Copied to
clipboard

G = C427S3order 96 = 25·3

6th semidirect product of C42 and S3 acting via S3/C3=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C427S3, C4.5D12, C12.28D4, (C4×C12)⋊5C2, D6⋊C41C2, C6.4(C2×D4), (C2×C4).77D6, C2.6(C2×D12), (C2×Dic6)⋊1C2, (C2×D12).2C2, C6.5(C4○D4), C31(C4.4D4), C2.7(C4○D12), (C2×C6).16C23, (C2×C12).74C22, (C22×S3).2C22, C22.37(C22×S3), (C2×Dic3).3C22, SmallGroup(96,82)

Series: Derived Chief Lower central Upper central

C1C2×C6 — C427S3
C1C3C6C2×C6C22×S3D6⋊C4 — C427S3
C3C2×C6 — C427S3
C1C22C42

Generators and relations for C427S3
 G = < a,b,c,d | a4=b4=c3=d2=1, ab=ba, ac=ca, dad=ab2, bc=cb, dbd=a2b, dcd=c-1 >

Subgroups: 202 in 76 conjugacy classes, 33 normal (13 characteristic)
C1, C2, C2 [×2], C2 [×2], C3, C4 [×2], C4 [×4], C22, C22 [×6], S3 [×2], C6, C6 [×2], C2×C4, C2×C4 [×2], C2×C4 [×2], D4 [×2], Q8 [×2], C23 [×2], Dic3 [×2], C12 [×2], C12 [×2], D6 [×6], C2×C6, C42, C22⋊C4 [×4], C2×D4, C2×Q8, Dic6 [×2], D12 [×2], C2×Dic3 [×2], C2×C12, C2×C12 [×2], C22×S3 [×2], C4.4D4, D6⋊C4 [×4], C4×C12, C2×Dic6, C2×D12, C427S3
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D6 [×3], C2×D4, C4○D4 [×2], D12 [×2], C22×S3, C4.4D4, C2×D12, C4○D12 [×2], C427S3

Character table of C427S3

 class 12A2B2C2D2E34A4B4C4D4E4F4G4H6A6B6C12A12B12C12D12E12F12G12H12I12J12K12L
 size 1111121222222221212222222222222222
ρ1111111111111111111111111111111    trivial
ρ21111-1-111-1-1-11-111111-1-1-111-111-1-1-1-1    linear of order 2
ρ311111-11-1-111-1-1-1111111-1-1-1-1-1-1-111-1    linear of order 2
ρ41111-111-11-1-1-11-11111-1-11-1-11-1-11-1-11    linear of order 2
ρ51111-1-11111111-1-1111111111111111    linear of order 2
ρ611111111-1-1-11-1-1-1111-1-1-111-111-1-1-1-1    linear of order 2
ρ71111-111-1-111-1-11-111111-1-1-1-1-1-1-111-1    linear of order 2
ρ811111-11-11-1-1-111-1111-1-11-1-11-1-11-1-11    linear of order 2
ρ9222200-12-2-2-22-200-1-1-1111-1-11-1-11111    orthogonal lifted from D6
ρ102-2-220022000-2000-22-2000-2-20220000    orthogonal lifted from D4
ρ11222200-1-2-222-2-200-1-1-1-1-11111111-1-11    orthogonal lifted from D6
ρ12222200-1-22-2-2-2200-1-1-111-111-111-111-1    orthogonal lifted from D6
ρ13222200-122222200-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ142-2-22002-20002000-22-2000220-2-20000    orthogonal lifted from D4
ρ152-2-2200-12000-20001-11-33-3113-1-1-33-33    orthogonal lifted from D12
ρ162-2-2200-12000-20001-113-3311-3-1-13-33-3    orthogonal lifted from D12
ρ172-2-2200-1-200020001-11-333-1-1-31133-3-3    orthogonal lifted from D12
ρ182-2-2200-1-200020001-113-3-3-1-1311-3-333    orthogonal lifted from D12
ρ1922-2-200200-2i2i00002-2-22i2i0000000-2i-2i0    complex lifted from C4○D4
ρ202-22-20020-2i0002i00-2-2200-2i00-2i002i002i    complex lifted from C4○D4
ρ2122-2-2002002i-2i00002-2-2-2i-2i00000002i2i0    complex lifted from C4○D4
ρ222-22-200202i000-2i00-2-22002i002i00-2i00-2i    complex lifted from C4○D4
ρ232-22-200-102i000-2i0011-1-3--3-i3-3-i-33i-3--3i    complex lifted from C4○D12
ρ2422-2-200-100-2i2i0000-111-i-i--3-33-3-33-3ii--3    complex lifted from C4○D12
ρ252-22-200-102i000-2i0011-1--3-3-i-33-i3-3i--3-3i    complex lifted from C4○D12
ρ262-22-200-10-2i0002i0011-1-3--3i-33i3-3-i-3--3-i    complex lifted from C4○D12
ρ272-22-200-10-2i0002i0011-1--3-3i3-3i-33-i--3-3-i    complex lifted from C4○D12
ρ2822-2-200-100-2i2i0000-111-i-i-33-3--33-3--3ii-3    complex lifted from C4○D12
ρ2922-2-200-1002i-2i0000-111ii--33-3-33-3-3-i-i--3    complex lifted from C4○D12
ρ3022-2-200-1002i-2i0000-111ii-3-33--3-33--3-i-i-3    complex lifted from C4○D12

Smallest permutation representation of C427S3
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 8 19 41)(2 5 20 42)(3 6 17 43)(4 7 18 44)(9 30 47 15)(10 31 48 16)(11 32 45 13)(12 29 46 14)(21 33 39 26)(22 34 40 27)(23 35 37 28)(24 36 38 25)
(1 23 9)(2 24 10)(3 21 11)(4 22 12)(5 36 31)(6 33 32)(7 34 29)(8 35 30)(13 43 26)(14 44 27)(15 41 28)(16 42 25)(17 39 45)(18 40 46)(19 37 47)(20 38 48)
(2 20)(4 18)(5 44)(6 8)(7 42)(9 23)(10 38)(11 21)(12 40)(13 28)(14 36)(15 26)(16 34)(22 46)(24 48)(25 29)(27 31)(30 33)(32 35)(37 47)(39 45)(41 43)

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

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

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

C427S3 is a maximal subgroup of
C42.D6  C8.8D12  C42.264D6  C8⋊D12  C42.20D6  C8.D12  C425D6  D4.10D12  D12.19D4  C42.36D6  D4.1D12  Q8.6D12  C42.214D6  C42.216D6  C42.74D6  C42.80D6  C42.82D6  C42.276D6  C42.277D6  C4211D6  C42.92D6  C4212D6  C42.97D6  C42.99D6  D1223D4  Dic623D4  D45D12  C42.114D6  C4219D6  C42.122D6  Q86D12  C42.133D6  C42.135D6  C42.136D6  C42.233D6  S3×C4.4D4  C4224D6  C42.145D6  C42.237D6  C42.157D6  C42.158D6  C4225D6  C42.164D6  C4228D6  C42.171D6  C42.178D6  C427D9  Dic3.D12  C12.27D12  C12.28D12  C1226C2  Dic5.8D12  C60.69D4  C60.70D4  C427D15
C427S3 is a maximal quotient of
(C2×C4).17D12  C6.C22≀C2  (C22×S3)⋊Q8  (C2×C4).21D12  C12.14Q16  C4.5D24  C42.264D6  C42.14D6  C42.19D6  C42.20D6  (C2×Dic6)⋊7C4  C4211Dic3  (C2×C4)⋊6D12  (C2×C42)⋊3S3  C427D9  Dic3.D12  C12.27D12  C12.28D12  C1226C2  Dic5.8D12  C60.69D4  C60.70D4  C427D15

Matrix representation of C427S3 in GL4(𝔽13) generated by

3600
71000
0080
0008
,
3600
71000
00119
0042
,
121200
1000
001212
0010
,
12000
1100
0010
001212
G:=sub<GL(4,GF(13))| [3,7,0,0,6,10,0,0,0,0,8,0,0,0,0,8],[3,7,0,0,6,10,0,0,0,0,11,4,0,0,9,2],[12,1,0,0,12,0,0,0,0,0,12,1,0,0,12,0],[12,1,0,0,0,1,0,0,0,0,1,12,0,0,0,12] >;

C427S3 in GAP, Magma, Sage, TeX

C_4^2\rtimes_7S_3
% in TeX

G:=Group("C4^2:7S3");
// GroupNames label

G:=SmallGroup(96,82);
// by ID

G=gap.SmallGroup(96,82);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-3,217,55,218,86,2309]);
// Polycyclic

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

Export

Character table of C427S3 in TeX

׿
×
𝔽