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G = C24.5D6  order 192 = 26·3

4th non-split extension by C24 of D6 acting via D6/C2=S3

non-abelian, soluble, monomial

Aliases: C24.5D6, C23.12D12, (C2×S4)⋊C4, (C2×C4)⋊1S4, C2.2(C4⋊S4), C2.11(C4×S4), C22⋊(D6⋊C4), (C23×C4)⋊1S3, (C2×A4).6D4, (C22×S4).C2, C23.5(C4×S3), A4⋊1(C22⋊C4), C22.16(C2×S4), C2.2(A4⋊D4), C23.16(C3⋊D4), (C22×A4).6C22, (C2×C4×A4)⋊1C2, (C2×A4⋊C4)⋊1C2, (C2×A4).5(C2×C4), SmallGroup(192,972)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C22 — C2×A4 — C24.5D6
Chief series
C1 — C22 — A4 — C2×A4 — C22×A4 — C22×S4 — C24.5D6
Lower central
A4 — C2×A4 — C24.5D6
Upper central
C1 — C22 — C2×C4

Generators and relations for C24.5D6
 G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e6=a, f2=ba=ab, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, fcf-1=ede-1=cd=dc, ece-1=d, df=fd, fef-1=be5 >

Subgroups: 626 in 143 conjugacy classes, 23 normal (21 characteristic)
C1, C2, C2, C3, C4, C22, C22, S3, C6, C2×C4, C2×C4, D4, C23, C23, Dic3, C12, A4, D6, C2×C6, C22⋊C4, C22×C4, C2×D4, C24, C24, C2×Dic3, C2×C12, S4, C2×A4, C22×S3, C2.C42, C2×C22⋊C4, C23×C4, C22×D4, D6⋊C4, A4⋊C4, C4×A4, C2×S4, C2×S4, C22×A4, C23.23D4, C2×A4⋊C4, C2×C4×A4, C22×S4, C24.5D6
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, D6, C22⋊C4, C4×S3, D12, C3⋊D4, S4, D6⋊C4, C2×S4, C4×S4, C4⋊S4, A4⋊D4, C24.5D6

Character table of C24.5D6

 class 12A2B2C2D2E2F2G2H2I34A4B4C4D4E4F4G4H4I4J6A6B6C12A12B12C12D
 size 111133331212822661212121212128888888
ρ11111111111111111111111111111    trivial
ρ211111111111-1-1-1-1-11-1-1-11111-1-1-1-1    linear of order 2
ρ311111111-1-111111-1-1-1-1-1-11111111    linear of order 2
ρ411111111-1-11-1-1-1-11-1111-1111-1-1-1-1    linear of order 2
ρ51-1-111-11-11-11-iii-ii1-i-ii-1-1-11i-i-ii    linear of order 4
ρ61-1-111-11-1-111-iii-i-i-1ii-i1-1-11i-i-ii    linear of order 4
ρ71-1-111-11-11-11i-i-ii-i1ii-i-1-1-11-iii-i    linear of order 4
ρ81-1-111-11-1-111i-i-iii-1-i-ii1-1-11-iii-i    linear of order 4
ρ92222222200-1-2-2-2-2000000-1-1-11111    orthogonal lifted from D6
ρ1022-2-2-2-2220020000000000-22-20000    orthogonal lifted from D4
ρ112-22-2-222-200200000000002-2-20000    orthogonal lifted from D4
ρ122222222200-12222000000-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ1322-2-2-2-22200-100000000001-11√3√3-√3-√3    orthogonal lifted from D12
ρ1422-2-2-2-22200-100000000001-11-√3-√3√3√3    orthogonal lifted from D12
ρ152-2-222-22-200-12i-2i-2i2i00000011-1i-i-ii    complex lifted from C4×S3
ρ162-2-222-22-200-1-2i2i2i-2i00000011-1-iii-i    complex lifted from C4×S3
ρ172-22-2-222-200-10000000000-111-√-3√-3-√-3√-3    complex lifted from C3⋊D4
ρ182-22-2-222-200-10000000000-111√-3-√-3√-3-√-3    complex lifted from C3⋊D4
ρ193333-1-1-1-111033-1-1-1-11-11-10000000    orthogonal lifted from S4
ρ203333-1-1-1-1-1-10-3-311-111-1110000000    orthogonal lifted from C2×S4
ρ213333-1-1-1-1-1-1033-1-111-11-110000000    orthogonal lifted from S4
ρ223333-1-1-1-1110-3-3111-1-11-1-10000000    orthogonal lifted from C2×S4
ρ233-3-33-11-111-10-3i3i-ii-i-1-iii10000000    complex lifted from C4×S4
ρ243-3-33-11-11-110-3i3i-iii1i-i-i-10000000    complex lifted from C4×S4
ρ253-3-33-11-11-1103i-3ii-i-i1-iii-10000000    complex lifted from C4×S4
ρ263-3-33-11-111-103i-3ii-ii-1i-i-i10000000    complex lifted from C4×S4
ρ2766-6-622-2-200000000000000000000    orthogonal lifted from C4⋊S4
ρ286-66-62-2-2200000000000000000000    orthogonal lifted from A4⋊D4

Permutation representations of C24.5D6
►On 24 points - transitive group 24T414
Generators in S24
(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 19)(2 20)(3 21)(4 22)(5 23)(6 24)(7 13)(8 14)(9 15)(10 16)(11 17)(12 18)

(1 13)(2 14)(4 16)(5 17)(7 19)(8 20)(10 22)(11 23)

(1 13)(3 15)(4 16)(6 18)(7 19)(9 21)(10 22)(12 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 13 18)(2 17 14 5)(3 4 15 16)(7 12 19 24)(8 23 20 11)(9 10 21 22)
 
G:=sub<Sym(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), (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,13)(8,14)(9,15)(10,16)(11,17)(12,18), (1,13)(2,14)(4,16)(5,17)(7,19)(8,20)(10,22)(11,23), (1,13)(3,15)(4,16)(6,18)(7,19)(9,21)(10,22)(12,24), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24), (1,6,13,18)(2,17,14,5)(3,4,15,16)(7,12,19,24)(8,23,20,11)(9,10,21,22)>;
 
G:=Group( (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24), (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,13)(8,14)(9,15)(10,16)(11,17)(12,18), (1,13)(2,14)(4,16)(5,17)(7,19)(8,20)(10,22)(11,23), (1,13)(3,15)(4,16)(6,18)(7,19)(9,21)(10,22)(12,24), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24), (1,6,13,18)(2,17,14,5)(3,4,15,16)(7,12,19,24)(8,23,20,11)(9,10,21,22) );
 
G=PermutationGroup([[(1,7),(2,8),(3,9),(4,10),(5,11),(6,12),(13,19),(14,20),(15,21),(16,22),(17,23),(18,24)], [(1,19),(2,20),(3,21),(4,22),(5,23),(6,24),(7,13),(8,14),(9,15),(10,16),(11,17),(12,18)], [(1,13),(2,14),(4,16),(5,17),(7,19),(8,20),(10,22),(11,23)], [(1,13),(3,15),(4,16),(6,18),(7,19),(9,21),(10,22),(12,24)], [(1,2,3,4,5,6,7,8,9,10,11,12),(13,14,15,16,17,18,19,20,21,22,23,24)], [(1,6,13,18),(2,17,14,5),(3,4,15,16),(7,12,19,24),(8,23,20,11),(9,10,21,22)]])
 
G:=TransitiveGroup(24,414);
 

►On 24 points - transitive group 24T415
Generators in S24
(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 20)(2 21)(3 22)(4 23)(5 24)(6 13)(7 14)(8 15)(9 16)(10 17)(11 18)(12 19)

(1 20)(2 21)(4 23)(5 24)(7 14)(8 15)(10 17)(11 18)

(1 20)(3 22)(4 23)(6 13)(7 14)(9 16)(10 17)(12 19)

(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 12 14 13)(2 24 15 11)(3 10 16 23)(4 22 17 9)(5 8 18 21)(6 20 19 7)
 
G:=sub<Sym(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), (1,20)(2,21)(3,22)(4,23)(5,24)(6,13)(7,14)(8,15)(9,16)(10,17)(11,18)(12,19), (1,20)(2,21)(4,23)(5,24)(7,14)(8,15)(10,17)(11,18), (1,20)(3,22)(4,23)(6,13)(7,14)(9,16)(10,17)(12,19), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24), (1,12,14,13)(2,24,15,11)(3,10,16,23)(4,22,17,9)(5,8,18,21)(6,20,19,7)>;
 
G:=Group( (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24), (1,20)(2,21)(3,22)(4,23)(5,24)(6,13)(7,14)(8,15)(9,16)(10,17)(11,18)(12,19), (1,20)(2,21)(4,23)(5,24)(7,14)(8,15)(10,17)(11,18), (1,20)(3,22)(4,23)(6,13)(7,14)(9,16)(10,17)(12,19), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24), (1,12,14,13)(2,24,15,11)(3,10,16,23)(4,22,17,9)(5,8,18,21)(6,20,19,7) );
 
G=PermutationGroup([[(1,7),(2,8),(3,9),(4,10),(5,11),(6,12),(13,19),(14,20),(15,21),(16,22),(17,23),(18,24)], [(1,20),(2,21),(3,22),(4,23),(5,24),(6,13),(7,14),(8,15),(9,16),(10,17),(11,18),(12,19)], [(1,20),(2,21),(4,23),(5,24),(7,14),(8,15),(10,17),(11,18)], [(1,20),(3,22),(4,23),(6,13),(7,14),(9,16),(10,17),(12,19)], [(1,2,3,4,5,6,7,8,9,10,11,12),(13,14,15,16,17,18,19,20,21,22,23,24)], [(1,12,14,13),(2,24,15,11),(3,10,16,23),(4,22,17,9),(5,8,18,21),(6,20,19,7)]])
 
G:=TransitiveGroup(24,415);
 

►On 24 points - transitive group 24T417
Generators in S24
(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 22)(2 23)(3 24)(4 13)(5 14)(6 15)(7 16)(8 17)(9 18)(10 19)(11 20)(12 21)

(1 7)(2 8)(4 10)(5 11)(13 19)(14 20)(16 22)(17 23)

(1 7)(3 9)(4 10)(6 12)(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 6 16 21)(2 20 17 5)(3 4 18 19)(7 12 22 15)(8 14 23 11)(9 10 24 13)
 
G:=sub<Sym(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), (1,22)(2,23)(3,24)(4,13)(5,14)(6,15)(7,16)(8,17)(9,18)(10,19)(11,20)(12,21), (1,7)(2,8)(4,10)(5,11)(13,19)(14,20)(16,22)(17,23), (1,7)(3,9)(4,10)(6,12)(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), (1,6,16,21)(2,20,17,5)(3,4,18,19)(7,12,22,15)(8,14,23,11)(9,10,24,13)>;
 
G:=Group( (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24), (1,22)(2,23)(3,24)(4,13)(5,14)(6,15)(7,16)(8,17)(9,18)(10,19)(11,20)(12,21), (1,7)(2,8)(4,10)(5,11)(13,19)(14,20)(16,22)(17,23), (1,7)(3,9)(4,10)(6,12)(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), (1,6,16,21)(2,20,17,5)(3,4,18,19)(7,12,22,15)(8,14,23,11)(9,10,24,13) );
 
G=PermutationGroup([[(1,7),(2,8),(3,9),(4,10),(5,11),(6,12),(13,19),(14,20),(15,21),(16,22),(17,23),(18,24)], [(1,22),(2,23),(3,24),(4,13),(5,14),(6,15),(7,16),(8,17),(9,18),(10,19),(11,20),(12,21)], [(1,7),(2,8),(4,10),(5,11),(13,19),(14,20),(16,22),(17,23)], [(1,7),(3,9),(4,10),(6,12),(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)], [(1,6,16,21),(2,20,17,5),(3,4,18,19),(7,12,22,15),(8,14,23,11),(9,10,24,13)]])
 
G:=TransitiveGroup(24,417);
 

Matrix representation of C24.5D6 ►in GL7(𝔽13)

12000000
01200000
0010000
0001000
0000100
0000010
0000001
,
12000000
01200000
00120000
00012000
0000100
0000010
0000001
,
1000000
0100000
0010000
0001000
00000121
00000120
00001120
,
1000000
0100000
0010000
0001000
00001200
00001201
00001210
,
01200000
1000000
00109000
00410000
00001210
00001200
00001201
,
12000000
0100000
0043000
0039000
00001200
00001210
00001201

G:=sub<GL(7,GF(13))| [12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,12,12,12,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,12,12,0,0,0,0,0,0,1,0,0,0,0,0,1,0],[0,1,0,0,0,0,0,12,0,0,0,0,0,0,0,0,10,4,0,0,0,0,0,9,10,0,0,0,0,0,0,0,12,12,12,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,4,3,0,0,0,0,0,3,9,0,0,0,0,0,0,0,12,12,12,0,0,0,0,0,1,0,0,0,0,0,0,0,1] >;
 

C24.5D6 in GAP, Magma, Sage, TeX 

C_2^4._5D_6
 
% in TeX
 
G:=Group("C2^4.5D6");
 
// GroupNames label
 
G:=SmallGroup(192,972);
 
// by ID
 
G=gap.SmallGroup(192,972);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-3,-2,2,141,36,1124,4037,285,2358,475]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^6=a,f^2=b*a=a*b,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*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=b*e^5>;
 
// generators/relations
 

Export

Character table of C24.5D6 in TeX

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