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G = C127D4order 96 = 25·3

1st semidirect product of C12 and D4 acting via D4/C22=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C127D4, C222D12, C23.29D6, (C2×C6)⋊5D4, D6⋊C43C2, (C2×D12)⋊6C2, C33(C4⋊D4), C43(C3⋊D4), C4⋊Dic39C2, (C22×C4)⋊6S3, C6.43(C2×D4), (C2×C4).85D6, (C22×C12)⋊6C2, C2.17(C2×D12), C6.19(C4○D4), (C2×C6).48C23, C2.19(C4○D12), (C2×C12).94C22, C22.56(C22×S3), (C22×C6).40C22, (C22×S3).10C22, (C2×Dic3).16C22, (C2×C3⋊D4)⋊3C2, C2.7(C2×C3⋊D4), SmallGroup(96,137)

Series: Derived Chief Lower central Upper central

C1C2×C6 — C127D4
C1C3C6C2×C6C22×S3C2×D12 — C127D4
C3C2×C6 — C127D4
C1C22C22×C4

Generators and relations for C127D4
 G = < a,b,c | a12=b4=c2=1, bab-1=cac=a-1, cbc=b-1 >

Subgroups: 242 in 94 conjugacy classes, 37 normal (21 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×2], C4 [×3], C22, C22 [×2], C22 [×8], S3 [×2], C6 [×3], C6 [×2], C2×C4 [×2], C2×C4 [×4], D4 [×6], C23, C23 [×2], Dic3 [×2], C12 [×2], C12, D6 [×6], C2×C6, C2×C6 [×2], C2×C6 [×2], C22⋊C4 [×2], C4⋊C4, C22×C4, C2×D4 [×3], D12 [×2], C2×Dic3 [×2], C3⋊D4 [×4], C2×C12 [×2], C2×C12 [×2], C22×S3 [×2], C22×C6, C4⋊D4, C4⋊Dic3, D6⋊C4 [×2], C2×D12, C2×C3⋊D4 [×2], C22×C12, C127D4
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×4], C23, D6 [×3], C2×D4 [×2], C4○D4, D12 [×2], C3⋊D4 [×2], C22×S3, C4⋊D4, C2×D12, C4○D12, C2×C3⋊D4, C127D4

Character table of C127D4

 class 12A2B2C2D2E2F2G34A4B4C4D4E4F6A6B6C6D6E6F6G12A12B12C12D12E12F12G12H
 size 1111221212222221212222222222222222
ρ1111111111111111111111111111111    trivial
ρ21111-1-1-11111-1-11-111-1-11-1-11-111-1-11-1    linear of order 2
ρ3111111-1-111111-1-1111111111111111    linear of order 2
ρ41111-1-11-1111-1-1-1111-1-11-1-11-111-1-11-1    linear of order 2
ρ5111111-111-1-1-1-1-111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ61111-1-1111-1-111-1-111-1-11-1-1-11-1-111-11    linear of order 2
ρ71111111-11-1-1-1-11-11111111-1-1-1-1-1-1-1-1    linear of order 2
ρ81111-1-1-1-11-1-1111111-1-11-1-1-11-1-111-11    linear of order 2
ρ922222200-1222200-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ102222-2-200-122-2-200-1-111-111-11-1-111-11    orthogonal lifted from D6
ρ1122-2-20000200-2200-2-20020002002-20-2    orthogonal lifted from D4
ρ122-22-22-2002000000-22-2-2-22200000000    orthogonal lifted from D4
ρ132222-2-200-1-2-22200-1-111-1111-111-1-11-1    orthogonal lifted from D6
ρ1422-2-200002002-200-2-2002000-200-2202    orthogonal lifted from D4
ρ1522222200-1-2-2-2-200-1-1-1-1-1-1-111111111    orthogonal lifted from D6
ρ162-22-2-22002000000-2222-2-2-200000000    orthogonal lifted from D4
ρ172-22-2-2200-10000001-1-1-1111-33-33-3-333    orthogonal lifted from D12
ρ182-22-22-200-10000001-1111-1-1333-3-3-3-33    orthogonal lifted from D12
ρ192-22-2-2200-10000001-1-1-11113-33-333-3-3    orthogonal lifted from D12
ρ202-22-22-200-10000001-1111-1-1-3-3-33333-3    orthogonal lifted from D12
ρ2122-2-20000-1002-20011-3--3-1-3--3--31-3--31-1-3-1    complex lifted from C3⋊D4
ρ2222-2-20000-100-220011-3--3-1-3--3-3-1--3-3-11--31    complex lifted from C3⋊D4
ρ232-2-220000-12i-2i0000-11-3--31--3-3-i3ii-33-i-3    complex lifted from C4○D12
ρ2422-2-20000-100-220011--3-3-1--3-3--3-1-3--3-11-31    complex lifted from C3⋊D4
ρ2522-2-20000-1002-20011--3-3-1--3-3-31--3-31-1--3-1    complex lifted from C3⋊D4
ρ262-2-22000022i-2i00002-200-2002i0-2i-2i002i0    complex lifted from C4○D4
ρ272-2-220000-1-2i2i0000-11-3--31--3-3i-3-i-i3-3i3    complex lifted from C4○D12
ρ282-2-220000-12i-2i0000-11--3-31-3--3-i-3ii3-3-i3    complex lifted from C4○D12
ρ292-2-2200002-2i2i00002-200-200-2i02i2i00-2i0    complex lifted from C4○D4
ρ302-2-220000-1-2i2i0000-11--3-31-3--3i3-i-i-33i-3    complex lifted from C4○D12

Smallest permutation representation of C127D4
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 20 29 43)(2 19 30 42)(3 18 31 41)(4 17 32 40)(5 16 33 39)(6 15 34 38)(7 14 35 37)(8 13 36 48)(9 24 25 47)(10 23 26 46)(11 22 27 45)(12 21 28 44)
(2 12)(3 11)(4 10)(5 9)(6 8)(13 38)(14 37)(15 48)(16 47)(17 46)(18 45)(19 44)(20 43)(21 42)(22 41)(23 40)(24 39)(25 33)(26 32)(27 31)(28 30)(34 36)

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

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

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

C127D4 is a maximal subgroup of
C6.C4≀C2  C22.2D24  D1213D4  D1214D4  C23.43D12  C22.D24  C23.18D12  Dic614D4  (C2×C6).40D8  C4⋊C4.228D6  C4⋊C4.236D6  C3⋊C822D4  C4⋊D4⋊S3  C3⋊C824D4  C3⋊C86D4  C2430D4  C2429D4  C242D4  C243D4  (C2×C6)⋊8D8  (C3×Q8)⋊13D4  (C3×D4)⋊14D4  C42.276D6  C42.277D6  C24.38D6  C234D12  C24.41D6  C6.2- 1+4  C6.2+ 1+4  C6.112+ 1+4  C6.62- 1+4  C42.95D6  C42.97D6  C42.99D6  C42.100D6  C42.104D6  D4×D12  D1223D4  Dic623D4  Dic624D4  D45D12  D46D12  C4219D6  C42.116D6  C42.117D6  C42.119D6  C12⋊(C4○D4)  C6.322+ 1+4  S3×C4⋊D4  C6.372+ 1+4  C6.382+ 1+4  C6.472+ 1+4  C6.482+ 1+4  C4⋊C4.187D6  C4⋊C426D6  C6.172- 1+4  C6.242- 1+4  C6.562+ 1+4  C6.782- 1+4  C6.592+ 1+4  C6.612+ 1+4  C6.662+ 1+4  C6.682+ 1+4  C6.692+ 1+4  C24.83D6  D4×C3⋊D4  C6.452- 1+4  C6.1452+ 1+4  C6.1462+ 1+4  C6.1082- 1+4  C6.1482+ 1+4  C367D4  C22⋊D36  D6⋊D12  D62D12  C127D12  C626D4  C6219D4  C12⋊S4  D10⋊D12  C60⋊D4  C127D20  (C2×C10)⋊4D12  C6029D4
C127D4 is a maximal quotient of
C124(C4⋊C4)  (C2×C4)⋊6D12  (C2×C42)⋊3S3  C24.21D6  C233D12  C24.27D6  (C2×C4).44D12  (C2×C4)⋊3D12  (C2×C12).56D4  C127D8  D4.1D12  D4.2D12  Q82D12  Q8.6D12  C127Q16  C2430D4  C2429D4  C24.82D4  C242D4  C243D4  C24.4D4  Q8.8D12  Q8.9D12  Q8.10D12  C24.75D6  C24.76D6  C367D4  D6⋊D12  D62D12  C127D12  C626D4  C6219D4  D10⋊D12  C60⋊D4  C127D20  (C2×C10)⋊4D12  C6029D4

Matrix representation of C127D4 in GL4(𝔽13) generated by

101000
3700
0063
00103
,
9200
11400
0010
001212
,
0100
1000
0010
001212
G:=sub<GL(4,GF(13))| [10,3,0,0,10,7,0,0,0,0,6,10,0,0,3,3],[9,11,0,0,2,4,0,0,0,0,1,12,0,0,0,12],[0,1,0,0,1,0,0,0,0,0,1,12,0,0,0,12] >;

C127D4 in GAP, Magma, Sage, TeX

C_{12}\rtimes_7D_4
% in TeX

G:=Group("C12:7D4");
// GroupNames label

G:=SmallGroup(96,137);
// by ID

G=gap.SmallGroup(96,137);
# by ID

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

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

Export

Character table of C127D4 in TeX

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