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G = C3⋊S3.4D8order 288 = 25·32

The non-split extension by C3⋊S3 of D8 acting via D8/C8=C2

metabelian, soluble, monomial

Aliases: (C3×C24)⋊4C4, C3⋊S3.4D8, C81(C32⋊C4), C3⋊S3.4Q16, C3⋊Dic3.7Q8, C324C812C4, C323(C2.D8), C4.9(C2×C32⋊C4), (C8×C3⋊S3).11C2, (C2×C3⋊S3).39D4, (C3×C6).12(C4⋊C4), (C3×C12).13(C2×C4), C4⋊(C32⋊C4).7C2, C2.5(C4⋊(C32⋊C4)), (C4×C3⋊S3).81C22, SmallGroup(288,417)

Series: Derived Chief Lower central Upper central

C1C3×C12 — C3⋊S3.4D8
C1C32C3×C6C2×C3⋊S3C4×C3⋊S3C4⋊(C32⋊C4) — C3⋊S3.4D8
C32C3×C6C3×C12 — C3⋊S3.4D8
C1C2C4C8

Generators and relations for C3⋊S3.4D8
 G = < a,b,c,d,e | a3=b3=c2=d8=1, e2=c, ab=ba, cac=a-1, ad=da, eae-1=ab-1, cbc=b-1, bd=db, ebe-1=a-1b-1, cd=dc, ce=ec, ede-1=d-1 >

Subgroups: 328 in 58 conjugacy classes, 18 normal (16 characteristic)
C1, C2, C2 [×2], C3 [×2], C4, C4 [×3], C22, S3 [×4], C6 [×2], C8, C8, C2×C4 [×3], C32, Dic3 [×2], C12 [×2], D6 [×2], C4⋊C4 [×2], C2×C8, C3⋊S3 [×2], C3×C6, C3⋊C8 [×2], C24 [×2], C4×S3 [×2], C2.D8, C3⋊Dic3, C3×C12, C32⋊C4 [×2], C2×C3⋊S3, S3×C8 [×2], C324C8, C3×C24, C4×C3⋊S3, C2×C32⋊C4 [×2], C8×C3⋊S3, C4⋊(C32⋊C4) [×2], C3⋊S3.4D8
Quotients: C1, C2 [×3], C4 [×2], C22, C2×C4, D4, Q8, C4⋊C4, D8, Q16, C2.D8, C32⋊C4, C2×C32⋊C4, C4⋊(C32⋊C4), C3⋊S3.4D8

Character table of C3⋊S3.4D8

 class 12A2B2C3A3B4A4B4C4D4E4F6A6B8A8B8C8D12A12B12C12D24A24B24C24D24E24F24G24H
 size 1199442183636363644221818444444444444
ρ1111111111111111111111111111111    trivial
ρ211111111-11-1111-1-1-1-11111-1-1-1-1-1-1-1-1    linear of order 2
ρ3111111111-11-111-1-1-1-11111-1-1-1-1-1-1-1-1    linear of order 2
ρ411111111-1-1-1-1111111111111111111    linear of order 2
ρ511-1-1111-1i-i-ii11-1-1111111-1-1-1-1-1-1-1-1    linear of order 4
ρ611-1-1111-1-i-iii1111-1-1111111111111    linear of order 4
ρ711-1-1111-1ii-i-i1111-1-1111111111111    linear of order 4
ρ811-1-1111-1-iii-i11-1-1111111-1-1-1-1-1-1-1-1    linear of order 4
ρ9222222-2-20000220000-2-2-2-200000000    orthogonal lifted from D4
ρ102-2-2222000000-2-2-22-22000022-2-2-2-222    orthogonal lifted from D8
ρ112-2-2222000000-2-22-22-20000-2-22222-2-2    orthogonal lifted from D8
ρ122-22-222000000-2-22-2-220000-2-22222-2-2    symplectic lifted from Q16, Schur index 2
ρ132-22-222000000-2-2-222-2000022-2-2-2-222    symplectic lifted from Q16, Schur index 2
ρ1422-2-222-220000220000-2-2-2-200000000    symplectic lifted from Q8, Schur index 2
ρ154400-214000001-244001-21-2-21-2-2111-2    orthogonal lifted from C32⋊C4
ρ164400-214000001-2-4-4001-21-22-122-1-1-12    orthogonal lifted from C2×C32⋊C4
ρ1744001-2400000-214400-21-211-211-2-2-21    orthogonal lifted from C32⋊C4
ρ1844001-2400000-21-4-400-21-21-12-1-1222-1    orthogonal lifted from C2×C32⋊C4
ρ1944001-2-400000-2100002-12-1-3i03i-3i0003i    complex lifted from C4⋊(C32⋊C4)
ρ204400-21-4000001-20000-12-1203i003i-3i-3i0    complex lifted from C4⋊(C32⋊C4)
ρ214400-21-4000001-20000-12-120-3i00-3i3i3i0    complex lifted from C4⋊(C32⋊C4)
ρ2244001-2-400000-2100002-12-13i0-3i3i000-3i    complex lifted from C4⋊(C32⋊C4)
ρ234-4001-20000002-122-22000-3i03iζ87+2ζ8528785ζ83+2ζ8-2-22838    complex faithful
ρ244-400-21000000-1222-22003i0-3i02ζ87+2ζ85-2-2ζ83+2ζ887858382    complex faithful
ρ254-4001-20000002-1-22220003i0-3i8785-2ζ87+2ζ8583822-2ζ83+2ζ8    complex faithful
ρ264-400-21000000-1222-2200-3i03i02838-2-28785ζ83+2ζ8ζ87+2ζ852    complex faithful
ρ274-4001-20000002-1-2222000-3i03iζ83+2ζ8-2838ζ87+2ζ8522-28785    complex faithful
ρ284-400-21000000-12-2222003i0-3i0-2ζ83+2ζ822ζ87+2ζ858388785-2    complex faithful
ρ294-400-21000000-12-222200-3i03i0-2878522838ζ87+2ζ85ζ83+2ζ8-2    complex faithful
ρ304-4001-20000002-122-220003i0-3i8382ζ83+2ζ88785-2-22ζ87+2ζ85    complex faithful

Smallest permutation representation of C3⋊S3.4D8
On 48 points
Generators in S48
(1 25 38)(2 26 39)(3 27 40)(4 28 33)(5 29 34)(6 30 35)(7 31 36)(8 32 37)(9 46 20)(10 47 21)(11 48 22)(12 41 23)(13 42 24)(14 43 17)(15 44 18)(16 45 19)
(9 20 46)(10 21 47)(11 22 48)(12 23 41)(13 24 42)(14 17 43)(15 18 44)(16 19 45)
(17 43)(18 44)(19 45)(20 46)(21 47)(22 48)(23 41)(24 42)(25 38)(26 39)(27 40)(28 33)(29 34)(30 35)(31 36)(32 37)
(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 12)(2 11)(3 10)(4 9)(5 16)(6 15)(7 14)(8 13)(17 31 43 36)(18 30 44 35)(19 29 45 34)(20 28 46 33)(21 27 47 40)(22 26 48 39)(23 25 41 38)(24 32 42 37)

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

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

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

Matrix representation of C3⋊S3.4D8 in GL6(𝔽73)

100000
010000
0072100
0072000
0000072
0000172
,
100000
010000
001000
000100
0000721
0000720
,
7200000
0720000
0017200
0007200
0000172
0000072
,
0410000
16410000
0046000
0004600
0000270
0000027
,
37400000
57360000
0000270
0000027
00462700
0002700

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,72,72],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,1,0],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,72,72,0,0,0,0,0,0,1,0,0,0,0,0,72,72],[0,16,0,0,0,0,41,41,0,0,0,0,0,0,46,0,0,0,0,0,0,46,0,0,0,0,0,0,27,0,0,0,0,0,0,27],[37,57,0,0,0,0,40,36,0,0,0,0,0,0,0,0,46,0,0,0,0,0,27,27,0,0,27,0,0,0,0,0,0,27,0,0] >;

C3⋊S3.4D8 in GAP, Magma, Sage, TeX

C_3\rtimes S_3._4D_8
% in TeX

G:=Group("C3:S3.4D8");
// GroupNames label

G:=SmallGroup(288,417);
// by ID

G=gap.SmallGroup(288,417);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,28,141,176,675,80,9413,691,12550,2372]);
// Polycyclic

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

Export

Character table of C3⋊S3.4D8 in TeX

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