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G = C62.(C2×C4)  order 288 = 25·32

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

metabelian, soluble, monomial

Aliases: C62.(C2×C4), D4.(C32⋊C4), (D4×C32).C4, C327D4.C4, C324Q8.C4, C327(C8○D4), C62.C45C2, C12.D6.4C2, C32⋊M4(2)⋊6C2, C3⋊Dic3.32C23, C322C8.11C22, C3⋊S33C84C2, C4.5(C2×C32⋊C4), (C3×C12).5(C2×C4), (C2×C322C8)⋊8C2, C22.1(C2×C32⋊C4), C2.8(C22×C32⋊C4), (C4×C3⋊S3).37C22, C3⋊Dic3.23(C2×C4), (C3×C6).30(C22×C4), (C2×C3⋊Dic3).100C22, (C2×C3⋊S3).20(C2×C4), SmallGroup(288,935)

Series: Derived Chief Lower central Upper central

C1C3×C6 — C62.(C2×C4)
C1C32C3×C6C3⋊Dic3C322C8C2×C322C8 — C62.(C2×C4)
C32C3×C6 — C62.(C2×C4)
C1C2D4

Generators and relations for C62.(C2×C4)
 G = < a,b,c,d | a6=b6=c2=1, d4=b3, ab=ba, cac=ab3, dad-1=a-1b4, bc=cb, dbd-1=a4b, cd=dc >

Subgroups: 456 in 102 conjugacy classes, 34 normal (18 characteristic)
C1, C2, C2 [×3], C3 [×2], C4, C4 [×3], C22 [×2], C22, S3 [×2], C6 [×6], C8 [×4], C2×C4 [×3], D4, D4 [×2], Q8, C32, Dic3 [×6], C12 [×2], D6 [×2], C2×C6 [×4], C2×C8 [×3], M4(2) [×3], C4○D4, C3⋊S3, C3×C6, C3×C6 [×2], Dic6 [×2], C4×S3 [×2], C2×Dic3 [×4], C3⋊D4 [×4], C3×D4 [×2], C8○D4, C3⋊Dic3, C3⋊Dic3 [×2], C3×C12, C2×C3⋊S3, C62 [×2], D42S3 [×2], C322C8 [×2], C322C8 [×2], C324Q8, C4×C3⋊S3, C2×C3⋊Dic3 [×2], C327D4 [×2], D4×C32, C3⋊S33C8, C32⋊M4(2), C2×C322C8 [×2], C62.C4 [×2], C12.D6, C62.(C2×C4)
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], C23, C22×C4, C8○D4, C32⋊C4, C2×C32⋊C4 [×3], C22×C32⋊C4, C62.(C2×C4)

Character table of C62.(C2×C4)

 class 12A2B2C2D3A3B4A4B4C4D4E6A6B6C6D6E6F8A8B8C8D8E8F8G8H8I8J12A12B
 size 112218442991818448888999918181818181888
ρ1111111111111111111111111111111    trivial
ρ2111111111111111111-1-1-1-1-1-1-1-1-1-111    linear of order 2
ρ3111-1-111-111-1111-111-11111-111-1-1-1-1-1    linear of order 2
ρ4111-1-111-111-1111-111-1-1-1-1-11-1-1111-1-1    linear of order 2
ρ511-1-1111111-1-111-1-1-1-1-1-1-1-1-11111-111    linear of order 2
ρ611-1-1111111-1-111-1-1-1-111111-1-1-1-1111    linear of order 2
ρ711-11-111-1111-1111-1-11-1-1-1-1111-1-11-1-1    linear of order 2
ρ811-11-111-1111-1111-1-111111-1-1-111-1-1-1    linear of order 2
ρ911-1-1-1111-1-11111-1-1-1-1ii-i-i-i-ii-iii11    linear of order 4
ρ101111-1111-1-1-1-1111111-i-iiii-ii-ii-i11    linear of order 4
ρ11111-1111-1-1-11-111-111-1-i-iii-i-iii-ii-1-1    linear of order 4
ρ1211-11111-1-1-1-11111-1-11ii-i-ii-iii-i-i-1-1    linear of order 4
ρ1311-1-1-1111-1-11111-1-1-1-1-i-iiiii-ii-i-i11    linear of order 4
ρ141111-1111-1-1-1-1111111ii-i-i-ii-ii-ii11    linear of order 4
ρ15111-1111-1-1-11-111-111-1ii-i-iii-i-ii-i-1-1    linear of order 4
ρ1611-11111-1-1-1-11111-1-11-i-iii-ii-i-iii-1-1    linear of order 4
ρ172-2000220-2i2i00-2-20000878388500000000    complex lifted from C8○D4
ρ182-20002202i-2i00-2-20000858838700000000    complex lifted from C8○D4
ρ192-2000220-2i2i00-2-20000838785800000000    complex lifted from C8○D4
ρ202-20002202i-2i00-2-20000885878300000000    complex lifted from C8○D4
ρ2144440-21400001-2-21-2100000000001-2    orthogonal lifted from C32⋊C4
ρ2244-440-21-400001-2-2-1210000000000-12    orthogonal lifted from C2×C32⋊C4
ρ23444401-240000-211-21-20000000000-21    orthogonal lifted from C32⋊C4
ρ2444-4401-2-40000-2112-1-200000000002-1    orthogonal lifted from C2×C32⋊C4
ρ25444-401-2-40000-21-1-21200000000002-1    orthogonal lifted from C2×C32⋊C4
ρ2644-4-401-240000-21-12-120000000000-21    orthogonal lifted from C2×C32⋊C4
ρ27444-40-21-400001-221-2-10000000000-12    orthogonal lifted from C2×C32⋊C4
ρ2844-4-40-21400001-22-12-100000000001-2    orthogonal lifted from C2×C32⋊C4
ρ298-8000-4200000-240000000000000000    symplectic faithful, Schur index 2
ρ308-80002-4000004-20000000000000000    symplectic faithful, Schur index 2

Smallest permutation representation of C62.(C2×C4)
On 48 points
Generators in S48
(1 39 47 28 13 24)(2 17 14 29 48 40)(3 18 15 30 41 33)(4 34 42 31 16 19)(5 35 43 32 9 20)(6 21 10 25 44 36)(7 22 11 26 45 37)(8 38 46 27 12 23)
(1 5)(2 10 48 6 14 44)(3 7)(4 46 16 8 42 12)(9 13)(11 15)(17 25 40 21 29 36)(18 22)(19 38 31 23 34 27)(20 24)(26 30)(28 32)(33 37)(35 39)(41 45)(43 47)
(1 30)(2 31)(3 32)(4 25)(5 26)(6 27)(7 28)(8 29)(9 22)(10 23)(11 24)(12 17)(13 18)(14 19)(15 20)(16 21)(33 47)(34 48)(35 41)(36 42)(37 43)(38 44)(39 45)(40 46)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)(25 26 27 28 29 30 31 32)(33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48)

G:=sub<Sym(48)| (1,39,47,28,13,24)(2,17,14,29,48,40)(3,18,15,30,41,33)(4,34,42,31,16,19)(5,35,43,32,9,20)(6,21,10,25,44,36)(7,22,11,26,45,37)(8,38,46,27,12,23), (1,5)(2,10,48,6,14,44)(3,7)(4,46,16,8,42,12)(9,13)(11,15)(17,25,40,21,29,36)(18,22)(19,38,31,23,34,27)(20,24)(26,30)(28,32)(33,37)(35,39)(41,45)(43,47), (1,30)(2,31)(3,32)(4,25)(5,26)(6,27)(7,28)(8,29)(9,22)(10,23)(11,24)(12,17)(13,18)(14,19)(15,20)(16,21)(33,47)(34,48)(35,41)(36,42)(37,43)(38,44)(39,45)(40,46), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)>;

G:=Group( (1,39,47,28,13,24)(2,17,14,29,48,40)(3,18,15,30,41,33)(4,34,42,31,16,19)(5,35,43,32,9,20)(6,21,10,25,44,36)(7,22,11,26,45,37)(8,38,46,27,12,23), (1,5)(2,10,48,6,14,44)(3,7)(4,46,16,8,42,12)(9,13)(11,15)(17,25,40,21,29,36)(18,22)(19,38,31,23,34,27)(20,24)(26,30)(28,32)(33,37)(35,39)(41,45)(43,47), (1,30)(2,31)(3,32)(4,25)(5,26)(6,27)(7,28)(8,29)(9,22)(10,23)(11,24)(12,17)(13,18)(14,19)(15,20)(16,21)(33,47)(34,48)(35,41)(36,42)(37,43)(38,44)(39,45)(40,46), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48) );

G=PermutationGroup([(1,39,47,28,13,24),(2,17,14,29,48,40),(3,18,15,30,41,33),(4,34,42,31,16,19),(5,35,43,32,9,20),(6,21,10,25,44,36),(7,22,11,26,45,37),(8,38,46,27,12,23)], [(1,5),(2,10,48,6,14,44),(3,7),(4,46,16,8,42,12),(9,13),(11,15),(17,25,40,21,29,36),(18,22),(19,38,31,23,34,27),(20,24),(26,30),(28,32),(33,37),(35,39),(41,45),(43,47)], [(1,30),(2,31),(3,32),(4,25),(5,26),(6,27),(7,28),(8,29),(9,22),(10,23),(11,24),(12,17),(13,18),(14,19),(15,20),(16,21),(33,47),(34,48),(35,41),(36,42),(37,43),(38,44),(39,45),(40,46)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32),(33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48)])

Matrix representation of C62.(C2×C4) in GL6(𝔽73)

0460000
2700000
0007200
0017200
0000072
0000172
,
7200000
0720000
001000
000100
0000721
0000720
,
010000
100000
001000
000100
000010
000001
,
2200000
0220000
000010
000001
000100
001000

G:=sub<GL(6,GF(73))| [0,27,0,0,0,0,46,0,0,0,0,0,0,0,0,1,0,0,0,0,72,72,0,0,0,0,0,0,0,1,0,0,0,0,72,72],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,1,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[22,0,0,0,0,0,0,22,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0] >;

C62.(C2×C4) in GAP, Magma, Sage, TeX

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

G:=Group("C6^2.(C2xC4)");
// GroupNames label

G:=SmallGroup(288,935);
// by ID

G=gap.SmallGroup(288,935);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,56,219,80,9413,362,12550,1203]);
// Polycyclic

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

Export

Character table of C62.(C2×C4) in TeX

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