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G = C2×A4⋊D4order 192 = 26·3

Direct product of C2 and A4⋊D4

direct product, non-abelian, soluble, monomial

Aliases: C2×A4⋊D4, C234S4, C251S3, C243D6, (C2×A4)⋊2D4, A43(C2×D4), C223(C2×S4), A4⋊C42C22, (C23×A4)⋊2C2, (C22×S4)⋊3C2, (C2×S4)⋊3C22, C233(C3⋊D4), C2.21(C22×S4), (C2×A4).10C23, (C22×A4)⋊3C22, C23.10(C22×S3), (C2×A4⋊C4)⋊4C2, C22⋊(C2×C3⋊D4), SmallGroup(192,1488)

Series: Derived Chief Lower central Upper central

C1C22C2×A4 — C2×A4⋊D4
C1C22A4C2×A4C2×S4C22×S4 — C2×A4⋊D4
A4C2×A4 — C2×A4⋊D4
C1C22C23

Generators and relations for C2×A4⋊D4
 G = < a,b,c,d,e,f | a2=b2=c2=d3=e4=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, dbd-1=ebe-1=fbf=bc=cb, dcd-1=b, ce=ec, cf=fc, ede-1=fdf=d-1, fef=e-1 >

Subgroups: 1134 in 271 conjugacy classes, 35 normal (15 characteristic)
C1, C2, C2 [×2], C2 [×10], C3, C4 [×6], C22 [×2], C22 [×2], C22 [×44], S3 [×2], C6 [×5], C2×C4 [×12], D4 [×24], C23 [×2], C23 [×2], C23 [×42], Dic3 [×2], A4, D6 [×4], C2×C6 [×5], C22⋊C4 [×12], C22×C4 [×3], C2×D4 [×24], C24, C24 [×2], C24 [×9], C2×Dic3, C3⋊D4 [×4], S4 [×2], C2×A4, C2×A4 [×2], C2×A4 [×2], C22×S3, C22×C6, C2×C22⋊C4 [×3], C22≀C2 [×8], C22×D4 [×3], C25, A4⋊C4 [×2], C2×C3⋊D4, C2×S4 [×2], C2×S4 [×2], C22×A4, C22×A4 [×2], C22×A4 [×2], C2×C22≀C2, C2×A4⋊C4, A4⋊D4 [×4], C22×S4, C23×A4, C2×A4⋊D4
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D6 [×3], C2×D4, C3⋊D4 [×2], S4, C22×S3, C2×C3⋊D4, C2×S4 [×3], A4⋊D4 [×2], C22×S4, C2×A4⋊D4

Character table of C2×A4⋊D4

 class 12A2B2C2D2E2F2G2H2I2J2K2L2M34A4B4C4D4E4F6A6B6C6D6E6F6G
 size 111122333366121281212121212128888888
ρ11111111111111111111111111111    trivial
ρ21111-1-11111-1-1-1-11111-1-11-1-11-11-11    linear of order 2
ρ31-1-11-11-111-1-11-1111-1-11-111-1-111-1-1    linear of order 2
ρ41-1-111-1-111-11-11-111-1-1-111-11-1-111-1    linear of order 2
ρ51111-1-11111-1-1111-1-1-111-1-1-11-11-11    linear of order 2
ρ6111111111111-1-11-1-1-1-1-1-11111111    linear of order 2
ρ71-1-11-11-111-1-111-11-111-11-11-1-111-1-1    linear of order 2
ρ81-1-111-1-111-11-1-111-1111-1-1-11-1-111-1    linear of order 2
ρ922-2-200-22-22000020000000020-20-2    orthogonal lifted from D4
ρ102-2-22-22-222-2-2200-1000000-111-1-111    orthogonal lifted from D6
ρ112222-2-22222-2-200-100000011-11-11-1    orthogonal lifted from D6
ρ1222222222222200-1000000-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ132-2-222-2-222-22-200-10000001-111-1-11    orthogonal lifted from D6
ρ142-22-20022-2-20000200000000-20-202    orthogonal lifted from D4
ρ152-22-20022-2-20000-1000000--3-31-31--3-1    complex lifted from C3⋊D4
ρ1622-2-200-22-220000-1000000-3-3-1--31--31    complex lifted from C3⋊D4
ρ172-22-20022-2-20000-1000000-3--31--31-3-1    complex lifted from C3⋊D4
ρ1822-2-200-22-220000-1000000--3--3-1-31-31    complex lifted from C3⋊D4
ρ193-3-333-31-1-11-11-11011-1-11-10000000    orthogonal lifted from C2×S4
ρ203-3-33-331-1-111-1-110-1-11-1110000000    orthogonal lifted from C2×S4
ρ21333333-1-1-1-1-1-1-1-101-1111-10000000    orthogonal lifted from S4
ρ223333-3-3-1-1-1-111-1-10-11-11110000000    orthogonal lifted from C2×S4
ρ233-3-333-31-1-11-111-10-1-111-110000000    orthogonal lifted from C2×S4
ρ243-3-33-331-1-111-11-1011-11-1-10000000    orthogonal lifted from C2×S4
ρ25333333-1-1-1-1-1-1110-11-1-1-110000000    orthogonal lifted from S4
ρ263333-3-3-1-1-1-1111101-11-1-1-10000000    orthogonal lifted from C2×S4
ρ276-66-600-2-222000000000000000000    orthogonal lifted from A4⋊D4
ρ2866-6-6002-22-2000000000000000000    orthogonal lifted from A4⋊D4

Permutation representations of C2×A4⋊D4
On 24 points - transitive group 24T398
Generators in S24
(1 8)(2 5)(3 6)(4 7)(9 16)(10 13)(11 14)(12 15)(17 21)(18 22)(19 23)(20 24)
(1 8)(2 5)(3 6)(4 7)(9 16)(11 14)(17 21)(19 23)
(9 16)(10 13)(11 14)(12 15)(17 21)(18 22)(19 23)(20 24)
(1 11 22)(2 23 12)(3 9 24)(4 21 10)(5 19 15)(6 16 20)(7 17 13)(8 14 18)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)
(2 4)(5 7)(9 24)(10 23)(11 22)(12 21)(13 19)(14 18)(15 17)(16 20)

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

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

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

G:=TransitiveGroup(24,398);

On 24 points - transitive group 24T399
Generators in S24
(1 13)(2 14)(3 15)(4 16)(5 22)(6 23)(7 24)(8 21)(9 19)(10 20)(11 17)(12 18)
(1 2)(3 4)(5 7)(6 11)(8 9)(10 12)(13 14)(15 16)(17 23)(18 20)(19 21)(22 24)
(1 3)(2 4)(5 12)(6 9)(7 10)(8 11)(13 15)(14 16)(17 21)(18 22)(19 23)(20 24)
(1 8 10)(2 11 5)(3 6 12)(4 9 7)(13 21 20)(14 17 22)(15 23 18)(16 19 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 13)(2 16)(3 15)(4 14)(5 19)(6 18)(7 17)(8 20)(9 22)(10 21)(11 24)(12 23)

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

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

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

G:=TransitiveGroup(24,399);

On 24 points - transitive group 24T405
Generators in S24
(1 11)(2 12)(3 9)(4 10)(5 16)(6 13)(7 14)(8 15)(17 21)(18 22)(19 23)(20 24)
(1 3)(2 12)(4 10)(5 14)(6 8)(7 16)(9 11)(13 15)(17 21)(18 24)(19 23)(20 22)
(1 9)(2 10)(3 11)(4 12)(5 16)(6 13)(7 14)(8 15)(17 19)(18 20)(21 23)(22 24)
(1 19 7)(2 8 20)(3 17 5)(4 6 18)(9 21 16)(10 13 22)(11 23 14)(12 15 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 4)(2 3)(5 8)(6 7)(9 12)(10 11)(13 14)(15 16)(17 20)(18 19)(21 24)(22 23)

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

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

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

G:=TransitiveGroup(24,405);

On 24 points - transitive group 24T406
Generators in S24
(1 9)(2 10)(3 11)(4 12)(5 13)(6 14)(7 15)(8 16)(17 21)(18 22)(19 23)(20 24)
(2 4)(5 7)(6 8)(10 12)(13 15)(14 16)(17 19)(21 23)
(1 3)(2 4)(9 11)(10 12)(17 19)(18 20)(21 23)(22 24)
(1 13 19)(2 20 14)(3 15 17)(4 18 16)(5 23 9)(6 10 24)(7 21 11)(8 12 22)
(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 16)(6 15)(7 14)(8 13)(17 24)(18 23)(19 22)(20 21)

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

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

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

G:=TransitiveGroup(24,406);

On 24 points - transitive group 24T407
Generators in S24
(1 16)(2 13)(3 14)(4 15)(5 10)(6 11)(7 12)(8 9)(17 22)(18 23)(19 24)(20 21)
(2 13)(4 15)(5 10)(6 11)(7 12)(8 9)(18 23)(20 21)
(1 16)(2 13)(3 14)(4 15)(17 22)(18 23)(19 24)(20 21)
(1 9 23)(2 24 10)(3 11 21)(4 22 12)(5 13 19)(6 20 14)(7 15 17)(8 18 16)
(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 4)(2 3)(5 6)(7 8)(9 12)(10 11)(13 14)(15 16)(17 18)(19 20)(21 24)(22 23)

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

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

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

G:=TransitiveGroup(24,407);

On 24 points - transitive group 24T416
Generators in S24
(1 9)(2 10)(3 11)(4 12)(5 17)(6 18)(7 19)(8 20)(13 21)(14 22)(15 23)(16 24)
(1 3)(2 4)(6 8)(9 11)(10 12)(14 16)(18 20)(22 24)
(5 7)(6 8)(13 15)(14 16)(17 19)(18 20)(21 23)(22 24)
(1 20 23)(2 24 17)(3 18 21)(4 22 19)(5 10 16)(6 13 11)(7 12 14)(8 15 9)
(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 9)(2 12)(3 11)(4 10)(5 22)(6 21)(7 24)(8 23)(13 18)(14 17)(15 20)(16 19)

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

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

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

G:=TransitiveGroup(24,416);

Matrix representation of C2×A4⋊D4 in GL7(𝔽13)

12000000
01200000
0010000
0001000
0000100
0000010
0000001
,
1000000
0100000
0010000
0001000
00001200
00000120
0000001
,
1000000
0100000
0010000
0001000
0000100
00000120
00000012
,
1000000
0100000
0096000
0003000
0000001
0000100
0000010
,
1200000
121200000
001211000
0011000
00001200
00000012
00000120
,
1000000
121200000
0010000
001212000
0000100
0000001
0000010

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],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,9,0,0,0,0,0,0,6,3,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0],[1,12,0,0,0,0,0,2,12,0,0,0,0,0,0,0,12,1,0,0,0,0,0,11,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,12,0],[1,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0] >;

C2×A4⋊D4 in GAP, Magma, Sage, TeX

C_2\times A_4\rtimes D_4
% in TeX

G:=Group("C2xA4:D4");
// GroupNames label

G:=SmallGroup(192,1488);
// by ID

G=gap.SmallGroup(192,1488);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-2,2,254,1124,4037,285,2358,475]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^3=e^4=f^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,d*b*d^-1=e*b*e^-1=f*b*f=b*c=c*b,d*c*d^-1=b,c*e=e*c,c*f=f*c,e*d*e^-1=f*d*f=d^-1,f*e*f=e^-1>;
// generators/relations

Export

Character table of C2×A4⋊D4 in TeX

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