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G = (C2×C62).C4order 288 = 25·32

5th non-split extension by C2×C62 of C4 acting faithfully

metabelian, soluble, monomial

Aliases: (C2×C62).5C4, C23.(C32⋊C4), C3⋊Dic3.12D4, C62.12(C2×C4), C62.C44C2, C324(C4.D4), C2.11(C62⋊C4), (C22×C3⋊S3).3C4, C22.5(C2×C32⋊C4), (C2×C327D4).4C2, (C3×C6).22(C22⋊C4), (C2×C3⋊Dic3).8C22, SmallGroup(288,436)

Series: Derived Chief Lower central Upper central

C1C62 — (C2×C62).C4
C1C32C3×C6C3⋊Dic3C2×C3⋊Dic3C62.C4 — (C2×C62).C4
C32C3×C6C62 — (C2×C62).C4
C1C2C22C23

Generators and relations for (C2×C62).C4
 G = < a,b,c,d | a2=b6=c6=1, d4=c3, ab=ba, ac=ca, dad-1=ab3c3, bc=cb, dbd-1=b-1c, dcd-1=b4c >

Subgroups: 584 in 100 conjugacy classes, 16 normal (12 characteristic)
C1, C2, C2 [×3], C3 [×2], C4 [×2], C22, C22 [×4], S3 [×4], C6 [×10], C8 [×2], C2×C4, D4 [×2], C23, C23, C32, Dic3 [×4], D6 [×8], C2×C6 [×10], M4(2) [×2], C2×D4, C3⋊S3, C3×C6, C3×C6 [×2], C2×Dic3 [×2], C3⋊D4 [×8], C22×S3 [×2], C22×C6 [×2], C4.D4, C3⋊Dic3 [×2], C2×C3⋊S3 [×2], C62, C62 [×2], C2×C3⋊D4 [×2], C322C8 [×2], C2×C3⋊Dic3, C327D4 [×2], C22×C3⋊S3, C2×C62, C62.C4 [×2], C2×C327D4, (C2×C62).C4
Quotients: C1, C2 [×3], C4 [×2], C22, C2×C4, D4 [×2], C22⋊C4, C4.D4, C32⋊C4, C2×C32⋊C4, C62⋊C4, (C2×C62).C4

Character table of (C2×C62).C4

 class 12A2B2C2D3A3B4A4B6A6B6C6D6E6F6G6H6I6J6K6L6M6N8A8B8C8D
 size 1124364418184444444444444436363636
ρ1111111111111111111111111111    trivial
ρ2111-1-1111111111-1-1-1-1-1-1-1-1111-1-1    linear of order 2
ρ311111111111111111111111-1-1-1-1    linear of order 2
ρ4111-1-1111111111-1-1-1-1-1-1-1-11-1-111    linear of order 2
ρ51111-111-1-111111111111111-ii-ii    linear of order 4
ρ6111-1111-1-111111-1-1-1-1-1-1-1-11-iii-i    linear of order 4
ρ71111-111-1-111111111111111i-ii-i    linear of order 4
ρ8111-1111-1-111111-1-1-1-1-1-1-1-11i-i-ii    linear of order 4
ρ922-200222-22-22-2-200000000-20000    orthogonal lifted from D4
ρ1022-20022-222-22-2-200000000-20000    orthogonal lifted from D4
ρ1144-4001-200-2212-1003-3030-3-10000    orthogonal lifted from C62⋊C4
ρ1244440-210011-21-211-2-21-21-2-20000    orthogonal lifted from C32⋊C4
ρ1344-400-21001-1-2-12-3-300303020000    orthogonal lifted from C62⋊C4
ρ1444-4001-200-2212-100-330-303-10000    orthogonal lifted from C62⋊C4
ρ15444-401-200-2-21-2122-1-12-12-110000    orthogonal lifted from C2×C32⋊C4
ρ16444401-200-2-21-21-2-211-21-2110000    orthogonal lifted from C32⋊C4
ρ17444-40-210011-21-2-1-122-12-12-20000    orthogonal lifted from C2×C32⋊C4
ρ1844-400-21001-1-2-123300-30-3020000    orthogonal lifted from C62⋊C4
ρ194-40004400-40-4000000000000000    orthogonal lifted from C4.D4
ρ204-40001-20020-10-300-3-3-2-3--32-3--330000    complex faithful
ρ214-4000-2100-132-30--3-3-2-30--32-3-3000000    complex faithful
ρ224-40001-20020-103-2-32-3-3--30--30-3-30000    complex faithful
ρ234-4000-2100-1-3230--3-302-3-30--3-2-300000    complex faithful
ρ244-40001-20020-1032-3-2-3--3-30-30--3-30000    complex faithful
ρ254-40001-20020-10-300--3--32-3-3-2-3-330000    complex faithful
ρ264-4000-2100-1-3230-3--30-2-3--30-32-300000    complex faithful
ρ274-4000-2100-132-30-3--32-30-3-2-3--3000000    complex faithful

Permutation representations of (C2×C62).C4
On 24 points - transitive group 24T585
Generators in S24
(2 6)(3 7)(10 14)(11 15)(18 22)(19 23)
(1 21 13)(2 18 14 6 22 10)(3 15 23)(4 12 24 8 16 20)(5 17 9)(7 11 19)
(1 9 21 5 13 17)(2 6)(3 19 15 7 23 11)(4 8)(10 14)(12 16)(18 22)(20 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)

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

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

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

G:=TransitiveGroup(24,585);

On 24 points - transitive group 24T625
Generators in S24
(1 7)(2 4)(3 5)(6 8)(9 20)(10 21)(11 18)(12 19)(13 24)(14 17)(15 22)(16 23)
(1 21 12)(2 18 13 6 22 9)(3 14 23)(4 11 24 8 15 20)(5 17 16)(7 10 19)
(1 16 21 5 12 17)(2 6)(3 19 14 7 23 10)(4 8)(9 13)(11 15)(18 22)(20 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)

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

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

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

G:=TransitiveGroup(24,625);

Matrix representation of (C2×C62).C4 in GL4(𝔽7) generated by

4526
0602
3331
0001
,
3532
3126
4406
0003
,
4534
3221
4416
0006
,
2156
0215
3424
3321
G:=sub<GL(4,GF(7))| [4,0,3,0,5,6,3,0,2,0,3,0,6,2,1,1],[3,3,4,0,5,1,4,0,3,2,0,0,2,6,6,3],[4,3,4,0,5,2,4,0,3,2,1,0,4,1,6,6],[2,0,3,3,1,2,4,3,5,1,2,2,6,5,4,1] >;

(C2×C62).C4 in GAP, Magma, Sage, TeX

(C_2\times C_6^2).C_4
% in TeX

G:=Group("(C2xC6^2).C4");
// GroupNames label

G:=SmallGroup(288,436);
// by ID

G=gap.SmallGroup(288,436);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,28,141,219,100,675,9413,691,12550,2372]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^6=c^6=1,d^4=c^3,a*b=b*a,a*c=c*a,d*a*d^-1=a*b^3*c^3,b*c=c*b,d*b*d^-1=b^-1*c,d*c*d^-1=b^4*c>;
// generators/relations

Export

Character table of (C2×C62).C4 in TeX

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