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G = C4⋊C47S3order 96 = 25·3

1st semidirect product of C4⋊C4 and S3 acting through Inn(C4⋊C4)

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C4⋊C47S3, (C4×S3)⋊2C4, C4.14(C4×S3), D6⋊C4.4C2, D6.4(C2×C4), (C2×C4).44D6, C12.11(C2×C4), Dic32(C4⋊C4), C4⋊Dic312C2, C33(C42⋊C2), (C4×Dic3)⋊13C2, C6.26(C4○D4), (C2×C6).33C23, C6.10(C22×C4), Dic3.9(C2×C4), C2.4(D42S3), (C2×C12).56C22, C2.1(Q83S3), C22.17(C22×S3), (C22×S3).19C22, (C2×Dic3).30C22, (C3×C4⋊C4)⋊3C2, (S3×C2×C4).2C2, C2.12(S3×C2×C4), SmallGroup(96,99)

Series: Derived Chief Lower central Upper central

C1C6 — C4⋊C47S3
C1C3C6C2×C6C22×S3S3×C2×C4 — C4⋊C47S3
C3C6 — C4⋊C47S3
C1C22C4⋊C4

Generators and relations for C4⋊C47S3
 G = < a,b,c,d | a4=b4=c3=d2=1, bab-1=a-1, ac=ca, ad=da, bc=cb, dbd=a2b, dcd=c-1 >

Subgroups: 154 in 76 conjugacy classes, 41 normal (19 characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×2], C4 [×6], C22, C22 [×4], S3 [×2], C6 [×3], C2×C4, C2×C4 [×2], C2×C4 [×7], C23, Dic3 [×2], Dic3 [×2], C12 [×2], C12 [×2], D6 [×2], D6 [×2], C2×C6, C42 [×2], C22⋊C4 [×2], C4⋊C4, C4⋊C4, C22×C4, C4×S3 [×4], C2×Dic3, C2×Dic3 [×2], C2×C12, C2×C12 [×2], C22×S3, C42⋊C2, C4×Dic3 [×2], C4⋊Dic3, D6⋊C4 [×2], C3×C4⋊C4, S3×C2×C4, C4⋊C47S3
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], C23, D6 [×3], C22×C4, C4○D4 [×2], C4×S3 [×2], C22×S3, C42⋊C2, S3×C2×C4, D42S3, Q83S3, C4⋊C47S3

Character table of C4⋊C47S3

 class 12A2B2C2D2E34A4B4C4D4E4F4G4H4I4J4K4L4M4N6A6B6C12A12B12C12D12E12F
 size 111166222222233336666222444444
ρ1111111111111111111111111111111    trivial
ρ21111111-1-111-1-1-1-1-1-11-11-1111-1-11-11-1    linear of order 2
ρ31111-1-11111111-1-1-1-1-1-1-1-1111111111    linear of order 2
ρ41111-1-11-1-111-1-11111-11-11111-1-11-11-1    linear of order 2
ρ51111111-11-1-11-1-1-1-1-1-11-1111111-1-1-1-1    linear of order 2
ρ611111111-1-1-1-111111-1-1-1-1111-1-1-11-11    linear of order 2
ρ71111-1-11-11-1-11-111111-11-111111-1-1-1-1    linear of order 2
ρ81111-1-111-1-1-1-11-1-1-1-11111111-1-1-11-11    linear of order 2
ρ911-1-1-111-1-i-iii111-1-1ii-i-i-1-11i-i-i1i-1    linear of order 4
ρ1011-1-11-111i-ii-i-111-1-1-iii-i-1-11-ii-i-1i1    linear of order 4
ρ1111-1-11-11-1-i-iii1-1-111-i-iii-1-11i-i-i1i-1    linear of order 4
ρ1211-1-1-1111i-ii-i-1-1-111i-i-ii-1-11-ii-i-1i1    linear of order 4
ρ1311-1-1-1111-ii-ii-1-1-111-iii-i-1-11i-ii-1-i1    linear of order 4
ρ1411-1-11-11-1ii-i-i1-1-111ii-i-i-1-11-iii1-i-1    linear of order 4
ρ1511-1-11-111-ii-ii-111-1-1i-i-ii-1-11i-ii-1-i1    linear of order 4
ρ1611-1-1-111-1ii-i-i111-1-1-i-iii-1-11-iii1-i-1    linear of order 4
ρ17222200-12-2-2-2-2200000000-1-1-1111-11-1    orthogonal lifted from D6
ρ18222200-122222200000000-1-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ19222200-1-22-2-22-200000000-1-1-1-1-11111    orthogonal lifted from D6
ρ20222200-1-2-222-2-200000000-1-1-111-11-11    orthogonal lifted from D6
ρ2122-2-200-1-2-2i-2i2i2i20000000011-1-iii-1-i1    complex lifted from C4×S3
ρ2222-2-200-1-22i2i-2i-2i20000000011-1i-i-i-1i1    complex lifted from C4×S3
ρ232-22-2002000000-2i2i-2i2i0000-22-2000000    complex lifted from C4○D4
ρ2422-2-200-12-2i2i-2i2i-20000000011-1-ii-i1i-1    complex lifted from C4×S3
ρ252-22-20020000002i-2i2i-2i0000-22-2000000    complex lifted from C4○D4
ρ2622-2-200-122i-2i2i-2i-20000000011-1i-ii1-i-1    complex lifted from C4×S3
ρ272-2-220020000002i-2i-2i2i00002-2-2000000    complex lifted from C4○D4
ρ282-2-22002000000-2i2i2i-2i00002-2-2000000    complex lifted from C4○D4
ρ294-4-4400-200000000000000-222000000    orthogonal lifted from Q83S3, Schur index 2
ρ304-44-400-2000000000000002-22000000    symplectic lifted from D42S3, Schur index 2

Smallest permutation representation of C4⋊C47S3
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 47 43 8)(2 46 44 7)(3 45 41 6)(4 48 42 5)(9 29 15 23)(10 32 16 22)(11 31 13 21)(12 30 14 24)(17 38 26 33)(18 37 27 36)(19 40 28 35)(20 39 25 34)
(1 16 17)(2 13 18)(3 14 19)(4 15 20)(5 29 34)(6 30 35)(7 31 36)(8 32 33)(9 25 42)(10 26 43)(11 27 44)(12 28 41)(21 37 46)(22 38 47)(23 39 48)(24 40 45)
(5 7)(6 8)(9 25)(10 26)(11 27)(12 28)(13 18)(14 19)(15 20)(16 17)(21 39)(22 40)(23 37)(24 38)(29 36)(30 33)(31 34)(32 35)(45 47)(46 48)

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

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

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

C4⋊C47S3 is a maximal subgroup of
C4⋊C419D6  D42S3⋊C4  D6⋊C811C2  C241C4⋊C2  (S3×Q8)⋊C4  C4⋊C4.150D6  D6⋊C8.C2  C8⋊Dic3⋊C2  (S3×C8)⋊C4  C8⋊(C4×S3)  C4.Q8⋊S3  C6.(C4○D8)  C8.27(C4×S3)  C8⋊S3⋊C4  C2.D8⋊S3  C2.D87S3  C6.82+ 1+4  C6.52- 1+4  C6.112+ 1+4  S3×C42⋊C2  C42.91D6  C42.97D6  C42.98D6  C4×D42S3  C4213D6  C42.113D6  C42.117D6  C42.125D6  C4×Q83S3  C42.132D6  C42.135D6  C4⋊C421D6  C6.382+ 1+4  C6.452+ 1+4  C6.1152+ 1+4  C4⋊C4.187D6  C4⋊C426D6  C6.162- 1+4  C6.532+ 1+4  C6.212- 1+4  C6.222- 1+4  C6.232- 1+4  C6.772- 1+4  C4⋊C4.197D6  C6.1222+ 1+4  C6.622+ 1+4  C6.662+ 1+4  C42.236D6  C42.148D6  C42.237D6  C42.151D6  C42.152D6  C42.153D6  C42.154D6  C4226D6  C42.189D6  C42.162D6  C42.164D6  C42.241D6  C42.174D6  C42.177D6  C42.179D6  C4⋊C47D9  C62.6C23  C62.11C23  C62.19C23  C62.48C23  C62.236C23  (S3×C20)⋊5C4  (C4×D15)⋊8C4  (S3×Dic5)⋊C4  D30.23(C2×C4)  C4⋊C47D15  C4⋊F53S3
C4⋊C47S3 is a maximal quotient of
Dic3.5C42  C3⋊(C425C4)  Dic3⋊C4⋊C4  C22.58(S3×D4)  D6⋊C42  D6⋊C43C4  C42.200D6  C42.202D6  C42.31D6  Dic3×C4⋊C4  (C4×Dic3)⋊9C4  C6.67(C4×D4)  C4⋊(D6⋊C4)  D6⋊C47C4  C4⋊C47D9  C62.6C23  C62.11C23  C62.19C23  C62.48C23  C62.236C23  (S3×C20)⋊5C4  (C4×D15)⋊8C4  (S3×Dic5)⋊C4  D30.23(C2×C4)  C4⋊C47D15  C4⋊F53S3

Matrix representation of C4⋊C47S3 in GL4(𝔽13) generated by

5000
3800
00120
00012
,
8800
0500
0080
0008
,
1000
0100
00012
00112
,
1000
111200
0001
0010
G:=sub<GL(4,GF(13))| [5,3,0,0,0,8,0,0,0,0,12,0,0,0,0,12],[8,0,0,0,8,5,0,0,0,0,8,0,0,0,0,8],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,12,12],[1,11,0,0,0,12,0,0,0,0,0,1,0,0,1,0] >;

C4⋊C47S3 in GAP, Magma, Sage, TeX

C_4\rtimes C_4\rtimes_7S_3
% in TeX

G:=Group("C4:C4:7S3");
// GroupNames label

G:=SmallGroup(96,99);
// by ID

G=gap.SmallGroup(96,99);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-3,103,362,188,50,2309]);
// Polycyclic

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

Export

Character table of C4⋊C47S3 in TeX

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