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G = C3⋊S3.5Q16order 288 = 25·32

The non-split extension by C3⋊S3 of Q16 acting via Q16/Q8=C2

metabelian, soluble, monomial

Aliases: C3⋊S3.5Q16, C3⋊S3.7SD16, Q81(C32⋊C4), (Q8×C32)⋊1C4, C324Q82C4, C3⋊Dic3.11D4, C327(Q8⋊C4), C2.8(C62⋊C4), C4.3(C2×C32⋊C4), (Q8×C3⋊S3).1C2, (C3×C12).3(C2×C4), (C2×C3⋊S3).52D4, C3⋊S33C8.1C2, C4⋊(C32⋊C4).2C2, (C4×C3⋊S3).7C22, (C3×C6).18(C22⋊C4), SmallGroup(288,432)

Series: Derived Chief Lower central Upper central

C1C3×C12 — C3⋊S3.5Q16
C1C32C3×C6C2×C3⋊S3C4×C3⋊S3C4⋊(C32⋊C4) — C3⋊S3.5Q16
C32C3×C6C3×C12 — C3⋊S3.5Q16
C1C2C4Q8

Generators and relations for C3⋊S3.5Q16
 G = < a,b,c,d,e | a3=b3=c2=d8=1, e2=d4, ab=ba, cac=a-1, dad-1=ab-1, ae=ea, cbc=b-1, dbd-1=a-1b-1, be=eb, cd=dc, ce=ec, ede-1=cd-1 >

Subgroups: 432 in 80 conjugacy classes, 18 normal (all characteristic)
C1, C2, C2 [×2], C3 [×2], C4, C4 [×4], C22, S3 [×4], C6 [×2], C8, C2×C4 [×3], Q8, Q8 [×2], C32, Dic3 [×6], C12 [×6], D6 [×2], C4⋊C4, C2×C8, C2×Q8, C3⋊S3 [×2], C3×C6, Dic6 [×6], C4×S3 [×6], C3×Q8 [×2], Q8⋊C4, C3⋊Dic3, C3⋊Dic3, C3×C12, C3×C12, C32⋊C4, C2×C3⋊S3, S3×Q8 [×2], C322C8, C324Q8, C324Q8, C4×C3⋊S3, C4×C3⋊S3, Q8×C32, C2×C32⋊C4, C3⋊S33C8, C4⋊(C32⋊C4), Q8×C3⋊S3, C3⋊S3.5Q16
Quotients: C1, C2 [×3], C4 [×2], C22, C2×C4, D4 [×2], C22⋊C4, SD16, Q16, Q8⋊C4, C32⋊C4, C2×C32⋊C4, C62⋊C4, C3⋊S3.5Q16

Character table of C3⋊S3.5Q16

 class 12A2B2C3A3B4A4B4C4D4E4F6A6B8A8B8C8D12A12B12C12D12E12F
 size 11994424183636364418181818888888
ρ1111111111111111111111111    trivial
ρ2111111111-11-111-1-1-1-1111111    linear of order 2
ρ31111111-111-1111-1-1-1-1-1-1-111-1    linear of order 2
ρ41111111-11-1-1-1111111-1-1-111-1    linear of order 2
ρ511-1-11111-1i-1-i11i-i-ii111111    linear of order 4
ρ611-1-1111-1-1i1-i11-iii-i-1-1-111-1    linear of order 4
ρ711-1-1111-1-1-i1i11i-i-ii-1-1-111-1    linear of order 4
ρ811-1-11111-1-i-1i11-iii-i111111    linear of order 4
ρ922-2-222-202000220000000-2-20    orthogonal lifted from D4
ρ10222222-20-2000220000000-2-20    orthogonal lifted from D4
ρ112-2-2222000000-2-2-2-222000000    symplectic lifted from Q16, Schur index 2
ρ122-2-2222000000-2-222-2-2000000    symplectic lifted from Q16, Schur index 2
ρ132-22-222000000-2-2-2--2-2--2000000    complex lifted from SD16
ρ142-22-222000000-2-2--2-2--2-2000000    complex lifted from SD16
ρ1544001-2440000-210000-2111-2-2    orthogonal lifted from C32⋊C4
ρ1644001-2-400000-21000003-3-120    orthogonal lifted from C62⋊C4
ρ1744001-2-400000-2100000-33-120    orthogonal lifted from C62⋊C4
ρ184400-214-400001-20000-122-21-1    orthogonal lifted from C2×C32⋊C4
ρ194400-21-4000001-200003002-1-3    orthogonal lifted from C62⋊C4
ρ204400-214400001-200001-2-2-211    orthogonal lifted from C32⋊C4
ρ214400-21-4000001-20000-3002-13    orthogonal lifted from C62⋊C4
ρ2244001-24-40000-2100002-1-11-22    orthogonal lifted from C2×C32⋊C4
ρ238-800-42000000-240000000000    symplectic faithful, Schur index 2
ρ248-8002-40000004-20000000000    symplectic faithful, Schur index 2

Smallest permutation representation of C3⋊S3.5Q16
On 48 points
Generators in S48
(1 30 46)(2 31 47)(3 48 32)(4 41 25)(5 26 42)(6 27 43)(7 44 28)(8 45 29)(9 35 19)(10 36 20)(11 21 37)(12 22 38)(13 39 23)(14 40 24)(15 17 33)(16 18 34)
(2 47 31)(4 25 41)(6 43 27)(8 29 45)(10 20 36)(12 38 22)(14 24 40)(16 34 18)
(9 35)(10 36)(11 37)(12 38)(13 39)(14 40)(15 33)(16 34)(25 41)(26 42)(27 43)(28 44)(29 45)(30 46)(31 47)(32 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 23 5 19)(2 22 6 18)(3 21 7 17)(4 20 8 24)(9 30 13 26)(10 45 14 41)(11 28 15 32)(12 43 16 47)(25 36 29 40)(27 34 31 38)(33 48 37 44)(35 46 39 42)

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

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

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

Matrix representation of C3⋊S3.5Q16 in GL6(𝔽73)

100000
010000
0072100
0072000
00017272
0072010
,
100000
010000
001000
000100
000001
00117272
,
7200000
0720000
000100
001000
000010
00117272
,
6670000
660000
0000721
00117172
002525720
002524720
,
13660000
66600000
001000
000100
002525720
002525072

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,72,0,0,1,0,1,0,0,0,0,0,72,1,0,0,0,0,72,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,1,0,1,0,0,0,0,0,72,0,0,0,0,1,72],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,1,0,1,0,0,1,0,0,1,0,0,0,0,1,72,0,0,0,0,0,72],[6,6,0,0,0,0,67,6,0,0,0,0,0,0,0,1,25,25,0,0,0,1,25,24,0,0,72,71,72,72,0,0,1,72,0,0],[13,66,0,0,0,0,66,60,0,0,0,0,0,0,1,0,25,25,0,0,0,1,25,25,0,0,0,0,72,0,0,0,0,0,0,72] >;

C3⋊S3.5Q16 in GAP, Magma, Sage, TeX

C_3\rtimes S_3._5Q_{16}
% in TeX

G:=Group("C3:S3.5Q16");
// GroupNames label

G:=SmallGroup(288,432);
// by ID

G=gap.SmallGroup(288,432);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,28,141,120,675,346,80,9413,691,12550,2372]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^3=c^2=d^8=1,e^2=d^4,a*b=b*a,c*a*c=a^-1,d*a*d^-1=a*b^-1,a*e=e*a,c*b*c=b^-1,d*b*d^-1=a^-1*b^-1,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=c*d^-1>;
// generators/relations

Export

Character table of C3⋊S3.5Q16 in TeX

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