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G = C246D6order 192 = 26·3

1st semidirect product of C24 and D6 acting via D6/C3=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C246D6, C32C2≀C22, (C2×C12)⋊2D4, C22⋊C43D6, (C22×C6)⋊3D4, C22≀C22S3, D46D63C2, (C2×Dic3)⋊2D4, (C2×D4).33D6, (C22×S3)⋊2D4, C244S31C2, C6.43C22≀C2, C231(C3⋊D4), (C23×C6)⋊7C22, C22.33(S3×D4), (C6×D4).49C22, C23.6D65C2, C23.7D65C2, C2.11(C232D6), C6.D44C22, C23.84(C22×S3), (C22×C6).113C23, (C2×C4)⋊1(C3⋊D4), (C2×C6).30(C2×D4), (C3×C22≀C2)⋊1C2, (C2×C3⋊D4).5C22, C22.29(C2×C3⋊D4), (C3×C22⋊C4)⋊34C22, SmallGroup(192,591)

Series: Derived Chief Lower central Upper central

C1C22×C6 — C246D6
C1C3C6C2×C6C22×C6C2×C3⋊D4D46D6 — C246D6
C3C6C22×C6 — C246D6
C1C2C23C22≀C2

Generators and relations for C246D6
 G = < a,b,c,d,e,f | a2=b2=c2=d2=e6=f2=1, ab=ba, eae-1=ac=ca, ad=da, faf=abcd, bc=cb, ebe-1=fbf=bd=db, fcf=cd=dc, ce=ec, de=ed, df=fd, fef=e-1 >

Subgroups: 688 in 198 conjugacy classes, 39 normal (23 characteristic)
C1, C2, C2 [×8], C3, C4 [×6], C22, C22 [×2], C22 [×18], S3 [×2], C6, C6 [×6], C2×C4, C2×C4 [×8], D4 [×15], Q8, C23 [×2], C23 [×8], Dic3 [×4], C12 [×2], D6 [×5], C2×C6, C2×C6 [×2], C2×C6 [×13], C22⋊C4, C22⋊C4 [×5], C2×D4, C2×D4 [×8], C4○D4 [×3], C24, Dic6, C4×S3 [×2], D12, C2×Dic3 [×2], C2×Dic3 [×3], C3⋊D4 [×10], C2×C12, C2×C12, C3×D4 [×4], C22×S3 [×2], C22×S3, C22×C6 [×2], C22×C6 [×5], C23⋊C4 [×3], C22≀C2, C22≀C2 [×2], 2+ 1+4, C6.D4 [×2], C6.D4 [×2], C3×C22⋊C4, C3×C22⋊C4, C4○D12, S3×D4 [×2], D42S3 [×2], C2×C3⋊D4 [×2], C2×C3⋊D4 [×3], C6×D4, C6×D4, C23×C6, C2≀C22, C23.6D6 [×2], C23.7D6, C244S3 [×2], C3×C22≀C2, D46D6, C246D6
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×6], C23, D6 [×3], C2×D4 [×3], C3⋊D4 [×2], C22×S3, C22≀C2, S3×D4 [×2], C2×C3⋊D4, C2≀C22, C232D6, C246D6

Character table of C246D6

 class 12A2B2C2D2E2F2G2H2I34A4B4C4D4E4F6A6B6C6D6E6F6G6H6I6J12A12B12C
 size 112224441212248121224242224444448888
ρ1111111111111111111111111111111    trivial
ρ21111111-11-11-1-1-111-1111111111-1-1-1-1    linear of order 2
ρ311111-1-1-1-111-111-11-11111-1-11-1-1-11-11    linear of order 2
ρ411111-1-11-1-111-1-1-1111111-1-11-1-11-11-1    linear of order 2
ρ511111-1-1-11-11-11-11-111111-1-11-1-1-11-11    linear of order 2
ρ611111-1-111111-111-1-11111-1-11-1-11-11-1    linear of order 2
ρ711111111-1-1111-1-1-1-11111111111111    linear of order 2
ρ81111111-1-111-1-11-1-11111111111-1-1-1-1    linear of order 2
ρ922-22-20000-220020002-2-2200-2000000    orthogonal lifted from D4
ρ10222-2-200-2002200000222-200-200-2020    orthogonal lifted from D4
ρ1122-2-22000202000-2002-2-2-2002000000    orthogonal lifted from D4
ρ1222-22-200002200-20002-2-2200-2000000    orthogonal lifted from D4
ρ1322222-2-2200-12-20000-1-1-1-111-111-11-11    orthogonal lifted from D6
ρ142222222-200-1-2-20000-1-1-1-1-1-1-1-1-11111    orthogonal lifted from D6
ρ1522222-2-2-200-1-220000-1-1-1-111-1111-11-1    orthogonal lifted from D6
ρ162222222200-1220000-1-1-1-1-1-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ17222-2-2002002-200000222-200-20020-20    orthogonal lifted from D4
ρ1822-2-22000-2020002002-2-2-2002000000    orthogonal lifted from D4
ρ19222-2-200-200-1200000-1-1-11--3-31--3-31--3-1-3    complex lifted from C3⋊D4
ρ20222-2-200-200-1200000-1-1-11-3--31-3--31-3-1--3    complex lifted from C3⋊D4
ρ21222-2-200200-1-200000-1-1-11--3-31--3-3-1-31--3    complex lifted from C3⋊D4
ρ22222-2-200200-1-200000-1-1-11-3--31-3--3-1--31-3    complex lifted from C3⋊D4
ρ2344-4-4400000-2000000-222200-2000000    orthogonal lifted from S3×D4
ρ244-40002-20004000000-4000-2-20220000    orthogonal lifted from C2≀C22
ρ2544-44-400000-2000000-222-2002000000    orthogonal lifted from S3×D4
ρ264-4000-220004000000-4000220-2-20000    orthogonal lifted from C2≀C22
ρ274-4000-22000-20000002-2-32-30-1--3-1+-301+-31--30000    complex faithful
ρ284-4000-22000-200000022-3-2-30-1+-3-1--301--31+-30000    complex faithful
ρ294-40002-2000-20000002-2-32-301+-31--30-1--3-1+-30000    complex faithful
ρ304-40002-2000-200000022-3-2-301--31+-30-1+-3-1--30000    complex faithful

Permutation representations of C246D6
On 24 points - transitive group 24T287
Generators in S24
(8 19)(10 21)(12 23)
(2 13)(4 15)(6 17)(8 19)(10 21)(12 23)
(7 24)(8 19)(9 20)(10 21)(11 22)(12 23)
(1 18)(2 13)(3 14)(4 15)(5 16)(6 17)(7 24)(8 19)(9 20)(10 21)(11 22)(12 23)
(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 23)(2 22)(3 21)(4 20)(5 19)(6 24)(7 17)(8 16)(9 15)(10 14)(11 13)(12 18)

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

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

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

G:=TransitiveGroup(24,287);

On 24 points - transitive group 24T357
Generators in S24
(7 14)(8 18)(9 16)(10 17)(11 15)(12 13)
(1 20)(2 24)(3 22)(4 23)(5 21)(6 19)(7 14)(8 18)(9 16)(10 17)(11 15)(12 13)
(7 10)(8 11)(9 12)(13 16)(14 17)(15 18)
(1 4)(2 5)(3 6)(7 10)(8 11)(9 12)(13 16)(14 17)(15 18)(19 22)(20 23)(21 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 12)(2 11)(3 10)(4 9)(5 8)(6 7)(13 23)(14 22)(15 21)(16 20)(17 19)(18 24)

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

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

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

G:=TransitiveGroup(24,357);

On 24 points - transitive group 24T364
Generators in S24
(1 19)(2 16)(3 21)(4 18)(5 23)(6 14)(7 20)(8 17)(9 22)(10 13)(11 24)(12 15)
(1 15)(2 7)(3 17)(4 9)(5 13)(6 11)(8 21)(10 23)(12 19)(14 24)(16 20)(18 22)
(1 12)(2 7)(3 8)(4 9)(5 10)(6 11)(13 23)(14 24)(15 19)(16 20)(17 21)(18 22)
(1 19)(2 20)(3 21)(4 22)(5 23)(6 24)(7 16)(8 17)(9 18)(10 13)(11 14)(12 15)
(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 3)(4 6)(7 16)(8 15)(9 14)(10 13)(11 18)(12 17)(19 21)(22 24)

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

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

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

G:=TransitiveGroup(24,364);

Matrix representation of C246D6 in GL4(𝔽7) generated by

1040
0150
0060
0001
,
2604
2641
0060
5210
,
0632
6042
0060
0001
,
6000
0600
0060
0006
,
4066
4155
6662
0003
,
6251
1121
4355
6642
G:=sub<GL(4,GF(7))| [1,0,0,0,0,1,0,0,4,5,6,0,0,0,0,1],[2,2,0,5,6,6,0,2,0,4,6,1,4,1,0,0],[0,6,0,0,6,0,0,0,3,4,6,0,2,2,0,1],[6,0,0,0,0,6,0,0,0,0,6,0,0,0,0,6],[4,4,6,0,0,1,6,0,6,5,6,0,6,5,2,3],[6,1,4,6,2,1,3,6,5,2,5,4,1,1,5,2] >;

C246D6 in GAP, Magma, Sage, TeX

C_2^4\rtimes_6D_6
% in TeX

G:=Group("C2^4:6D6");
// GroupNames label

G:=SmallGroup(192,591);
// by ID

G=gap.SmallGroup(192,591);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,254,219,570,1684,6278]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=e^6=f^2=1,a*b=b*a,e*a*e^-1=a*c=c*a,a*d=d*a,f*a*f=a*b*c*d,b*c=c*b,e*b*e^-1=f*b*f=b*d=d*b,f*c*f=c*d=d*c,c*e=e*c,d*e=e*d,d*f=f*d,f*e*f=e^-1>;
// generators/relations

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Character table of C246D6 in TeX

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