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## G = C4.3PSU3(𝔽2)  order 288 = 25·32

### 3rd non-split extension by C4 of PSU3(𝔽2) acting via PSU3(𝔽2)/C32⋊C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C2×C3⋊S3 — C4.3PSU3(𝔽2)
 Chief series C1 — C32 — C3⋊S3 — C2×C3⋊S3 — C2×C32⋊C4 — C2.PSU3(𝔽2) — C4.3PSU3(𝔽2)
 Lower central C32 — C2×C3⋊S3 — C4.3PSU3(𝔽2)
 Upper central C1 — C2 — C4

Generators and relations for C4.3PSU3(𝔽2)
G = < a,b,c,d,e | a4=b3=c3=d4=1, e2=a2d2, ab=ba, ac=ca, ad=da, eae-1=a-1, ece-1=bc=cb, dbd-1=c-1, ebe-1=b-1c, dcd-1=b, ede-1=a2d-1 >

Subgroups: 388 in 64 conjugacy classes, 25 normal (11 characteristic)
C1, C2, C2, C3, C4, C4, C22, S3, C6, C2×C4, C32, Dic3, C12, D6, C42, C4⋊C4, C3⋊S3, C3×C6, C4×S3, C42.C2, C3⋊Dic3, C3×C12, C32⋊C4, C2×C3⋊S3, C4×C3⋊S3, C2×C32⋊C4, C2.PSU3(𝔽2), C4×C32⋊C4, C4⋊(C32⋊C4), C4.3PSU3(𝔽2)
Quotients: C1, C2, C22, Q8, C23, C2×Q8, C4○D4, C42.C2, PSU3(𝔽2), C2×PSU3(𝔽2), C4.3PSU3(𝔽2)

Character table of C4.3PSU3(𝔽2)

 class 1 2A 2B 2C 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 6 12A 12B size 1 1 9 9 8 2 18 18 18 18 18 36 36 36 36 8 8 8 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 1 1 -1 1 1 1 linear of order 2 ρ3 1 1 1 1 1 -1 -1 1 1 -1 -1 1 -1 1 -1 1 -1 -1 linear of order 2 ρ4 1 1 1 1 1 -1 -1 1 1 -1 -1 -1 1 -1 1 1 -1 -1 linear of order 2 ρ5 1 1 1 1 1 1 1 -1 -1 -1 -1 1 -1 -1 1 1 1 1 linear of order 2 ρ6 1 1 1 1 1 -1 -1 -1 -1 1 1 -1 -1 1 1 1 -1 -1 linear of order 2 ρ7 1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 -1 -1 1 -1 -1 linear of order 2 ρ8 1 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 1 1 1 linear of order 2 ρ9 2 2 -2 -2 2 -2 2 0 0 0 0 0 0 0 0 2 -2 -2 symplectic lifted from Q8, Schur index 2 ρ10 2 2 -2 -2 2 2 -2 0 0 0 0 0 0 0 0 2 2 2 symplectic lifted from Q8, Schur index 2 ρ11 2 -2 -2 2 2 0 0 0 0 2i -2i 0 0 0 0 -2 0 0 complex lifted from C4○D4 ρ12 2 -2 2 -2 2 0 0 2i -2i 0 0 0 0 0 0 -2 0 0 complex lifted from C4○D4 ρ13 2 -2 -2 2 2 0 0 0 0 -2i 2i 0 0 0 0 -2 0 0 complex lifted from C4○D4 ρ14 2 -2 2 -2 2 0 0 -2i 2i 0 0 0 0 0 0 -2 0 0 complex lifted from C4○D4 ρ15 8 8 0 0 -1 -8 0 0 0 0 0 0 0 0 0 -1 1 1 orthogonal lifted from C2×PSU3(𝔽2) ρ16 8 8 0 0 -1 8 0 0 0 0 0 0 0 0 0 -1 -1 -1 orthogonal lifted from PSU3(𝔽2) ρ17 8 -8 0 0 -1 0 0 0 0 0 0 0 0 0 0 1 -3i 3i complex faithful ρ18 8 -8 0 0 -1 0 0 0 0 0 0 0 0 0 0 1 3i -3i complex faithful

Smallest permutation representation of C4.3PSU3(𝔽2)
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)
(5 45 10)(6 46 11)(7 47 12)(8 48 9)(21 25 30)(22 26 31)(23 27 32)(24 28 29)(33 44 39)(34 41 40)(35 42 37)(36 43 38)
(1 17 16)(2 18 13)(3 19 14)(4 20 15)(5 10 45)(6 11 46)(7 12 47)(8 9 48)(33 44 39)(34 41 40)(35 42 37)(36 43 38)
(1 22)(2 23)(3 24)(4 21)(5 37 10 42)(6 38 11 43)(7 39 12 44)(8 40 9 41)(13 27 18 32)(14 28 19 29)(15 25 20 30)(16 26 17 31)(33 47)(34 48)(35 45)(36 46)
(1 36 3 34)(2 35 4 33)(5 32 12 25)(6 31 9 28)(7 30 10 27)(8 29 11 26)(13 37 20 44)(14 40 17 43)(15 39 18 42)(16 38 19 41)(21 45 23 47)(22 48 24 46)

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

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

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

Matrix representation of C4.3PSU3(𝔽2) in GL8(𝔽13)

 5 0 0 0 0 0 0 0 0 5 0 0 0 0 0 0 0 0 5 0 0 0 0 0 0 0 0 5 0 0 0 0 0 0 0 0 8 0 0 0 0 0 0 0 0 8 0 0 1 1 12 12 0 0 8 0 1 1 12 12 0 0 0 8
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 12 1 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 1 12 0 0 0 0 12 0 0 12 0 1 12 12 0 1 1 0 12 12
,
 12 1 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 1 12 0 0 0 12 1 1 1 0 12 12 1 0 0 0 0 12 1 0
,
 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 12 12 12 12 2 1 0 0 0 0 0 0 1 12 5 5 8 8 0 0 1 0 5 5 8 8 12 0 1 0
,
 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 12 1 12 12 1 1 1 1 11 12 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 0 8 8 5 5 0 0 12 0 8 8 5 4 0 0 12 0

G:=sub<GL(8,GF(13))| [5,0,0,0,0,0,1,1,0,5,0,0,0,0,1,1,0,0,5,0,0,0,12,12,0,0,0,5,0,0,12,12,0,0,0,0,8,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,8],[1,0,0,0,0,0,0,12,0,1,0,0,0,0,0,12,0,0,12,12,0,0,12,0,0,0,1,0,0,0,0,1,0,0,0,0,0,1,0,1,0,0,0,0,12,12,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1,12],[12,12,0,0,0,0,0,1,1,0,0,0,0,0,12,0,0,0,1,0,0,0,1,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,0,12,12,0,12,0,0,0,0,0,0,12,1,0,0,0,0,0,0,12,0],[0,0,0,1,1,0,5,5,0,0,1,0,1,0,5,5,1,0,0,0,12,0,8,8,0,1,0,0,12,0,8,8,0,0,0,0,12,0,0,12,0,0,0,0,12,0,0,0,0,0,0,0,2,1,1,1,0,0,0,0,1,12,0,0],[0,0,0,12,0,12,8,8,0,0,0,12,12,0,8,8,0,0,0,1,0,0,5,5,0,0,0,1,0,0,5,4,1,0,0,1,0,0,0,0,0,1,0,1,0,0,0,0,0,0,12,11,0,0,12,12,0,0,1,12,0,0,0,0] >;

C4.3PSU3(𝔽2) in GAP, Magma, Sage, TeX

C_4._3{\rm PSU}_3({\mathbb F}_2)
% in TeX

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

G:=SmallGroup(288,891);
// by ID

G=gap.SmallGroup(288,891);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,56,141,176,422,100,9413,2028,362,12550,1581,1203]);
// Polycyclic

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

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