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G = C12⋊S3.C4order 288 = 25·32

The non-split extension by C12⋊S3 of C4 acting faithfully

metabelian, soluble, monomial

Aliases: C12⋊S3.C4, Q8.(C32⋊C4), (Q8×C32).C4, C328(C8○D4), C12.26D6.4C2, C32⋊M4(2)⋊7C2, C3⋊Dic3.33C23, C322C8.12C22, C3⋊S33C85C2, C4.6(C2×C32⋊C4), (C3×C12).6(C2×C4), (C4×C3⋊S3).39C22, (C3×C6).32(C22×C4), C2.10(C22×C32⋊C4), (C2×C3⋊S3).21(C2×C4), SmallGroup(288,937)

Series: Derived Chief Lower central Upper central

C1C3×C6 — C12⋊S3.C4
C1C32C3×C6C3⋊Dic3C322C8C3⋊S33C8 — C12⋊S3.C4
C32C3×C6 — C12⋊S3.C4
C1C2Q8

Generators and relations for C12⋊S3.C4
 G = < a,b,c,d | a12=b3=c2=1, d4=a6, ab=ba, cac=a-1, dad-1=ab-1, cbc=b-1, dbd-1=a8b-1, cd=dc >

Subgroups: 520 in 102 conjugacy classes, 34 normal (10 characteristic)
C1, C2, C2 [×3], C3 [×2], C4 [×3], C4, C22 [×3], S3 [×6], C6 [×2], C8 [×4], C2×C4 [×3], D4 [×3], Q8, C32, Dic3 [×2], C12 [×6], D6 [×6], C2×C8 [×3], M4(2) [×3], C4○D4, C3⋊S3 [×3], C3×C6, C4×S3 [×6], D12 [×6], C3×Q8 [×2], C8○D4, C3⋊Dic3, C3×C12 [×3], C2×C3⋊S3 [×3], Q83S3 [×2], C322C8, C322C8 [×3], C4×C3⋊S3 [×3], C12⋊S3 [×3], Q8×C32, C3⋊S33C8 [×3], C32⋊M4(2) [×3], C12.26D6, C12⋊S3.C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], C23, C22×C4, C8○D4, C32⋊C4, C2×C32⋊C4 [×3], C22×C32⋊C4, C12⋊S3.C4

Character table of C12⋊S3.C4

 class 12A2B2C2D3A3B4A4B4C4D4E6A6B8A8B8C8D8E8F8G8H8I8J12A12B12C12D12E12F
 size 111818184422299449999181818181818888888
ρ1111111111111111111111111111111    trivial
ρ211-1-1111-1-11111111111-1-1-11-11-1-1-1-11    linear of order 2
ρ311-11-1111-1-111111111-11-1-1-11-1-1-111-1    linear of order 2
ρ4111-1-111-11-111111111-1-111-1-1-111-1-1-1    linear of order 2
ρ511-1-1111-1-111111-1-1-1-1-1111-111-1-1-1-11    linear of order 2
ρ611111111111111-1-1-1-1-1-1-1-1-1-1111111    linear of order 2
ρ7111-1-111-11-11111-1-1-1-111-1-111-111-1-1-1    linear of order 2
ρ811-11-1111-1-11111-1-1-1-11-1111-1-1-1-111-1    linear of order 2
ρ911-1-1-111111-1-111-i-iiiii-ii-i-i111111    linear of order 4
ρ1011-1-1-111111-1-111ii-i-i-i-ii-iii111111    linear of order 4
ρ1111-11111-11-1-1-111ii-i-iiii-i-i-i-111-1-1-1    linear of order 4
ρ1211-11111-11-1-1-111-i-iii-i-i-iiii-111-1-1-1    linear of order 4
ρ131111-111-1-11-1-111ii-i-i-ii-iii-i1-1-1-1-11    linear of order 4
ρ141111-111-1-11-1-111-i-iiii-ii-i-ii1-1-1-1-11    linear of order 4
ρ15111-11111-1-1-1-111-i-iii-iii-ii-i-1-1-111-1    linear of order 4
ρ16111-11111-1-1-1-111ii-i-ii-i-ii-ii-1-1-111-1    linear of order 4
ρ172-200022000-2i2i-2-28387885000000000000    complex lifted from C8○D4
ρ182-2000220002i-2i-2-28858387000000000000    complex lifted from C8○D4
ρ192-200022000-2i2i-2-28783858000000000000    complex lifted from C8○D4
ρ202-2000220002i-2i-2-28588783000000000000    complex lifted from C8○D4
ρ21440001-24-4-400-2100000000002-121-2-1    orthogonal lifted from C2×C32⋊C4
ρ22440001-2-44-400-21000000000021-2-12-1    orthogonal lifted from C2×C32⋊C4
ρ2344000-21444001-200000000001-21-21-2    orthogonal lifted from C32⋊C4
ρ2444000-21-4-44001-2000000000012-12-1-2    orthogonal lifted from C2×C32⋊C4
ρ2544000-21-44-4001-20000000000-1-212-12    orthogonal lifted from C2×C32⋊C4
ρ2644000-214-4-4001-20000000000-12-1-212    orthogonal lifted from C2×C32⋊C4
ρ27440001-2-4-4400-210000000000-2-12-121    orthogonal lifted from C2×C32⋊C4
ρ28440001-244400-210000000000-21-21-21    orthogonal lifted from C32⋊C4
ρ298-8000-4200000-240000000000000000    orthogonal faithful, Schur index 2
ρ308-80002-4000004-20000000000000000    orthogonal faithful, Schur index 2

Smallest permutation representation of C12⋊S3.C4
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)
(1 13 12)(2 14 9)(3 15 10)(4 16 11)(5 21 20)(6 22 17)(7 23 18)(8 24 19)(25 29 33)(26 30 34)(27 31 35)(28 32 36)(37 41 45)(38 42 46)(39 43 47)(40 44 48)
(1 15)(2 14)(3 13)(4 16)(5 7)(10 12)(17 22)(18 21)(19 24)(20 23)(25 29)(26 28)(30 36)(31 35)(32 34)(37 45)(38 44)(39 43)(40 42)(46 48)
(1 40 18 28 3 46 20 34)(2 37 19 25 4 43 17 31)(5 30 12 44 7 36 10 38)(6 27 9 41 8 33 11 47)(13 48 23 32 15 42 21 26)(14 45 24 29 16 39 22 35)

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

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

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

Matrix representation of C12⋊S3.C4 in GL6(𝔽73)

51710000
60220000
0072000
0007200
00464611
0000720
,
100000
010000
0007200
0017200
002707272
0004610
,
100000
51720000
000100
001000
000010
0027277272
,
2200000
0220000
0000721
0027277172
004848460
004849460

G:=sub<GL(6,GF(73))| [51,60,0,0,0,0,71,22,0,0,0,0,0,0,72,0,46,0,0,0,0,72,46,0,0,0,0,0,1,72,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,27,0,0,0,72,72,0,46,0,0,0,0,72,1,0,0,0,0,72,0],[1,51,0,0,0,0,0,72,0,0,0,0,0,0,0,1,0,27,0,0,1,0,0,27,0,0,0,0,1,72,0,0,0,0,0,72],[22,0,0,0,0,0,0,22,0,0,0,0,0,0,0,27,48,48,0,0,0,27,48,49,0,0,72,71,46,46,0,0,1,72,0,0] >;

C12⋊S3.C4 in GAP, Magma, Sage, TeX

C_{12}\rtimes S_3.C_4
% in TeX

G:=Group("C12:S3.C4");
// GroupNames label

G:=SmallGroup(288,937);
// by ID

G=gap.SmallGroup(288,937);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,56,120,219,100,80,9413,362,12550,1203]);
// Polycyclic

G:=Group<a,b,c,d|a^12=b^3=c^2=1,d^4=a^6,a*b=b*a,c*a*c=a^-1,d*a*d^-1=a*b^-1,c*b*c=b^-1,d*b*d^-1=a^8*b^-1,c*d=d*c>;
// generators/relations

Export

Character table of C12⋊S3.C4 in TeX

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