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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, C22:3Dic6, C23.25D6, (C2xC6):3Q8, C6.8(C2xQ8), C4:Dic3:8C2, (C2xC4).83D6, C6.39(C2xD4), C3:4(C22:Q8), Dic3:C4:2C2, (C2xDic6):6C2, (C22xC4).7S3, C2.9(C2xDic6), C6.15(C4oD4), C4.23(C3:D4), (C22xC12).6C2, (C2xC6).42C23, C2.17(C4oD12), (C2xC12).91C22, C6.D4.4C2, C22.54(C22xS3), (C22xC6).34C22, (C2xDic3).14C22, C2.5(C2xC3:D4), SmallGroup(96,131)

Series: Derived Chief Lower central Upper central

C1C2xC6 — C12.48D4
C1C3C6C2xC6C2xDic3C2xDic6 — C12.48D4
C3C2xC6 — C12.48D4
C1C22C22xC4

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, C2, C3, C4, C4, C22, C22, C22, C6, C6, C2xC4, C2xC4, Q8, C23, Dic3, C12, C12, C2xC6, C2xC6, C2xC6, C22:C4, C4:C4, C22xC4, C2xQ8, Dic6, C2xDic3, C2xC12, C2xC12, C22xC6, C22:Q8, Dic3:C4, C4:Dic3, C6.D4, C2xDic6, C22xC12, C12.48D4
Quotients: C1, C2, C22, S3, D4, Q8, C23, D6, C2xD4, C2xQ8, C4oD4, Dic6, C3:D4, C22xS3, C22:Q8, C2xDic6, C4oD12, C2xC3: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 C4oD4
ρ262-2-22002002i-2i00002-200-2002i0-2i-2i002i0    complex lifted from C4oD4
ρ272-2-2200-100-2i2i0000-11-3--31--3-3i-3-i-i3-3i3    complex lifted from C4oD12
ρ282-2-2200-100-2i2i0000-11--3-31-3--3i3-i-i-33i-3    complex lifted from C4oD12
ρ292-2-2200-1002i-2i0000-11-3--31--3-3-i3ii-33-i-3    complex lifted from C4oD12
ρ302-2-2200-1002i-2i0000-11--3-31-3--3-i-3ii3-3-i3    complex lifted from C4oD12

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

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

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

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

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:C8:23D4  C3:C8:5D4  C3:C8.29D4  C3:C8.6D4  C24:30D4  C24.82D4  C24:2D4  C24.4D4  (C3xD4).31D4  (C2xC6):8Q16  (C3xD4).32D4  C42.274D6  C42.277D6  C23:3Dic6  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  D4xDic6  D4:5Dic6  C42.105D6  C42.106D6  D4:6Dic6  D12:23D4  D12:24D4  Dic6:23D4  C42:18D6  C42.115D6  C42.118D6  C4:C4.178D6  C6.702- 1+4  C6.712- 1+4  C4:C4:21D6  C6.722- 1+4  C6.462+ 1+4  C6.492+ 1+4  (Q8xDic3):C2  C6.752- 1+4  C6.152- 1+4  S3xC22: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  Q8xC3:D4  C6.1042- 1+4  C6.1052- 1+4  C6.1072- 1+4  C6.1082- 1+4  C36.49D4  C12.1S4  D6:Dic6  D6:6Dic6  D6:7Dic6  C62:3Q8  C62:10Q8  A4:Dic6  D10:Dic6  C60.67D4  C60.68D4  (C2xC30):Q8  C60.205D4
C12.48D4 is a maximal quotient of
C12:4(C4:C4)  (C2xDic6):7C4  (C2xC42).6S3  C23:2Dic6  C24.17D6  C24.18D6  (C2xDic3):Q8  (C2xC12).54D4  (C2xC12).55D4  C12.50D8  C12.38SD16  D4.3Dic6  Q8:4Dic6  Q8:5Dic6  Q8.5Dic6  C24.73D6  C24.75D6  C36.49D4  D6:Dic6  D6:6Dic6  D6:7Dic6  C62:3Q8  C62:10Q8  D10:Dic6  C60.67D4  C60.68D4  (C2xC30):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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