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G = C4×C32⋊C4order 144 = 24·32

Direct product of C4 and C32⋊C4

Aliases: C4×C32⋊C4, C321C42, (C3×C12)⋊1C4, C3⋊Dic34C4, (C4×C3⋊S3).8C2, C3⋊S3.5(C2×C4), (C3×C6).3(C2×C4), C2.2(C2×C32⋊C4), (C2×C32⋊C4).6C2, (C2×C3⋊S3).8C22, SmallGroup(144,132)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C4×C32⋊C4
 Chief series C1 — C32 — C3⋊S3 — C2×C3⋊S3 — C2×C32⋊C4 — C4×C32⋊C4
 Lower central C32 — C4×C32⋊C4
 Upper central C1 — C4

Generators and relations for C4×C32⋊C4
G = < a,b,c,d | a4=b3=c3=d4=1, ab=ba, ac=ca, ad=da, dcd-1=bc=cb, dbd-1=b-1c >

Character table of C4×C32⋊C4

 class 1 2A 2B 2C 3A 3B 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 6A 6B 12A 12B 12C 12D size 1 1 9 9 4 4 1 1 9 9 9 9 9 9 9 9 9 9 4 4 4 4 4 4 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 1 1 1 -1 -1 1 -1 -1 -1 -1 -1 1 1 1 -1 1 1 -1 -1 -1 -1 linear of order 2 ρ3 1 1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 -1 -1 -1 -1 1 1 -1 -1 -1 -1 linear of order 2 ρ4 1 1 1 1 1 1 1 1 -1 1 -1 -1 -1 -1 -1 -1 -1 1 1 1 1 1 1 1 linear of order 2 ρ5 1 -1 -1 1 1 1 -i i -1 -i -i i i -i -1 1 1 i -1 -1 -i i -i i linear of order 4 ρ6 1 -1 -1 1 1 1 i -i -1 i i -i -i i -1 1 1 -i -1 -1 i -i i -i linear of order 4 ρ7 1 1 -1 -1 1 1 1 1 -i -1 i i -i -i i i -i -1 1 1 1 1 1 1 linear of order 4 ρ8 1 -1 -1 1 1 1 i -i 1 i -i i i -i 1 -1 -1 -i -1 -1 i -i i -i linear of order 4 ρ9 1 1 -1 -1 1 1 -1 -1 i 1 i i -i -i -i -i i 1 1 1 -1 -1 -1 -1 linear of order 4 ρ10 1 -1 1 -1 1 1 -i i i i 1 -1 1 -1 -i i -i -i -1 -1 -i i -i i linear of order 4 ρ11 1 -1 1 -1 1 1 i -i -i -i 1 -1 1 -1 i -i i i -1 -1 i -i i -i linear of order 4 ρ12 1 -1 1 -1 1 1 i -i i -i -1 1 -1 1 -i i -i i -1 -1 i -i i -i linear of order 4 ρ13 1 -1 1 -1 1 1 -i i -i i -1 1 -1 1 i -i i -i -1 -1 -i i -i i linear of order 4 ρ14 1 -1 -1 1 1 1 -i i 1 -i i -i -i i 1 -1 -1 i -1 -1 -i i -i i linear of order 4 ρ15 1 1 -1 -1 1 1 -1 -1 -i 1 -i -i i i i i -i 1 1 1 -1 -1 -1 -1 linear of order 4 ρ16 1 1 -1 -1 1 1 1 1 i -1 -i -i i i -i -i i -1 1 1 1 1 1 1 linear of order 4 ρ17 4 4 0 0 -2 1 4 4 0 0 0 0 0 0 0 0 0 0 -2 1 1 1 -2 -2 orthogonal lifted from C32⋊C4 ρ18 4 4 0 0 1 -2 -4 -4 0 0 0 0 0 0 0 0 0 0 1 -2 2 2 -1 -1 orthogonal lifted from C2×C32⋊C4 ρ19 4 4 0 0 -2 1 -4 -4 0 0 0 0 0 0 0 0 0 0 -2 1 -1 -1 2 2 orthogonal lifted from C2×C32⋊C4 ρ20 4 4 0 0 1 -2 4 4 0 0 0 0 0 0 0 0 0 0 1 -2 -2 -2 1 1 orthogonal lifted from C32⋊C4 ρ21 4 -4 0 0 -2 1 -4i 4i 0 0 0 0 0 0 0 0 0 0 2 -1 -i i 2i -2i complex faithful ρ22 4 -4 0 0 1 -2 4i -4i 0 0 0 0 0 0 0 0 0 0 -1 2 -2i 2i i -i complex faithful ρ23 4 -4 0 0 -2 1 4i -4i 0 0 0 0 0 0 0 0 0 0 2 -1 i -i -2i 2i complex faithful ρ24 4 -4 0 0 1 -2 -4i 4i 0 0 0 0 0 0 0 0 0 0 -1 2 2i -2i -i i complex faithful

Permutation representations of C4×C32⋊C4
On 24 points - transitive group 24T240
Generators in S24
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)
(5 23 10)(6 24 11)(7 21 12)(8 22 9)
(1 16 17)(2 13 18)(3 14 19)(4 15 20)(5 23 10)(6 24 11)(7 21 12)(8 22 9)
(1 22 3 24)(2 23 4 21)(5 15 12 18)(6 16 9 19)(7 13 10 20)(8 14 11 17)

G:=sub<Sym(24)| (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (5,23,10)(6,24,11)(7,21,12)(8,22,9), (1,16,17)(2,13,18)(3,14,19)(4,15,20)(5,23,10)(6,24,11)(7,21,12)(8,22,9), (1,22,3,24)(2,23,4,21)(5,15,12,18)(6,16,9,19)(7,13,10,20)(8,14,11,17)>;

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), (5,23,10)(6,24,11)(7,21,12)(8,22,9), (1,16,17)(2,13,18)(3,14,19)(4,15,20)(5,23,10)(6,24,11)(7,21,12)(8,22,9), (1,22,3,24)(2,23,4,21)(5,15,12,18)(6,16,9,19)(7,13,10,20)(8,14,11,17) );

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)], [(5,23,10),(6,24,11),(7,21,12),(8,22,9)], [(1,16,17),(2,13,18),(3,14,19),(4,15,20),(5,23,10),(6,24,11),(7,21,12),(8,22,9)], [(1,22,3,24),(2,23,4,21),(5,15,12,18),(6,16,9,19),(7,13,10,20),(8,14,11,17)]])

G:=TransitiveGroup(24,240);

C4×C32⋊C4 is a maximal subgroup of
C32⋊C4≀C2  C32⋊C4⋊C8  C4.4PSU3(𝔽2)  (C3×C24)⋊C4  C326C4≀C2  C327C4≀C2  C4⋊F9  C4.4S3≀C2  C32⋊C4⋊Q8  C4⋊S3≀C2  C4.3PSU3(𝔽2)  C4⋊PSU3(𝔽2)  (C6×C12)⋊5C4
C4×C32⋊C4 is a maximal quotient of
(C3×C24)⋊C4  C322C8⋊C4  (C6×C12)⋊2C4

Matrix representation of C4×C32⋊C4 in GL4(𝔽5) generated by

 3 0 0 0 0 3 0 0 0 0 3 0 0 0 0 3
,
 1 0 3 1 3 0 0 3 0 0 0 3 4 2 0 0
,
 0 3 0 0 3 4 0 0 3 0 4 3 0 2 3 0
,
 4 0 0 4 0 0 0 2 0 4 1 2 0 0 2 0
G:=sub<GL(4,GF(5))| [3,0,0,0,0,3,0,0,0,0,3,0,0,0,0,3],[1,3,0,4,0,0,0,2,3,0,0,0,1,3,3,0],[0,3,3,0,3,4,0,2,0,0,4,3,0,0,3,0],[4,0,0,0,0,0,4,0,0,0,1,2,4,2,2,0] >;

C4×C32⋊C4 in GAP, Magma, Sage, TeX

C_4\times C_3^2\rtimes C_4
% in TeX

G:=Group("C4xC3^2:C4");
// GroupNames label

G:=SmallGroup(144,132);
// by ID

G=gap.SmallGroup(144,132);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,3,24,55,3364,256,4613,881]);
// Polycyclic

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

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