Copied to
clipboard

G = CSU2(𝔽3)⋊C4order 192 = 26·3

1st semidirect product of CSU2(𝔽3) and C4 acting via C4/C2=C2

non-abelian, soluble

Aliases: CSU2(𝔽3)⋊1C4, C2.4(C4×S4), (C2×C4).1S4, (C2×Q8).4D6, Q8.2(C4×S3), (C4×Q8).4S3, Q8⋊Dic3.4C2, C22.10(C2×S4), C2.1(C4.S4), C2.1(Q8.D6), SL2(𝔽3).2(C2×C4), (C4×SL2(𝔽3)).1C2, (C2×CSU2(𝔽3)).2C2, (C2×SL2(𝔽3)).4C22, SmallGroup(192,947)

Series: Derived Chief Lower central Upper central

C1C2Q8SL2(𝔽3) — CSU2(𝔽3)⋊C4
C1C2Q8SL2(𝔽3)C2×SL2(𝔽3)C2×CSU2(𝔽3) — CSU2(𝔽3)⋊C4
SL2(𝔽3) — CSU2(𝔽3)⋊C4
C1C22C2×C4

Generators and relations for CSU2(𝔽3)⋊C4
 G = < a,b,c,d,e | a4=c3=e4=1, b2=d2=a2, bab-1=dbd-1=a-1, cac-1=ab, dad-1=a2b, ae=ea, cbc-1=a, be=eb, dcd-1=c-1, ce=ec, ede-1=a2d >

Subgroups: 223 in 67 conjugacy classes, 17 normal (15 characteristic)
C1, C2, C3, C4, C22, C6, C8, C2×C4, C2×C4, Q8, Q8, Dic3, C12, C2×C6, C42, C4⋊C4, C2×C8, Q16, C2×Q8, C2×Q8, SL2(𝔽3), C2×Dic3, C2×C12, C8⋊C4, Q8⋊C4, C4.Q8, C4×Q8, C4×Q8, C2×Q16, Dic3⋊C4, CSU2(𝔽3), C2×SL2(𝔽3), Q16⋊C4, Q8⋊Dic3, C4×SL2(𝔽3), C2×CSU2(𝔽3), CSU2(𝔽3)⋊C4
Quotients: C1, C2, C4, C22, S3, C2×C4, D6, C4×S3, S4, C2×S4, C4×S4, Q8.D6, C4.S4, CSU2(𝔽3)⋊C4

Character table of CSU2(𝔽3)⋊C4

 class 12A2B2C34A4B4C4D4E4F4G4H4I4J6A6B6C8A8B8C8D12A12B12C12D
 size 1111822666612121212888121212128888
ρ111111111111111111111111111    trivial
ρ211111-1-11-11-1-11-11111-1-111-1-1-1-1    linear of order 2
ρ311111-1-11-11-11-11-111111-1-1-1-1-1-1    linear of order 2
ρ411111111111-1-1-1-1111-1-1-1-11111    linear of order 2
ρ51-1-111i-i1-i-1ii-1-i1-11-1i-i-11-i-iii    linear of order 4
ρ61-1-111-ii1i-1-i-i-1i1-11-1-ii-11ii-i-i    linear of order 4
ρ71-1-111i-i1-i-1i-i1i-1-11-1-ii1-1-i-iii    linear of order 4
ρ81-1-111-ii1i-1-ii1-i-1-11-1i-i1-1ii-i-i    linear of order 4
ρ92222-1-2-22-22-20000-1-1-100001111    orthogonal lifted from D6
ρ102222-12222220000-1-1-10000-1-1-1-1    orthogonal lifted from S3
ρ112-2-22-12i-2i2-2i-22i00001-110000ii-i-i    complex lifted from C4×S3
ρ122-2-22-1-2i2i22i-2-2i00001-110000-i-iii    complex lifted from C4×S3
ρ1333330-3-3-11-111-11-1000-1-1110000    orthogonal lifted from C2×S4
ρ143333033-1-1-1-11111000-1-1-1-10000    orthogonal lifted from S4
ρ1533330-3-3-11-11-11-1100011-1-10000    orthogonal lifted from C2×S4
ρ163333033-1-1-1-1-1-1-1-100011110000    orthogonal lifted from S4
ρ173-3-3303i-3i-1i1-i-i1i-1000i-i-110000    complex lifted from C4×S4
ρ183-3-330-3i3i-1-i1i-i-1i1000i-i1-10000    complex lifted from C4×S4
ρ193-3-3303i-3i-1i1-ii-1-i1000-ii1-10000    complex lifted from C4×S4
ρ203-3-330-3i3i-1-i1ii1-i-1000-ii-110000    complex lifted from C4×S4
ρ2144-4-4-20000000000-22200000000    symplectic lifted from C4.S4, Schur index 2
ρ224-44-4-2000000000022-200000000    symplectic lifted from Q8.D6, Schur index 2
ρ2344-4-4100000000001-1-100003-3-33    symplectic lifted from C4.S4, Schur index 2
ρ2444-4-4100000000001-1-10000-333-3    symplectic lifted from C4.S4, Schur index 2
ρ254-44-410000000000-1-110000--3-3--3-3    complex lifted from Q8.D6
ρ264-44-410000000000-1-110000-3--3-3--3    complex lifted from Q8.D6

Smallest permutation representation of CSU2(𝔽3)⋊C4
On 64 points
Generators in S64
(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)(49 50 51 52)(53 54 55 56)(57 58 59 60)(61 62 63 64)
(1 11 3 9)(2 10 4 12)(5 63 7 61)(6 62 8 64)(13 18 15 20)(14 17 16 19)(21 26 23 28)(22 25 24 27)(29 34 31 36)(30 33 32 35)(37 43 39 41)(38 42 40 44)(45 51 47 49)(46 50 48 52)(53 59 55 57)(54 58 56 60)
(2 11 10)(4 9 12)(5 8 62)(6 64 7)(13 19 18)(15 17 20)(21 27 26)(23 25 28)(29 35 34)(31 33 36)(38 43 42)(40 41 44)(46 51 50)(48 49 52)(54 59 58)(56 57 60)
(1 37 3 39)(2 41 4 43)(5 29 7 31)(6 36 8 34)(9 38 11 40)(10 44 12 42)(13 49 15 51)(14 47 16 45)(17 46 19 48)(18 52 20 50)(21 57 23 59)(22 55 24 53)(25 54 27 56)(26 60 28 58)(30 63 32 61)(33 62 35 64)
(1 32 16 24)(2 29 13 21)(3 30 14 22)(4 31 15 23)(5 49 59 41)(6 50 60 42)(7 51 57 43)(8 52 58 44)(9 33 17 25)(10 34 18 26)(11 35 19 27)(12 36 20 28)(37 63 45 55)(38 64 46 56)(39 61 47 53)(40 62 48 54)

G:=sub<Sym(64)| (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)(49,50,51,52)(53,54,55,56)(57,58,59,60)(61,62,63,64), (1,11,3,9)(2,10,4,12)(5,63,7,61)(6,62,8,64)(13,18,15,20)(14,17,16,19)(21,26,23,28)(22,25,24,27)(29,34,31,36)(30,33,32,35)(37,43,39,41)(38,42,40,44)(45,51,47,49)(46,50,48,52)(53,59,55,57)(54,58,56,60), (2,11,10)(4,9,12)(5,8,62)(6,64,7)(13,19,18)(15,17,20)(21,27,26)(23,25,28)(29,35,34)(31,33,36)(38,43,42)(40,41,44)(46,51,50)(48,49,52)(54,59,58)(56,57,60), (1,37,3,39)(2,41,4,43)(5,29,7,31)(6,36,8,34)(9,38,11,40)(10,44,12,42)(13,49,15,51)(14,47,16,45)(17,46,19,48)(18,52,20,50)(21,57,23,59)(22,55,24,53)(25,54,27,56)(26,60,28,58)(30,63,32,61)(33,62,35,64), (1,32,16,24)(2,29,13,21)(3,30,14,22)(4,31,15,23)(5,49,59,41)(6,50,60,42)(7,51,57,43)(8,52,58,44)(9,33,17,25)(10,34,18,26)(11,35,19,27)(12,36,20,28)(37,63,45,55)(38,64,46,56)(39,61,47,53)(40,62,48,54)>;

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)(49,50,51,52)(53,54,55,56)(57,58,59,60)(61,62,63,64), (1,11,3,9)(2,10,4,12)(5,63,7,61)(6,62,8,64)(13,18,15,20)(14,17,16,19)(21,26,23,28)(22,25,24,27)(29,34,31,36)(30,33,32,35)(37,43,39,41)(38,42,40,44)(45,51,47,49)(46,50,48,52)(53,59,55,57)(54,58,56,60), (2,11,10)(4,9,12)(5,8,62)(6,64,7)(13,19,18)(15,17,20)(21,27,26)(23,25,28)(29,35,34)(31,33,36)(38,43,42)(40,41,44)(46,51,50)(48,49,52)(54,59,58)(56,57,60), (1,37,3,39)(2,41,4,43)(5,29,7,31)(6,36,8,34)(9,38,11,40)(10,44,12,42)(13,49,15,51)(14,47,16,45)(17,46,19,48)(18,52,20,50)(21,57,23,59)(22,55,24,53)(25,54,27,56)(26,60,28,58)(30,63,32,61)(33,62,35,64), (1,32,16,24)(2,29,13,21)(3,30,14,22)(4,31,15,23)(5,49,59,41)(6,50,60,42)(7,51,57,43)(8,52,58,44)(9,33,17,25)(10,34,18,26)(11,35,19,27)(12,36,20,28)(37,63,45,55)(38,64,46,56)(39,61,47,53)(40,62,48,54) );

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),(49,50,51,52),(53,54,55,56),(57,58,59,60),(61,62,63,64)], [(1,11,3,9),(2,10,4,12),(5,63,7,61),(6,62,8,64),(13,18,15,20),(14,17,16,19),(21,26,23,28),(22,25,24,27),(29,34,31,36),(30,33,32,35),(37,43,39,41),(38,42,40,44),(45,51,47,49),(46,50,48,52),(53,59,55,57),(54,58,56,60)], [(2,11,10),(4,9,12),(5,8,62),(6,64,7),(13,19,18),(15,17,20),(21,27,26),(23,25,28),(29,35,34),(31,33,36),(38,43,42),(40,41,44),(46,51,50),(48,49,52),(54,59,58),(56,57,60)], [(1,37,3,39),(2,41,4,43),(5,29,7,31),(6,36,8,34),(9,38,11,40),(10,44,12,42),(13,49,15,51),(14,47,16,45),(17,46,19,48),(18,52,20,50),(21,57,23,59),(22,55,24,53),(25,54,27,56),(26,60,28,58),(30,63,32,61),(33,62,35,64)], [(1,32,16,24),(2,29,13,21),(3,30,14,22),(4,31,15,23),(5,49,59,41),(6,50,60,42),(7,51,57,43),(8,52,58,44),(9,33,17,25),(10,34,18,26),(11,35,19,27),(12,36,20,28),(37,63,45,55),(38,64,46,56),(39,61,47,53),(40,62,48,54)]])

Matrix representation of CSU2(𝔽3)⋊C4 in GL7(𝔽73)

0010000
7272720000
1000000
0000010
0000001
00072000
00007200
,
0100000
1000000
7272720000
00007200
0001000
0000001
00000720
,
1000000
0010000
7272720000
0001000
00000720
00000072
0000100
,
1000000
0010000
0100000
00032462746
00046273246
00027324646
00046464641
,
27000000
02700000
00270000
0000304330
0004303030
0003043030
0004343430

G:=sub<GL(7,GF(73))| [0,72,1,0,0,0,0,0,72,0,0,0,0,0,1,72,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,1,0,0,0,0,0,0,0,1,0,0],[0,1,72,0,0,0,0,1,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,1,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,1,0],[1,0,72,0,0,0,0,0,0,72,0,0,0,0,0,1,72,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,72,0,0,0,0,0,0,0,72,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,32,46,27,46,0,0,0,46,27,32,46,0,0,0,27,32,46,46,0,0,0,46,46,46,41],[27,0,0,0,0,0,0,0,27,0,0,0,0,0,0,0,27,0,0,0,0,0,0,0,0,43,30,43,0,0,0,30,0,43,43,0,0,0,43,30,0,43,0,0,0,30,30,30,0] >;

CSU2(𝔽3)⋊C4 in GAP, Magma, Sage, TeX

{\rm CSU}_2({\mathbb F}_3)\rtimes C_4
% in TeX

G:=Group("CSU(2,3):C4");
// GroupNames label

G:=SmallGroup(192,947);
// by ID

G=gap.SmallGroup(192,947);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,2,-2,672,1373,36,451,1684,655,172,1013,404,285,124]);
// Polycyclic

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

Export

Character table of CSU2(𝔽3)⋊C4 in TeX

׿
×
𝔽