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G = C4.D12order 96 = 25·3

5th non-split extension by C4 of D12 acting via D12/D6=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D62Q8, C12.11D4, C4.13D12, C4⋊C45S3, C6.8(C2×D4), C2.7(S3×Q8), D6⋊C4.3C2, C4⋊Dic36C2, (C2×C4).14D6, C33(C22⋊Q8), C6.14(C2×Q8), (C2×Dic6)⋊7C2, C2.10(C2×D12), C6.27(C4○D4), (C2×C6).38C23, (C2×C12).6C22, C2.13(D42S3), C22.52(C22×S3), (C22×S3).22C22, (C2×Dic3).13C22, (C3×C4⋊C4)⋊8C2, (S3×C2×C4).3C2, SmallGroup(96,104)

Series: Derived Chief Lower central Upper central

C1C2×C6 — C4.D12
C1C3C6C2×C6C22×S3S3×C2×C4 — C4.D12
C3C2×C6 — C4.D12
C1C22C4⋊C4

Generators and relations for C4.D12
 G = < a,b,c | a4=b12=1, c2=a2, bab-1=cac-1=a-1, cbc-1=a2b-1 >

Subgroups: 170 in 74 conjugacy classes, 35 normal (19 characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×2], C4 [×5], C22, C22 [×4], S3 [×2], C6 [×3], C2×C4, C2×C4 [×2], C2×C4 [×5], Q8 [×2], C23, Dic3 [×3], C12 [×2], C12 [×2], D6 [×2], D6 [×2], C2×C6, C22⋊C4 [×2], C4⋊C4, C4⋊C4 [×2], C22×C4, C2×Q8, Dic6 [×2], C4×S3 [×2], C2×Dic3, C2×Dic3 [×2], C2×C12, C2×C12 [×2], C22×S3, C22⋊Q8, C4⋊Dic3 [×2], D6⋊C4 [×2], C3×C4⋊C4, C2×Dic6, S3×C2×C4, C4.D12
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], Q8 [×2], C23, D6 [×3], C2×D4, C2×Q8, C4○D4, D12 [×2], C22×S3, C22⋊Q8, C2×D12, D42S3, S3×Q8, C4.D12

Character table of C4.D12

 class 12A2B2C2D2E34A4B4C4D4E4F4G4H6A6B6C12A12B12C12D12E12F
 size 11116622244661212222444444
ρ1111111111111111111111111    trivial
ρ2111111111-1-111-1-11111-1-11-1-1    linear of order 2
ρ31111-1-11-1-11-1111-1111-11-1-1-11    linear of order 2
ρ41111-1-11-1-1-1111-11111-1-11-11-1    linear of order 2
ρ51111-1-1111-1-1-1-1111111-1-11-1-1    linear of order 2
ρ61111-1-111111-1-1-1-1111111111    linear of order 2
ρ71111111-1-1-11-1-11-1111-1-11-11-1    linear of order 2
ρ81111111-1-11-1-1-1-11111-11-1-1-11    linear of order 2
ρ9222200-1-2-2-220000-1-1-111-11-11    orthogonal lifted from D6
ρ1022-2-20022-2000000-22-2-200200    orthogonal lifted from D4
ρ11222200-122-2-20000-1-1-1-111-111    orthogonal lifted from D6
ρ1222-2-2002-22000000-22-2200-200    orthogonal lifted from D4
ρ13222200-122220000-1-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ1422-2-200-1-220000001-11-1331-3-3    orthogonal lifted from D12
ρ15222200-1-2-22-20000-1-1-11-1111-1    orthogonal lifted from D6
ρ1622-2-200-12-20000001-1113-3-13-3    orthogonal lifted from D12
ρ1722-2-200-12-20000001-111-33-1-33    orthogonal lifted from D12
ρ1822-2-200-1-220000001-11-1-3-3133    orthogonal lifted from D12
ρ192-22-22-22000000002-2-2000000    symplectic lifted from Q8, Schur index 2
ρ202-22-2-222000000002-2-2000000    symplectic lifted from Q8, Schur index 2
ρ212-2-2200200002i-2i00-2-22000000    complex lifted from C4○D4
ρ222-2-220020000-2i2i00-2-22000000    complex lifted from C4○D4
ρ234-4-4400-20000000022-2000000    symplectic lifted from D42S3, Schur index 2
ρ244-44-400-200000000-222000000    symplectic lifted from S3×Q8, Schur index 2

Smallest permutation representation of C4.D12
On 48 points
Generators in S48
(1 27 39 14)(2 15 40 28)(3 29 41 16)(4 17 42 30)(5 31 43 18)(6 19 44 32)(7 33 45 20)(8 21 46 34)(9 35 47 22)(10 23 48 36)(11 25 37 24)(12 13 38 26)
(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 38 39 12)(2 11 40 37)(3 48 41 10)(4 9 42 47)(5 46 43 8)(6 7 44 45)(13 14 26 27)(15 24 28 25)(16 36 29 23)(17 22 30 35)(18 34 31 21)(19 20 32 33)

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

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

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

C4.D12 is a maximal subgroup of
D65SD16  D6⋊SD16  C3⋊C81D4  D4.D12  Q8.11D12  D6⋊Q16  D61Q16  C3⋊C8.D4  D6.2SD16  C88D12  C8.2D12  C6.(C4○D8)  D6.2Q16  C83D12  D62Q16  C2.D87S3  C6.2+ 1+4  C6.102+ 1+4  C6.52- 1+4  C4210D6  C42.92D6  C42.94D6  C42.98D6  D45D12  D46D12  C42.229D6  C42.115D6  Q8×D12  Q86D12  C42.232D6  C42.134D6  Dic619D4  C4⋊C421D6  C6.722- 1+4  D1220D4  S3×C22⋊Q8  C6.162- 1+4  D1222D4  Dic621D4  C6.512+ 1+4  C6.1182+ 1+4  C6.242- 1+4  C6.592+ 1+4  C6.612+ 1+4  C6.1222+ 1+4  C6.852- 1+4  C6.692+ 1+4  C42.148D6  D127Q8  C42.237D6  C42.150D6  C42.152D6  C42.156D6  C42.157D6  C4226D6  C42.161D6  C42.164D6  C42.165D6  C42.241D6  D129Q8  C42.177D6  C42.178D6  D182Q8  D67Dic6  C12.30D12  D62Dic6  C62.65C23  C12.31D12  C60.45D4  D309Q8  D62Dic10  D302Q8  D306Q8
C4.D12 is a maximal quotient of
C2.(C4×D12)  (C2×C4)⋊Dic6  (C2×C4).17D12  D6⋊C4⋊C4  (C22×S3)⋊Q8  (C2×C12).33D4  Dic6.3Q8  D123Q8  D124Q8  D12.3Q8  Dic63Q8  Dic64Q8  C4.(D6⋊C4)  (C2×C4).44D12  C4⋊C46Dic3  C4⋊(D6⋊C4)  (C2×C12).56D4  D182Q8  D67Dic6  C12.30D12  D62Dic6  C62.65C23  C12.31D12  C60.45D4  D309Q8  D62Dic10  D302Q8  D306Q8

Matrix representation of C4.D12 in GL4(𝔽13) generated by

1000
0100
0050
0008
,
61000
3300
0001
0010
,
3300
61000
0001
00120
G:=sub<GL(4,GF(13))| [1,0,0,0,0,1,0,0,0,0,5,0,0,0,0,8],[6,3,0,0,10,3,0,0,0,0,0,1,0,0,1,0],[3,6,0,0,3,10,0,0,0,0,0,12,0,0,1,0] >;

C4.D12 in GAP, Magma, Sage, TeX

C_4.D_{12}
% in TeX

G:=Group("C4.D12");
// GroupNames label

G:=SmallGroup(96,104);
// by ID

G=gap.SmallGroup(96,104);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-3,103,218,188,122,2309]);
// Polycyclic

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

Export

Character table of C4.D12 in TeX

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