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G = C2×C322SD16order 288 = 25·32

Direct product of C2 and C322SD16

direct product, non-abelian, soluble, monomial

Aliases: C2×C322SD16, C62.14D4, (C3×C6)⋊2SD16, C323(C2×SD16), C22.15S3≀C2, C3⋊Dic3.32D4, C322C86C22, C322Q88C22, D6⋊S3.7C22, C3⋊Dic3.10C23, C2.19(C2×S3≀C2), (C3×C6).19(C2×D4), (C2×C322C8)⋊6C2, (C2×C322Q8)⋊9C2, (C2×D6⋊S3).8C2, (C2×C3⋊Dic3).97C22, SmallGroup(288,886)

Series: Derived Chief Lower central Upper central

C1C32C3⋊Dic3 — C2×C322SD16
C1C32C3×C6C3⋊Dic3D6⋊S3C322SD16 — C2×C322SD16
C32C3×C6C3⋊Dic3 — C2×C322SD16
C1C22

Generators and relations for C2×C322SD16
 G = < a,b,c,d,e | a2=b3=c3=d8=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=c-1, ebe=b-1, dcd-1=b, ce=ec, ede=d3 >

Subgroups: 560 in 114 conjugacy classes, 27 normal (13 characteristic)
C1, C2, C2 [×2], C2 [×2], C3 [×2], C4 [×4], C22, C22 [×4], S3 [×2], C6 [×8], C8 [×2], C2×C4 [×2], D4 [×3], Q8 [×3], C23, C32, Dic3 [×6], C12 [×2], D6 [×4], C2×C6 [×6], C2×C8, SD16 [×4], C2×D4, C2×Q8, C3×S3 [×2], C3×C6, C3×C6 [×2], Dic6 [×4], C2×Dic3 [×3], C3⋊D4 [×4], C2×C12, C22×S3, C22×C6, C2×SD16, C3×Dic3 [×2], C3⋊Dic3 [×2], S3×C6 [×4], C62, C2×Dic6, C2×C3⋊D4, C322C8 [×2], D6⋊S3 [×2], D6⋊S3, C322Q8 [×2], C322Q8, C6×Dic3, C2×C3⋊Dic3, S3×C2×C6, C322SD16 [×4], C2×C322C8, C2×D6⋊S3, C2×C322Q8, C2×C322SD16
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, SD16 [×2], C2×D4, C2×SD16, S3≀C2, C322SD16 [×2], C2×S3≀C2, C2×C322SD16

Character table of C2×C322SD16

 class 12A2B2C2D2E3A3B4A4B4C4D6A6B6C6D6E6F6G6H6I6J8A8B8C8D12A12B12C12D
 size 111112124412121818444444121212121818181812121212
ρ1111111111111111111111111111111    trivial
ρ211-1-11-111-11-11-1-1-111-1-1-111-11-1111-1-1    linear of order 2
ρ31111-1-111-1-111111111-1-1-1-11111-1-1-1-1    linear of order 2
ρ411-1-1-11111-1-11-1-1-111-111-1-1-11-11-1-111    linear of order 2
ρ511-1-1-1111-11-11-1-1-111-111-1-11-11-111-1-1    linear of order 2
ρ61111-1-1111111111111-1-1-1-1-1-1-1-11111    linear of order 2
ρ711-1-11-1111-1-11-1-1-111-1-1-1111-11-1-1-111    linear of order 2
ρ811111111-1-1111111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ922-2-20022002-2-2-2-222-2000000000000    orthogonal lifted from D4
ρ102222002200-2-2222222000000000000    orthogonal lifted from D4
ρ112-2-22002200002-2-2-2-220000--2-2-2--20000    complex lifted from SD16
ρ122-22-200220000-222-2-2-20000--2--2-2-20000    complex lifted from SD16
ρ132-2-22002200002-2-2-2-220000-2--2--2-20000    complex lifted from SD16
ρ142-22-200220000-222-2-2-20000-2-2--2--20000    complex lifted from SD16
ρ154444221-2000011-21-2-2-1-1-1-100000000    orthogonal lifted from S3≀C2
ρ16444400-212200-2-21-21100000000-1-1-1-1    orthogonal lifted from S3≀C2
ρ17444400-21-2-200-2-21-211000000001111    orthogonal lifted from S3≀C2
ρ184444-2-21-2000011-21-2-2111100000000    orthogonal lifted from S3≀C2
ρ1944-4-4-221-20000-1-121-22-1-11100000000    orthogonal lifted from C2×S3≀C2
ρ2044-4-400-21-220022-1-21-100000000-1-111    orthogonal lifted from C2×S3≀C2
ρ2144-4-400-212-20022-1-21-10000000011-1-1    orthogonal lifted from C2×S3≀C2
ρ2244-4-42-21-20000-1-121-2211-1-100000000    orthogonal lifted from C2×S3≀C2
ρ234-44-400-2100002-212-1-1000000003-3-33    symplectic lifted from C322SD16, Schur index 2
ρ244-4-4400-210000-22-12-1100000000-33-33    symplectic lifted from C322SD16, Schur index 2
ρ254-44-400-2100002-212-1-100000000-333-3    symplectic lifted from C322SD16, Schur index 2
ρ264-4-4400-210000-22-12-11000000003-33-3    symplectic lifted from C322SD16, Schur index 2
ρ274-44-4001-20000-11-2-122--3-3-3--300000000    complex lifted from C322SD16
ρ284-4-44001-200001-12-12-2--3-3--3-300000000    complex lifted from C322SD16
ρ294-44-4001-20000-11-2-122-3--3--3-300000000    complex lifted from C322SD16
ρ304-4-44001-200001-12-12-2-3--3-3--300000000    complex lifted from C322SD16

Smallest permutation representation of C2×C322SD16
On 48 points
Generators in S48
(1 18)(2 19)(3 20)(4 21)(5 22)(6 23)(7 24)(8 17)(9 31)(10 32)(11 25)(12 26)(13 27)(14 28)(15 29)(16 30)(33 43)(34 44)(35 45)(36 46)(37 47)(38 48)(39 41)(40 42)
(1 45 31)(3 25 47)(5 41 27)(7 29 43)(9 18 35)(11 37 20)(13 22 39)(15 33 24)
(2 46 32)(4 26 48)(6 42 28)(8 30 44)(10 19 36)(12 38 21)(14 23 40)(16 34 17)
(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 22)(2 17)(3 20)(4 23)(5 18)(6 21)(7 24)(8 19)(9 41)(10 44)(11 47)(12 42)(13 45)(14 48)(15 43)(16 46)(25 37)(26 40)(27 35)(28 38)(29 33)(30 36)(31 39)(32 34)

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

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

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

Matrix representation of C2×C322SD16 in GL6(𝔽73)

7200000
0720000
0072000
0007200
0000720
0000072
,
100000
010000
0007200
0017200
0054010
0054001
,
100000
010000
001000
000100
0023387272
000010
,
60500000
55130000
0023387172
0000721
00495455
00618455
,
7200000
1710000
000100
001000
004603060
007041343

G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,5,5,0,0,72,72,40,40,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,23,0,0,0,0,1,38,0,0,0,0,0,72,1,0,0,0,0,72,0],[60,55,0,0,0,0,50,13,0,0,0,0,0,0,23,0,49,6,0,0,38,0,5,18,0,0,71,72,45,45,0,0,72,1,5,5],[72,17,0,0,0,0,0,1,0,0,0,0,0,0,0,1,4,70,0,0,1,0,60,4,0,0,0,0,30,13,0,0,0,0,60,43] >;

C2×C322SD16 in GAP, Magma, Sage, TeX

C_2\times C_3^2\rtimes_2{\rm SD}_{16}
% in TeX

G:=Group("C2xC3^2:2SD16");
// GroupNames label

G:=SmallGroup(288,886);
// by ID

G=gap.SmallGroup(288,886);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,141,120,675,346,80,2693,2028,362,797,1203]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^3=c^3=d^8=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,d*b*d^-1=c^-1,e*b*e=b^-1,d*c*d^-1=b,c*e=e*c,e*d*e=d^3>;
// generators/relations

Export

Character table of C2×C322SD16 in TeX

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