Copied to
clipboard

?

G = C24.10D6  order 192 = 26·3

9th non-split extension by C24 of D6 acting via D6/C2=S3

non-abelian, soluble, monomial

Aliases: C24.10D6, C4⋊S4⋊5C2, (C2×C4)⋊3S4, (C4×S4)⋊4C2, A4⋊Q8⋊5C2, C4.34(C2×S4), (C23×C4)⋊4S3, A4⋊1(C4○D4), A4⋊D4⋊3C2, C22⋊(C4○D12), C22.7(C2×S4), C2.5(C22×S4), A4⋊C4.2C22, (C2×A4).4C23, (C2×S4).1C22, (C22×C4).12D6, (C4×A4).16C22, C23.4(C22×S3), (C22×A4).11C22, (C2×C4×A4)⋊4C2, SmallGroup(192,1471)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C22 — C2×A4 — C24.10D6
Chief series
C1 — C22 — A4 — C2×A4 — C2×S4 — C4×S4 — C24.10D6
Lower central
A4 — C2×A4 — C24.10D6

Subgroups: 626 in 171 conjugacy classes, 29 normal (21 characteristic)
C1, C2, C2 [×6], C3, C4 [×2], C4 [×8], C22 [×2], C22 [×14], S3 [×2], C6 [×2], C2×C4, C2×C4 [×18], D4 [×14], Q8 [×2], C23, C23 [×6], Dic3 [×2], C12 [×2], A4, D6 [×2], C2×C6, C42 [×2], C22⋊C4 [×10], C4⋊C4 [×6], C22×C4 [×2], C22×C4 [×6], C2×D4 [×7], C2×Q8, C4○D4 [×4], C24, Dic6, C4×S3 [×2], D12, C3⋊D4 [×2], C2×C12, S4 [×2], C2×A4, C2×A4, C42⋊C2, C4×D4 [×4], C22≀C2 [×2], C4⋊D4 [×2], C22⋊Q8 [×2], C22.D4 [×2], C23×C4, C2×C4○D4, A4⋊C4 [×2], C4×A4 [×2], C4○D12, C2×S4 [×2], C22×A4, C22.19C24, A4⋊Q8, C4×S4 [×2], C4⋊S4, A4⋊D4 [×2], C2×C4×A4, C24.10D6

Quotients:
C1, C2 [×7], C22 [×7], S3, C23, D6 [×3], C4○D4, S4, C22×S3, C4○D12, C2×S4 [×3], C22×S4, C24.10D6

Generators and relations
 G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e6=f2=b, faf-1=ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, bf=fb, fcf-1=ede-1=cd=dc, ece-1=d, df=fd, fef-1=e5 >

Permutation representations
►On 24 points - transitive group 24T292
Generators in S24
(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)

(1 7)(2 8)(3 9)(4 10)(5 11)(6 12)(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)

(2 8)(3 9)(5 11)(6 12)(13 19)(14 20)(16 22)(17 23)

(1 7)(2 8)(4 10)(5 11)(13 19)(15 21)(16 22)(18 24)

(1 2 3 4 5 6 7 8 9 10 11 12)(13 14 15 16 17 18 19 20 21 22 23 24)

(1 22 7 16)(2 15 8 21)(3 20 9 14)(4 13 10 19)(5 18 11 24)(6 23 12 17)
 
G:=sub<Sym(24)| (13,19)(14,20)(15,21)(16,22)(17,23)(18,24), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24), (2,8)(3,9)(5,11)(6,12)(13,19)(14,20)(16,22)(17,23), (1,7)(2,8)(4,10)(5,11)(13,19)(15,21)(16,22)(18,24), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24), (1,22,7,16)(2,15,8,21)(3,20,9,14)(4,13,10,19)(5,18,11,24)(6,23,12,17)>;
 
G:=Group( (13,19)(14,20)(15,21)(16,22)(17,23)(18,24), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24), (2,8)(3,9)(5,11)(6,12)(13,19)(14,20)(16,22)(17,23), (1,7)(2,8)(4,10)(5,11)(13,19)(15,21)(16,22)(18,24), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24), (1,22,7,16)(2,15,8,21)(3,20,9,14)(4,13,10,19)(5,18,11,24)(6,23,12,17) );
 
G=PermutationGroup([(13,19),(14,20),(15,21),(16,22),(17,23),(18,24)], [(1,7),(2,8),(3,9),(4,10),(5,11),(6,12),(13,19),(14,20),(15,21),(16,22),(17,23),(18,24)], [(2,8),(3,9),(5,11),(6,12),(13,19),(14,20),(16,22),(17,23)], [(1,7),(2,8),(4,10),(5,11),(13,19),(15,21),(16,22),(18,24)], [(1,2,3,4,5,6,7,8,9,10,11,12),(13,14,15,16,17,18,19,20,21,22,23,24)], [(1,22,7,16),(2,15,8,21),(3,20,9,14),(4,13,10,19),(5,18,11,24),(6,23,12,17)])
 
G:=TransitiveGroup(24,292);
 

►On 24 points - transitive group 24T319
Generators in S24
(1 24)(2 13)(3 14)(4 15)(5 16)(6 17)(7 18)(8 19)(9 20)(10 21)(11 22)(12 23)

(1 7)(2 8)(3 9)(4 10)(5 11)(6 12)(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)

(1 7)(2 8)(4 10)(5 11)(13 19)(15 21)(16 22)(18 24)

(1 7)(3 9)(4 10)(6 12)(14 20)(15 21)(17 23)(18 24)

(1 2 3 4 5 6 7 8 9 10 11 12)(13 14 15 16 17 18 19 20 21 22 23 24)

(1 6 7 12)(2 11 8 5)(3 4 9 10)(13 16 19 22)(14 21 20 15)(17 24 23 18)
 
G:=sub<Sym(24)| (1,24)(2,13)(3,14)(4,15)(5,16)(6,17)(7,18)(8,19)(9,20)(10,21)(11,22)(12,23), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24), (1,7)(2,8)(4,10)(5,11)(13,19)(15,21)(16,22)(18,24), (1,7)(3,9)(4,10)(6,12)(14,20)(15,21)(17,23)(18,24), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24), (1,6,7,12)(2,11,8,5)(3,4,9,10)(13,16,19,22)(14,21,20,15)(17,24,23,18)>;
 
G:=Group( (1,24)(2,13)(3,14)(4,15)(5,16)(6,17)(7,18)(8,19)(9,20)(10,21)(11,22)(12,23), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24), (1,7)(2,8)(4,10)(5,11)(13,19)(15,21)(16,22)(18,24), (1,7)(3,9)(4,10)(6,12)(14,20)(15,21)(17,23)(18,24), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24), (1,6,7,12)(2,11,8,5)(3,4,9,10)(13,16,19,22)(14,21,20,15)(17,24,23,18) );
 
G=PermutationGroup([(1,24),(2,13),(3,14),(4,15),(5,16),(6,17),(7,18),(8,19),(9,20),(10,21),(11,22),(12,23)], [(1,7),(2,8),(3,9),(4,10),(5,11),(6,12),(13,19),(14,20),(15,21),(16,22),(17,23),(18,24)], [(1,7),(2,8),(4,10),(5,11),(13,19),(15,21),(16,22),(18,24)], [(1,7),(3,9),(4,10),(6,12),(14,20),(15,21),(17,23),(18,24)], [(1,2,3,4,5,6,7,8,9,10,11,12),(13,14,15,16,17,18,19,20,21,22,23,24)], [(1,6,7,12),(2,11,8,5),(3,4,9,10),(13,16,19,22),(14,21,20,15),(17,24,23,18)])
 
G:=TransitiveGroup(24,319);
 

►On 24 points - transitive group 24T395
Generators in S24
(1 23)(2 24)(3 13)(4 14)(5 15)(6 16)(7 17)(8 18)(9 19)(10 20)(11 21)(12 22)

(1 7)(2 8)(3 9)(4 10)(5 11)(6 12)(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)

(1 23)(2 18)(3 9)(4 14)(5 21)(6 12)(7 17)(8 24)(10 20)(11 15)(13 19)(16 22)

(1 17)(2 8)(3 13)(4 20)(5 11)(6 16)(7 23)(9 19)(10 14)(12 22)(15 21)(18 24)

(1 2 3 4 5 6 7 8 9 10 11 12)(13 14 15 16 17 18 19 20 21 22 23 24)

(1 6 7 12)(2 11 8 5)(3 4 9 10)(13 20 19 14)(15 18 21 24)(16 23 22 17)
 
G:=sub<Sym(24)| (1,23)(2,24)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,19)(10,20)(11,21)(12,22), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24), (1,23)(2,18)(3,9)(4,14)(5,21)(6,12)(7,17)(8,24)(10,20)(11,15)(13,19)(16,22), (1,17)(2,8)(3,13)(4,20)(5,11)(6,16)(7,23)(9,19)(10,14)(12,22)(15,21)(18,24), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24), (1,6,7,12)(2,11,8,5)(3,4,9,10)(13,20,19,14)(15,18,21,24)(16,23,22,17)>;
 
G:=Group( (1,23)(2,24)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,19)(10,20)(11,21)(12,22), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24), (1,23)(2,18)(3,9)(4,14)(5,21)(6,12)(7,17)(8,24)(10,20)(11,15)(13,19)(16,22), (1,17)(2,8)(3,13)(4,20)(5,11)(6,16)(7,23)(9,19)(10,14)(12,22)(15,21)(18,24), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24), (1,6,7,12)(2,11,8,5)(3,4,9,10)(13,20,19,14)(15,18,21,24)(16,23,22,17) );
 
G=PermutationGroup([(1,23),(2,24),(3,13),(4,14),(5,15),(6,16),(7,17),(8,18),(9,19),(10,20),(11,21),(12,22)], [(1,7),(2,8),(3,9),(4,10),(5,11),(6,12),(13,19),(14,20),(15,21),(16,22),(17,23),(18,24)], [(1,23),(2,18),(3,9),(4,14),(5,21),(6,12),(7,17),(8,24),(10,20),(11,15),(13,19),(16,22)], [(1,17),(2,8),(3,13),(4,20),(5,11),(6,16),(7,23),(9,19),(10,14),(12,22),(15,21),(18,24)], [(1,2,3,4,5,6,7,8,9,10,11,12),(13,14,15,16,17,18,19,20,21,22,23,24)], [(1,6,7,12),(2,11,8,5),(3,4,9,10),(13,20,19,14),(15,18,21,24),(16,23,22,17)])
 
G:=TransitiveGroup(24,395);
 

Matrix representation ►G ⊆ GL5(𝔽13)

05000
80000
001200
000120
000012
,
120000
012000
00100
00010
00001
,
10000
01000
001200
000120
00001
,
10000
01000
001200
00010
000012
,
80000
08000
000012
001200
000120
,
08000
80000
000012
000120
001200

G:=sub<GL(5,GF(13))| [0,8,0,0,0,5,0,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12],[12,0,0,0,0,0,12,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,12,0,0,0,0,0,1,0,0,0,0,0,12],[8,0,0,0,0,0,8,0,0,0,0,0,0,12,0,0,0,0,0,12,0,0,12,0,0],[0,8,0,0,0,8,0,0,0,0,0,0,0,0,12,0,0,0,12,0,0,0,12,0,0] >;
 

Character table of C24.10D6

 class 12A2B2C2D2E2F2G34A4B4C4D4E4F4G4H4I4J4K4L6A6B6C12A12B12C12D
 size 112336121281123361212121212128888888
ρ11111111111111111111111111111    trivial
ρ2111111-111-1-1-1-1-1-1-1111-1-1111-1-1-1-1    linear of order 2
ρ3111111-1-11111111-1-1-1-1-1-11111111    linear of order 2
ρ41111111-11-1-1-1-1-1-11-1-1-111111-1-1-1-1    linear of order 2
ρ511-111-1111-1-11-1-1111-1-1-1-1-1-111-1-11    linear of order 2
ρ611-111-1-11111-111-1-11-1-111-1-11-111-1    linear of order 2
ρ711-111-11-1111-111-11-111-1-1-1-11-111-1    linear of order 2
ρ811-111-1-1-11-1-11-1-11-1-11111-1-111-1-11    linear of order 2
ρ922-222-200-122-222-200000011-11-1-11    orthogonal lifted from D6
ρ1022-222-200-1-2-22-2-2200000011-1-111-1    orthogonal lifted from D6
ρ1122222200-1-2-2-2-2-2-2000000-1-1-11111    orthogonal lifted from D6
ρ1222222200-1222222000000-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ132-20-2200022i2i02i2i000000000-202i2i0    complex lifted from C4○D4
ρ142-20-2200022i2i02i2i000000000-202i2i0    complex lifted from C4○D4
ρ152-20-22000-12i2i02i2i0000000√-3√-31√3ζ2ζ2√3    complex lifted from C4○D12
ρ162-20-22000-12i2i02i2i0000000√-3√-31√3ζ2ζ2√3    complex lifted from C4○D12
ρ172-20-22000-12i2i02i2i0000000√-3√-31√3ζ2ζ2√3    complex lifted from C4○D12
ρ182-20-22000-12i2i02i2i0000000√-3√-31√3ζ2ζ2√3    complex lifted from C4○D12
ρ19333-1-1-1-110-3-3-31111-11-1-110000000    orthogonal lifted from C2×S4
ρ2033-3-1-11-11033-3-1-111-1-111-10000000    orthogonal lifted from C2×S4
ρ2133-3-1-111-1033-3-1-11-111-1-110000000    orthogonal lifted from C2×S4
ρ22333-1-1-11-10-3-3-3111-11-111-10000000    orthogonal lifted from C2×S4
ρ23333-1-1-1110333-1-1-1-1-11-11-10000000    orthogonal lifted from S4
ρ2433-3-1-11110-3-3311-1-1-1-11-110000000    orthogonal lifted from C2×S4
ρ2533-3-1-11-1-10-3-3311-1111-11-10000000    orthogonal lifted from C2×S4
ρ26333-1-1-1-1-10333-1-1-111-11-110000000    orthogonal lifted from S4
ρ276-602-200006i6i02i2i00000000000000    complex faithful
ρ286-602-200006i6i02i2i00000000000000    complex faithful

In GAP, Magma, Sage, TeX 

C_2^4._{10}D_6
 
% in TeX
 
G:=Group("C2^4.10D6");
 
// GroupNames label
 
G:=SmallGroup(192,1471);
 
// by ID
 
G=gap.SmallGroup(192,1471);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-3,-2,2,64,254,1124,4037,285,2358,475]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^6=f^2=b,f*a*f^-1=a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,f*c*f^-1=e*d*e^-1=c*d=d*c,e*c*e^-1=d,d*f=f*d,f*e*f^-1=e^5>;
 
// generators/relations
 

׿
×
⋊
ℤ
𝔽
○
≀