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G = C12.48D4order 96 = 25·3

5th non-split extension by C12 of D4 acting via D4/C22=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C12.48D4, C223Dic6, C23.25D6, (C2×C6)⋊3Q8, C6.8(C2×Q8), C4⋊Dic38C2, (C2×C4).83D6, C6.39(C2×D4), C34(C22⋊Q8), Dic3⋊C42C2, (C2×Dic6)⋊6C2, (C22×C4).7S3, C2.9(C2×Dic6), C6.15(C4○D4), C4.23(C3⋊D4), (C22×C12).6C2, (C2×C6).42C23, C2.17(C4○D12), (C2×C12).91C22, C6.D4.4C2, C22.54(C22×S3), (C22×C6).34C22, (C2×Dic3).14C22, C2.5(C2×C3⋊D4), SmallGroup(96,131)

Series: Derived Chief Lower central Upper central

C1C2×C6 — C12.48D4
C1C3C6C2×C6C2×Dic3C2×Dic6 — C12.48D4
C3C2×C6 — C12.48D4
C1C22C22×C4

Generators and relations for C12.48D4
 G = < a,b,c | a12=b4=1, c2=a6, bab-1=cac-1=a-1, cbc-1=a6b-1 >

Subgroups: 146 in 74 conjugacy classes, 37 normal (21 characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×2], C4 [×5], C22, C22 [×2], C22 [×2], C6 [×3], C6 [×2], C2×C4 [×2], C2×C4 [×6], Q8 [×2], C23, Dic3 [×4], C12 [×2], C12, C2×C6, C2×C6 [×2], C2×C6 [×2], C22⋊C4 [×2], C4⋊C4 [×3], C22×C4, C2×Q8, Dic6 [×2], C2×Dic3 [×4], C2×C12 [×2], C2×C12 [×2], C22×C6, C22⋊Q8, Dic3⋊C4 [×2], C4⋊Dic3, C6.D4 [×2], C2×Dic6, C22×C12, C12.48D4
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], Q8 [×2], C23, D6 [×3], C2×D4, C2×Q8, C4○D4, Dic6 [×2], C3⋊D4 [×2], C22×S3, C22⋊Q8, C2×Dic6, C4○D12, C2×C3⋊D4, C12.48D4

Character table of C12.48D4

 class 12A2B2C2D2E34A4B4C4D4E4F4G4H6A6B6C6D6E6F6G12A12B12C12D12E12F12G12H
 size 1111222222212121212222222222222222
ρ1111111111111111111111111111111    trivial
ρ21111-1-11-1-111-111-111-1-11-1-11-111-1-11-1    linear of order 2
ρ31111111-1-1-1-11-11-11111111-1-1-1-1-1-1-1-1    linear of order 2
ρ41111-1-1111-1-1-1-11111-1-11-1-1-11-1-111-11    linear of order 2
ρ51111111-1-1-1-1-11-111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ61111-1-1111-1-111-1-111-1-11-1-1-11-1-111-11    linear of order 2
ρ711111111111-1-1-1-1111111111111111    linear of order 2
ρ81111-1-11-1-1111-1-1111-1-11-1-11-111-1-11-1    linear of order 2
ρ9222222-122220000-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ102222-2-2-1-2-2220000-1-111-111-11-1-111-11    orthogonal lifted from D6
ρ1122-2-20022-2000000-2-2002000-200-2202    orthogonal lifted from D4
ρ122222-2-2-122-2-20000-1-111-1111-111-1-11-1    orthogonal lifted from D6
ρ1322-2-2002-22000000-2-20020002002-20-2    orthogonal lifted from D4
ρ14222222-1-2-2-2-20000-1-1-1-1-1-1-111111111    orthogonal lifted from D6
ρ152-22-22-2200000000-2222-2-2-200000000    symplectic lifted from Q8, Schur index 2
ρ162-22-2-22200000000-22-2-2-22200000000    symplectic lifted from Q8, Schur index 2
ρ172-22-22-2-1000000001-1-1-1111-33-33-3-333    symplectic lifted from Dic6, Schur index 2
ρ182-22-2-22-1000000001-1111-1-1-3-3-33333-3    symplectic lifted from Dic6, Schur index 2
ρ192-22-22-2-1000000001-1-1-11113-33-333-3-3    symplectic lifted from Dic6, Schur index 2
ρ202-22-2-22-1000000001-1111-1-1333-3-3-3-33    symplectic lifted from Dic6, Schur index 2
ρ2122-2-200-1-2200000011-3--3-1-3--3-3-1--3-3-11--31    complex lifted from C3⋊D4
ρ2222-2-200-12-200000011--3-3-1--3-3-31--3-31-1--3-1    complex lifted from C3⋊D4
ρ2322-2-200-1-2200000011--3-3-1--3-3--3-1-3--3-11-31    complex lifted from C3⋊D4
ρ2422-2-200-12-200000011-3--3-1-3--3--31-3--31-1-3-1    complex lifted from C3⋊D4
ρ252-2-2200200-2i2i00002-200-200-2i02i2i00-2i0    complex lifted from C4○D4
ρ262-2-22002002i-2i00002-200-2002i0-2i-2i002i0    complex lifted from C4○D4
ρ272-2-2200-100-2i2i0000-11-3--31--3-3i-3-i-i3-3i3    complex lifted from C4○D12
ρ282-2-2200-100-2i2i0000-11--3-31-3--3i3-i-i-33i-3    complex lifted from C4○D12
ρ292-2-2200-1002i-2i0000-11-3--31--3-3-i3ii-33-i-3    complex lifted from C4○D12
ρ302-2-2200-1002i-2i0000-11--3-31-3--3-i-3ii3-3-i3    complex lifted from C4○D12

Smallest permutation representation of C12.48D4
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 44 26 15)(2 43 27 14)(3 42 28 13)(4 41 29 24)(5 40 30 23)(6 39 31 22)(7 38 32 21)(8 37 33 20)(9 48 34 19)(10 47 35 18)(11 46 36 17)(12 45 25 16)
(1 21 7 15)(2 20 8 14)(3 19 9 13)(4 18 10 24)(5 17 11 23)(6 16 12 22)(25 39 31 45)(26 38 32 44)(27 37 33 43)(28 48 34 42)(29 47 35 41)(30 46 36 40)

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

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

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

C12.48D4 is a maximal subgroup of
C4⋊Dic3⋊C4  C23.35D12  C23.39D12  C23.40D12  C23.15D12  D12.31D4  D12.32D4  Dic6.32D4  C4⋊C4.230D6  C4⋊C4.231D6  C4⋊C4.233D6  C3⋊C823D4  C3⋊C85D4  C3⋊C8.29D4  C3⋊C8.6D4  C2430D4  C24.82D4  C242D4  C24.4D4  (C3×D4).31D4  (C2×C6)⋊8Q16  (C3×D4).32D4  C42.274D6  C42.277D6  C233Dic6  C24.41D6  C24.42D6  C6.72+ 1+4  C6.102+ 1+4  C6.52- 1+4  C6.62- 1+4  C42.89D6  C42.94D6  C42.98D6  C42.99D6  D4×Dic6  D45Dic6  C42.105D6  C42.106D6  D46Dic6  D1223D4  D1224D4  Dic623D4  C4218D6  C42.115D6  C42.118D6  C4⋊C4.178D6  C6.702- 1+4  C6.712- 1+4  C4⋊C421D6  C6.722- 1+4  C6.462+ 1+4  C6.492+ 1+4  (Q8×Dic3)⋊C2  C6.752- 1+4  C6.152- 1+4  S3×C22⋊Q8  C6.162- 1+4  C6.512+ 1+4  C6.252- 1+4  C6.812- 1+4  C6.632+ 1+4  C6.652+ 1+4  C6.692+ 1+4  C24.83D6  C24.53D6  Q8×C3⋊D4  C6.1042- 1+4  C6.1052- 1+4  C6.1072- 1+4  C6.1082- 1+4  C36.49D4  C12.1S4  D6⋊Dic6  D66Dic6  D67Dic6  C623Q8  C6210Q8  A4⋊Dic6  D10⋊Dic6  C60.67D4  C60.68D4  (C2×C30)⋊Q8  C60.205D4
C12.48D4 is a maximal quotient of
C124(C4⋊C4)  (C2×Dic6)⋊7C4  (C2×C42).6S3  C232Dic6  C24.17D6  C24.18D6  (C2×Dic3)⋊Q8  (C2×C12).54D4  (C2×C12).55D4  C12.50D8  C12.38SD16  D4.3Dic6  Q84Dic6  Q85Dic6  Q8.5Dic6  C24.73D6  C24.75D6  C36.49D4  D6⋊Dic6  D66Dic6  D67Dic6  C623Q8  C6210Q8  D10⋊Dic6  C60.67D4  C60.68D4  (C2×C30)⋊Q8  C60.205D4

Matrix representation of C12.48D4 in GL4(𝔽13) generated by

6000
01100
0055
0008
,
0100
1000
00107
0063
,
0100
12000
00107
0063
G:=sub<GL(4,GF(13))| [6,0,0,0,0,11,0,0,0,0,5,0,0,0,5,8],[0,1,0,0,1,0,0,0,0,0,10,6,0,0,7,3],[0,12,0,0,1,0,0,0,0,0,10,6,0,0,7,3] >;

C12.48D4 in GAP, Magma, Sage, TeX

C_{12}._{48}D_4
% in TeX

G:=Group("C12.48D4");
// GroupNames label

G:=SmallGroup(96,131);
// by ID

G=gap.SmallGroup(96,131);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-3,96,217,103,218,2309]);
// Polycyclic

G:=Group<a,b,c|a^12=b^4=1,c^2=a^6,b*a*b^-1=c*a*c^-1=a^-1,c*b*c^-1=a^6*b^-1>;
// generators/relations

Export

Character table of C12.48D4 in TeX

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