Copied to
clipboard

G = C62.C22order 144 = 24·32

5th non-split extension by C62 of C22 acting faithfully

metabelian, supersoluble, monomial

Aliases: C6.2Dic6, C62.5C22, C6.4(C4×S3), C22.7S32, (C3×C6).3Q8, C325(C4⋊C4), C3⋊Dic32C4, (C3×C6).16D4, (C2×C6).12D6, C32(Dic3⋊C4), C6.11(C3⋊D4), (C2×Dic3).2S3, (C6×Dic3).1C2, C2.2(D6⋊S3), C2.2(C322Q8), C2.5(C6.D6), (C3×C6).16(C2×C4), (C2×C3⋊Dic3).3C2, SmallGroup(144,67)

Series: Derived Chief Lower central Upper central

C1C3×C6 — C62.C22
C1C3C32C3×C6C62C6×Dic3 — C62.C22
C32C3×C6 — C62.C22
C1C22

Generators and relations for C62.C22
 G = < a,b,c,d | a6=b6=1, c2=d2=a3, ab=ba, ac=ca, dad-1=a-1, cbc-1=b-1, bd=db, dcd-1=b3c >

Subgroups: 160 in 60 conjugacy classes, 28 normal (12 characteristic)
C1, C2 [×3], C3 [×2], C3, C4 [×4], C22, C6 [×6], C6 [×3], C2×C4 [×3], C32, Dic3 [×8], C12 [×2], C2×C6 [×2], C2×C6, C4⋊C4, C3×C6 [×3], C2×Dic3 [×2], C2×Dic3 [×3], C2×C12 [×2], C3×Dic3 [×2], C3⋊Dic3 [×2], C62, Dic3⋊C4 [×2], C6×Dic3 [×2], C2×C3⋊Dic3, C62.C22
Quotients: C1, C2 [×3], C4 [×2], C22, S3 [×2], C2×C4, D4, Q8, D6 [×2], C4⋊C4, Dic6 [×2], C4×S3 [×2], C3⋊D4 [×2], S32, Dic3⋊C4 [×2], C6.D6, D6⋊S3, C322Q8, C62.C22

Character table of C62.C22

 class 12A2B2C3A3B3C4A4B4C4D4E4F6A6B6C6D6E6F6G6H6I12A12B12C12D12E12F12G12H
 size 11112246666181822222244466666666
ρ1111111111111111111111111111111    trivial
ρ211111111-11-1-1-1111111111-111-111-1-1    linear of order 2
ρ31111111-11-11-1-11111111111-1-11-1-111    linear of order 2
ρ41111111-1-1-1-111111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ51-11-1111i-i-ii-11-1-11-1-11-11-1i-i-iiii-i-i    linear of order 4
ρ61-11-1111ii-i-i1-1-1-11-1-11-11-1-i-i-i-iiiii    linear of order 4
ρ71-11-1111-i-iii1-1-1-11-1-11-11-1iiii-i-i-i-i    linear of order 4
ρ81-11-1111-iii-i-11-1-11-1-11-11-1-iii-i-i-iii    linear of order 4
ρ922222-1-1-20-20002-122-1-1-1-1-101101100    orthogonal lifted from D6
ρ1022-2-2222000000-2-2-222-2-2-2200000000    orthogonal lifted from D4
ρ112222-12-1020200-12-1-122-1-1-1-100-100-1-1    orthogonal lifted from S3
ρ122222-12-10-20-200-12-1-122-1-1-110010011    orthogonal lifted from D6
ρ1322222-1-12020002-122-1-1-1-1-10-1-10-1-100    orthogonal lifted from S3
ρ142-2-2222200000022-2-2-2-22-2-200000000    symplectic lifted from Q8, Schur index 2
ρ152-2-222-1-10000002-1-2-211-1110-3303-300    symplectic lifted from Dic6, Schur index 2
ρ162-2-22-12-1000000-1211-2-2-111-3003003-3    symplectic lifted from Dic6, Schur index 2
ρ172-2-222-1-10000002-1-2-211-11103-30-3300    symplectic lifted from Dic6, Schur index 2
ρ182-2-22-12-1000000-1211-2-2-111300-300-33    symplectic lifted from Dic6, Schur index 2
ρ1922-2-22-1-1000000-21-22-1111-10--3-30--3-300    complex lifted from C3⋊D4
ρ2022-2-22-1-1000000-21-22-1111-10-3--30-3--300    complex lifted from C3⋊D4
ρ2122-2-2-12-10000001-21-12-211-1--300-300--3-3    complex lifted from C3⋊D4
ρ2222-2-2-12-10000001-21-12-211-1-300--300-3--3    complex lifted from C3⋊D4
ρ232-22-2-12-10-2i02i001-2-11-221-11-i00-i00ii    complex lifted from C4×S3
ρ242-22-22-1-1-2i02i000-212-21-11-110-i-i0ii00    complex lifted from C4×S3
ρ252-22-22-1-12i0-2i000-212-21-11-110ii0-i-i00    complex lifted from C4×S3
ρ262-22-2-12-102i0-2i001-2-11-221-11i00i00-i-i    complex lifted from C4×S3
ρ274444-2-21000000-2-2-2-2-2-211100000000    orthogonal lifted from S32
ρ284-44-4-2-2100000022-222-2-11-100000000    orthogonal lifted from C6.D6
ρ2944-4-4-2-21000000222-2-22-1-1100000000    symplectic lifted from D6⋊S3, Schur index 2
ρ304-4-44-2-21000000-2-222221-1-100000000    symplectic lifted from C322Q8, Schur index 2

Smallest permutation representation of C62.C22
On 48 points
Generators in S48
(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)
(1 17 5 15 3 13)(2 18 6 16 4 14)(7 47 11 45 9 43)(8 48 12 46 10 44)(19 29 21 25 23 27)(20 30 22 26 24 28)(31 38 33 40 35 42)(32 39 34 41 36 37)
(1 33 4 36)(2 34 5 31)(3 35 6 32)(7 27 10 30)(8 28 11 25)(9 29 12 26)(13 40 16 37)(14 41 17 38)(15 42 18 39)(19 46 22 43)(20 47 23 44)(21 48 24 45)
(1 21 4 24)(2 20 5 23)(3 19 6 22)(7 33 10 36)(8 32 11 35)(9 31 12 34)(13 29 16 26)(14 28 17 25)(15 27 18 30)(37 47 40 44)(38 46 41 43)(39 45 42 48)

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

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

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

C62.C22 is a maximal subgroup of
C62.3D4  C62.4D4  C62.6D4  C62.7D4  C62.8C23  C62.9C23  Dic36Dic6  Dic3.Dic6  C62.17C23  C62.20C23  C62.24C23  D66Dic6  C62.29C23  C62.31C23  C62.35C23  C62.37C23  C62.38C23  C62.39C23  C62.40C23  C62.43C23  C62.44C23  S3×Dic3⋊C4  C62.47C23  C62.58C23  D63Dic6  D64Dic6  C62.70C23  C4×D6⋊S3  C62.72C23  C62.82C23  C4×C322Q8  C12⋊Dic6  C62.94C23  C62.98C23  C62.99C23  C62.56D4  C623Q8  C62.111C23  C62.113C23  C627D4  C624Q8  C18.Dic6  C62.D6  C62.81D6  C62.85D6
C62.C22 is a maximal quotient of
C12.15Dic6  C12.6Dic6  C12.8Dic6  C62.5Q8  C62.6Q8  C18.Dic6  C62.3D6  C62.81D6  C62.85D6

Matrix representation of C62.C22 in GL6(𝔽13)

100000
010000
0012000
0001200
0000121
0000120
,
1200000
0120000
000100
00121200
000010
000001
,
730000
1060000
008000
005500
000010
000001
,
010000
100000
005000
000500
000001
000010

G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,12,0,0,0,0,1,0],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,1,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[7,10,0,0,0,0,3,6,0,0,0,0,0,0,8,5,0,0,0,0,0,5,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,5,0,0,0,0,0,0,5,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

C62.C22 in GAP, Magma, Sage, TeX

C_6^2.C_2^2
% in TeX

G:=Group("C6^2.C2^2");
// GroupNames label

G:=SmallGroup(144,67);
// by ID

G=gap.SmallGroup(144,67);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-3,24,121,31,490,3461]);
// Polycyclic

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

Export

Character table of C62.C22 in TeX

׿
×
𝔽