear of order 2
ρ811111-1-111-11-1-1-1-1-1-111-11-1-1-11-1111    linear of order 2
ρ911111-11-111-1-1-1-1-1111-1-1-1-111-1-1-111    linear of order 2
ρ1011111111-1-1-1-1-1-1-11-1-1-111-11-1-111-11    linear of order 2
ρ11111111-1-11-1-11111-1-11-11-1-1111-11-1-1    linear of order 2
ρ1211111-1-11-11-11111-11-1-1-11-11-111-11-1    linear of order 2
ρ1311111111-1-1-1-1-1-1-11-1-1-1111-111-1-11-1    linear of order 2
ρ1411111-11-111-1-1-1-1-1111-1-1-11-1-1111-1-1    linear of order 2
ρ1511111-1-11-11-11111-11-1-1-111-11-1-11-11    linear of order 2
ρ16111111-1-11-1-11111-1-11-11-11-1-1-11-111    linear of order 2
ρ1722-22-2-200-200-222-200202000000000    orthogonal lifted from D4
ρ1822-2-2200-20-20-2-22202000200000000    orthogonal lifted from D4
ρ19222-2-20-200022-22-2200-20000000000    orthogonal lifted from D4
ρ2022-22-2200200-222-200-20-2000000000    orthogonal lifted from D4
ρ2122-2-2200-202022-2-20-2000200000000    orthogonal lifted from D4
ρ2222-22-2200-2002-2-220020-2000000000    orthogonal lifted from D4
ρ23222-2-202000-22-22-2-20020000000000    orthogonal lifted from D4
ρ2422-22-2-2002002-2-2200-202000000000    orthogonal lifted from D4
ρ25222-2-20-2000-2-22-2220020000000000    orthogonal lifted from D4
ρ2622-2-220020-2022-2-202000-200000000    orthogonal lifted from D4
ρ27222-2-2020002-22-22-200-20000000000    orthogonal lifted from D4
ρ2822-2-22002020-2-2220-2000-200000000    orthogonal lifted from D4
ρ298-8000000000000000000000000000    symplectic faithful, Schur index 2

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

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

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

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

Matrix representation of C23.10C24 in GL8(𝔽5)

00100000
11430000
10000000
00040000
00000010
10410001
00001000
10410100
,
01000000
10000000
11430000
00010000
24004300
14000100
24000043
14000001
,
40000000
04000000
00400000
00040000
00004000
00000400
00000040
00000004
,
00001000
24004300
00000010
00000401
40000000
21004100
00400000
21044100
,
03000000
20000000
33240000
20330000
12000024
12000033
12002400
12003300
,
10000000
01000000
00400000
11040000
24004300
00000100
31000012
24000004
,
30000000
03000000
00300000
00030000
00002000
12000200
00000020
12000002

G:=sub<GL(8,GF(5))| [0,1,1,0,0,1,0,1,0,1,0,0,0,0,0,0,1,4,0,0,0,4,0,4,0,3,0,4,0,1,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[0,1,1,0,2,1,2,1,1,0,1,0,4,4,4,4,0,0,4,0,0,0,0,0,0,0,3,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,3,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,3,1],[4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[0,2,0,0,4,2,0,2,0,4,0,0,0,1,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,1,4,0,0,0,4,0,4,0,3,0,4,0,1,0,1,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0],[0,2,3,2,1,1,1,1,3,0,3,0,2,2,2,2,0,0,2,3,0,0,0,0,0,0,4,3,0,0,0,0,0,0,0,0,0,0,2,3,0,0,0,0,0,0,4,3,0,0,0,0,2,3,0,0,0,0,0,0,4,3,0,0],[1,0,0,1,2,0,3,2,0,1,0,1,4,0,1,4,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,3,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,2,4],[3,0,0,0,0,1,0,1,0,3,0,0,0,2,0,2,0,0,3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,2] >;

C23.10C24 in GAP, Magma, Sage, TeX

C_2^3._{10}C_2^4
% in TeX

G:=Group("C2^3.10C2^4");
// GroupNames label

G:=SmallGroup(128,1760);
// by ID

G=gap.SmallGroup(128,1760);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,758,2019,248,718,2028]);
// Polycyclic

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

Export

Character table of C23.10C24 in TeX

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