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G = C12.53D4order 96 = 25·3

10th non-split extension by C12 of D4 acting via D4/C22=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C12.53D4, M4(2).1S3, C22.1Dic6, (C2×C6).Q8, C3⋊C8.1C4, C6.8(C4⋊C4), C4.13(C4×S3), C12.5(C2×C4), (C2×C4).37D6, C32(C8.C4), C4.28(C3⋊D4), C4.Dic3.2C2, C2.5(Dic3⋊C4), (C2×C12).12C22, (C3×M4(2)).1C2, (C2×C3⋊C8).4C2, SmallGroup(96,29)

Series: Derived Chief Lower central Upper central

C1C12 — C12.53D4
C1C3C6C12C2×C12C2×C3⋊C8 — C12.53D4
C3C6C12 — C12.53D4
C1C4C2×C4M4(2)

Generators and relations for C12.53D4
 G = < a,b,c | a12=1, b4=a6, c2=a9, bab-1=cac-1=a5, cbc-1=a6b3 >

2C2
2C6
2C8
3C8
3C8
6C8
3M4(2)
3C2×C8
2C24
2C3⋊C8
3C8.C4

Character table of C12.53D4

 class 12A2B34A4B4C6A6B8A8B8C8D8E8F8G8H12A12B12C24A24B24C24D
 size 11221122444666612122244444
ρ1111111111111111111111111    trivial
ρ2111111111-1-1-1-1-1-111111-1-1-1-1    linear of order 2
ρ311111111111-1-1-1-1-1-11111111    linear of order 2
ρ4111111111-1-11111-1-1111-1-1-1-1    linear of order 2
ρ511-11-1-111-1-ii-111-1i-i-1-11ii-i-i    linear of order 4
ρ611-11-1-111-1i-i1-1-11i-i-1-11-i-iii    linear of order 4
ρ711-11-1-111-1i-i-111-1-ii-1-11-i-iii    linear of order 4
ρ811-11-1-111-1-ii1-1-11-ii-1-11ii-i-i    linear of order 4
ρ9222-1222-1-122000000-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ10222-1222-1-1-2-2000000-1-1-11111    orthogonal lifted from D6
ρ1122-2222-22-20000000022-20000    orthogonal lifted from D4
ρ122222-2-2-22200000000-2-2-20000    symplectic lifted from Q8, Schur index 2
ρ13222-1-2-2-2-1-100000000111-33-33    symplectic lifted from Dic6, Schur index 2
ρ14222-1-2-2-2-1-1000000001113-33-3    symplectic lifted from Dic6, Schur index 2
ρ1522-2-1-2-22-112i-2i00000011-1ii-i-i    complex lifted from C4×S3
ρ1622-2-1-2-22-11-2i2i00000011-1-i-iii    complex lifted from C4×S3
ρ1722-2-122-2-1100000000-1-11--3-3-3--3    complex lifted from C3⋊D4
ρ1822-2-122-2-1100000000-1-11-3--3--3-3    complex lifted from C3⋊D4
ρ192-2022i-2i0-2000--22-2-2002i-2i00000    complex lifted from C8.C4
ρ202-202-2i2i0-2000--2-22-200-2i2i00000    complex lifted from C8.C4
ρ212-202-2i2i0-2000-22-2--200-2i2i00000    complex lifted from C8.C4
ρ222-2022i-2i0-2000-2-22--2002i-2i00000    complex lifted from C8.C4
ρ234-40-24i-4i02000000000-2i2i00000    complex faithful
ρ244-40-2-4i4i020000000002i-2i00000    complex faithful

Smallest permutation representation of C12.53D4
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 25 4 28 7 31 10 34)(2 30 5 33 8 36 11 27)(3 35 6 26 9 29 12 32)(13 40 22 37 19 46 16 43)(14 45 23 42 20 39 17 48)(15 38 24 47 21 44 18 41)
(1 40 10 37 7 46 4 43)(2 45 11 42 8 39 5 48)(3 38 12 47 9 44 6 41)(13 34 22 31 19 28 16 25)(14 27 23 36 20 33 17 30)(15 32 24 29 21 26 18 35)

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

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

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

C12.53D4 is a maximal subgroup of
D12.2D4  D12.3D4  D12.6D4  D12.7D4  M4(2).22D6  C42.196D6  S3×C8.C4  M4(2).25D6  C23.8Dic6  C24.100D4  C24.54D4  M4(2).D6  M4(2).13D6  M4(2).15D6  M4(2).16D6  C36.53D4  C12.82D12  C62.5Q8  C62.8Q8  C60.D4  C12.59D20  C60.210D4  D10.Dic6  D10.2Dic6
C12.53D4 is a maximal quotient of
C12.53D8  C12.39SD16  C12.4C42  C36.53D4  C12.82D12  C62.5Q8  C62.8Q8  C60.D4  C12.59D20  C60.210D4  D10.Dic6  D10.2Dic6

Matrix representation of C12.53D4 in GL4(𝔽5) generated by

3300
1400
0033
0014
,
0200
1000
0031
0042
,
0014
0044
4100
1100
G:=sub<GL(4,GF(5))| [3,1,0,0,3,4,0,0,0,0,3,1,0,0,3,4],[0,1,0,0,2,0,0,0,0,0,3,4,0,0,1,2],[0,0,4,1,0,0,1,1,1,4,0,0,4,4,0,0] >;

C12.53D4 in GAP, Magma, Sage, TeX

C_{12}._{53}D_4
% in TeX

G:=Group("C12.53D4");
// GroupNames label

G:=SmallGroup(96,29);
// by ID

G=gap.SmallGroup(96,29);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-3,48,121,31,86,297,69,2309]);
// Polycyclic

G:=Group<a,b,c|a^12=1,b^4=a^6,c^2=a^9,b*a*b^-1=c*a*c^-1=a^5,c*b*c^-1=a^6*b^3>;
// generators/relations

Export

Subgroup lattice of C12.53D4 in TeX
Character table of C12.53D4 in TeX

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