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G = C4⋊C4⋊S3  order 96 = 25·3

6th semidirect product of C4⋊C4 and S3 acting via S3/C3=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C4⋊C4⋊6S3, D6⋊C4.5C2, C4⋊Dic3⋊7C2, (C2×C4).32D6, Dic3⋊C4⋊13C2, C3⋊3(C42⋊2C2), (C4×Dic3)⋊14C2, C6.14(C4○D4), (C2×C12).7C22, (C2×C6).39C23, C2.16(C4○D12), C2.7(Q8⋊3S3), C2.14(D4⋊2S3), (C22×S3).8C22, C22.53(C22×S3), (C2×Dic3).32C22, (C3×C4⋊C4)⋊9C2, SmallGroup(96,105)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2×C6 — C4⋊C4⋊S3
Chief series
C1 — C3 — C6 — C2×C6 — C22×S3 — D6⋊C4 — C4⋊C4⋊S3
Lower central
C3 — C2×C6 — C4⋊C4⋊S3
Upper central
C1 — C22 — C4⋊C4

Generators and relations for C4⋊C4⋊S3
 G = < a,b,c,d | a4=b4=c3=d2=1, bab-1=a-1, ac=ca, dad=ab2, bc=cb, dbd=a2b, dcd=c-1 >

Subgroups: 138 in 60 conjugacy classes, 29 normal (all characteristic)
C1, C2, C2, C3, C4, C22, C22, S3, C6, C2×C4, C2×C4, C23, Dic3, C12, D6, C2×C6, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×Dic3, C2×C12, C22×S3, C42⋊2C2, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C3×C4⋊C4, C4⋊C4⋊S3
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C22×S3, C42⋊2C2, C4○D12, D4⋊2S3, Q8⋊3S3, C4⋊C4⋊S3

Character table of C4⋊C4⋊S3

 class 12A2B2C2D34A4B4C4D4E4F4G4H4I6A6B6C12A12B12C12D12E12F
 size 11111222244666612222444444
ρ1111111111111111111111111    trivial
ρ211111111-1-1-1-1-1-11111-11-1-11-1    linear of order 2
ρ3111111-1-1-11-11-11-11111-1-1-1-11    linear of order 2
ρ4111111-1-11-11-11-1-1111-1-111-1-1    linear of order 2
ρ51111-11-1-1-111-11-111111-1-1-1-11    linear of order 2
ρ61111-11-1-11-1-11-111111-1-111-1-1    linear of order 2
ρ71111-111111-1-1-1-1-1111111111    linear of order 2
ρ81111-1111-1-11111-1111-11-1-11-1    linear of order 2
ρ922220-1-2-22-200000-1-1-111-1-111    orthogonal lifted from D6
ρ1022220-1-2-2-2200000-1-1-1-11111-1    orthogonal lifted from D6
ρ1122220-122-2-200000-1-1-11-111-11    orthogonal lifted from D6
ρ1222220-1222200000-1-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ132-2-22020000-2i02i00-2-22000000    complex lifted from C4○D4
ρ142-22-202000002i0-2i02-2-2000000    complex lifted from C4○D4
ρ1522-2-2022i-2i0000000-22-202i00-2i0    complex lifted from C4○D4
ρ162-22-20200000-2i02i02-2-2000000    complex lifted from C4○D4
ρ1722-2-202-2i2i0000000-22-20-2i002i0    complex lifted from C4○D4
ρ182-2-220200002i0-2i00-2-22000000    complex lifted from C4○D4
ρ1922-2-20-12i-2i00000001-11-√-3-i√3-√3i√-3    complex lifted from C4○D12
ρ2022-2-20-1-2i2i00000001-11-√-3i-√3√3-i√-3    complex lifted from C4○D12
ρ2122-2-20-12i-2i00000001-11√-3-i-√3√3i-√-3    complex lifted from C4○D12
ρ2222-2-20-1-2i2i00000001-11√-3i√3-√3-i-√-3    complex lifted from C4○D12
ρ234-4-440-200000000022-2000000    orthogonal lifted from Q8⋊3S3, Schur index 2
ρ244-44-40-2000000000-222000000    symplectic lifted from D4⋊2S3, Schur index 2

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

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

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

C4⋊C4⋊S3 is a maximal subgroup of
 C6.52- 1+4  C6.112+ 1+4  C6.62- 1+4  C42⋊12D6  C42.93D6  C42.96D6  C42.99D6  C42.102D6  C42⋊18D6  C42⋊19D6  C42.119D6  C42.122D6  C42.131D6  C42.134D6  C42.135D6  C6.342+ 1+4  C6.422+ 1+4  C6.462+ 1+4  C6.482+ 1+4  C4⋊C4.187D6  C6.532+ 1+4  C6.202- 1+4  C6.222- 1+4  C6.232- 1+4  C6.772- 1+4  C6.242- 1+4  C6.562+ 1+4  C6.782- 1+4  C6.252- 1+4  C4⋊C4.197D6  C6.612+ 1+4  C6.1222+ 1+4  C6.642+ 1+4  C6.652+ 1+4  C6.852- 1+4  C6.682+ 1+4  C42.237D6  C42.150D6  C42.151D6  C42.152D6  C42.153D6  C42.154D6  C42.156D6  C42.157D6  C42.160D6  S3×C42⋊2C2  C42⋊25D6  C42.161D6  C42.163D6  C42.164D6  C42.165D6  C42.176D6  C42.177D6  C42.178D6  C42.180D6  C4⋊C4⋊D9  C62.18C23  C62.28C23  C62.31C23  C62.32C23  C62.38C23  C62.242C23  C4⋊Dic3⋊D5  D6⋊C4.D5  C60⋊5C4⋊C2  C4⋊Dic5⋊S3  D6⋊Dic5.C2  C10.D4⋊S3  C4⋊C4⋊D15
C4⋊C4⋊S3 is a maximal quotient of
 C3⋊(C42⋊5C4)  C2.(C4×Dic6)  (C2×C4).Dic6  (C22×C4).30D6  D6⋊C4⋊5C4  D6⋊C4⋊3C4  (C2×C4).21D12  C6.(C4⋊D4)  C6.67(C4×D4)  C4⋊C4⋊5Dic3  (C2×C12).288D4  (C2×C12).55D4  D6⋊C4⋊7C4  (C2×C12).289D4  (C2×C12).56D4  C4⋊C4⋊D9  C62.18C23  C62.28C23  C62.31C23  C62.32C23  C62.38C23  C62.242C23  C4⋊Dic3⋊D5  D6⋊C4.D5  C60⋊5C4⋊C2  C4⋊Dic5⋊S3  D6⋊Dic5.C2  C10.D4⋊S3  C4⋊C4⋊D15

Matrix representation of C4⋊C4⋊S3 ►in GL4(𝔽13) generated by

11900
4200
00710
0086
,
8000
0800
0042
00119
,
121200
1000
0010
0001
,
1000
121200
0010
00912
G:=sub<GL(4,GF(13))| [11,4,0,0,9,2,0,0,0,0,7,8,0,0,10,6],[8,0,0,0,0,8,0,0,0,0,4,11,0,0,2,9],[12,1,0,0,12,0,0,0,0,0,1,0,0,0,0,1],[1,12,0,0,0,12,0,0,0,0,1,9,0,0,0,12] >;
 

C4⋊C4⋊S3 in GAP, Magma, Sage, TeX 

C_4\rtimes C_4\rtimes S_3
 
% in TeX
 
G:=Group("C4:C4:S3");
 
// GroupNames label
 
G:=SmallGroup(96,105);
 
// by ID
 
G=gap.SmallGroup(96,105);
 
# by ID
 
G:=PCGroup([6,-2,-2,-2,-2,-2,-3,217,55,218,188,86,2309]);
 
// 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,d*a*d=a*b^2,b*c=c*b,d*b*d=a^2*b,d*c*d=c^-1>;
 
// generators/relations
 

Export

Character table of C4⋊C4⋊S3 in TeX

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