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G = C22×C23⋊C4order 128 = 27

Direct product of C22 and C23⋊C4

direct product, p-group, metabelian, nilpotent (class 3), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22 — C22×C23⋊C4
 Chief series C1 — C2 — C22 — C23 — C24 — C25 — D4×C23 — C22×C23⋊C4
 Lower central C1 — C2 — C22 — C22×C23⋊C4
 Upper central C1 — C23 — C25 — C22×C23⋊C4
 Jennings C1 — C2 — C23 — C22×C23⋊C4

Generators and relations for C22×C23⋊C4
G = < a,b,c,d,e,f | a2=b2=c2=d2=e2=f4=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, fcf-1=cde, fdf-1=de=ed, ef=fe >

Subgroups: 1228 in 556 conjugacy classes, 180 normal (11 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C2×C4, C2×C4, D4, C23, C23, C23, C22⋊C4, C22⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C24, C24, C24, C23⋊C4, C2×C22⋊C4, C2×C22⋊C4, C23×C4, C23×C4, C22×D4, C22×D4, C25, C2×C23⋊C4, C22×C22⋊C4, D4×C23, C22×C23⋊C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C24, C23⋊C4, C2×C22⋊C4, C23×C4, C22×D4, C2×C23⋊C4, C22×C22⋊C4, C22×C23⋊C4

Smallest permutation representation of C22×C23⋊C4
On 32 points
Generators in S32
(1 27)(2 28)(3 25)(4 26)(5 12)(6 9)(7 10)(8 11)(13 29)(14 30)(15 31)(16 32)(17 24)(18 21)(19 22)(20 23)
(1 20)(2 17)(3 18)(4 19)(5 32)(6 29)(7 30)(8 31)(9 13)(10 14)(11 15)(12 16)(21 25)(22 26)(23 27)(24 28)
(1 14)(2 31)(3 25)(5 12)(7 23)(8 17)(10 20)(11 24)(15 28)(16 32)(18 21)(27 30)
(1 14)(2 28)(3 16)(4 26)(5 21)(6 9)(7 23)(8 11)(10 20)(12 18)(13 29)(15 31)(17 24)(19 22)(25 32)(27 30)
(1 30)(2 31)(3 32)(4 29)(5 18)(6 19)(7 20)(8 17)(9 22)(10 23)(11 24)(12 21)(13 26)(14 27)(15 28)(16 25)
(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)

G:=sub<Sym(32)| (1,27)(2,28)(3,25)(4,26)(5,12)(6,9)(7,10)(8,11)(13,29)(14,30)(15,31)(16,32)(17,24)(18,21)(19,22)(20,23), (1,20)(2,17)(3,18)(4,19)(5,32)(6,29)(7,30)(8,31)(9,13)(10,14)(11,15)(12,16)(21,25)(22,26)(23,27)(24,28), (1,14)(2,31)(3,25)(5,12)(7,23)(8,17)(10,20)(11,24)(15,28)(16,32)(18,21)(27,30), (1,14)(2,28)(3,16)(4,26)(5,21)(6,9)(7,23)(8,11)(10,20)(12,18)(13,29)(15,31)(17,24)(19,22)(25,32)(27,30), (1,30)(2,31)(3,32)(4,29)(5,18)(6,19)(7,20)(8,17)(9,22)(10,23)(11,24)(12,21)(13,26)(14,27)(15,28)(16,25), (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)>;

G:=Group( (1,27)(2,28)(3,25)(4,26)(5,12)(6,9)(7,10)(8,11)(13,29)(14,30)(15,31)(16,32)(17,24)(18,21)(19,22)(20,23), (1,20)(2,17)(3,18)(4,19)(5,32)(6,29)(7,30)(8,31)(9,13)(10,14)(11,15)(12,16)(21,25)(22,26)(23,27)(24,28), (1,14)(2,31)(3,25)(5,12)(7,23)(8,17)(10,20)(11,24)(15,28)(16,32)(18,21)(27,30), (1,14)(2,28)(3,16)(4,26)(5,21)(6,9)(7,23)(8,11)(10,20)(12,18)(13,29)(15,31)(17,24)(19,22)(25,32)(27,30), (1,30)(2,31)(3,32)(4,29)(5,18)(6,19)(7,20)(8,17)(9,22)(10,23)(11,24)(12,21)(13,26)(14,27)(15,28)(16,25), (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) );

G=PermutationGroup([[(1,27),(2,28),(3,25),(4,26),(5,12),(6,9),(7,10),(8,11),(13,29),(14,30),(15,31),(16,32),(17,24),(18,21),(19,22),(20,23)], [(1,20),(2,17),(3,18),(4,19),(5,32),(6,29),(7,30),(8,31),(9,13),(10,14),(11,15),(12,16),(21,25),(22,26),(23,27),(24,28)], [(1,14),(2,31),(3,25),(5,12),(7,23),(8,17),(10,20),(11,24),(15,28),(16,32),(18,21),(27,30)], [(1,14),(2,28),(3,16),(4,26),(5,21),(6,9),(7,23),(8,11),(10,20),(12,18),(13,29),(15,31),(17,24),(19,22),(25,32),(27,30)], [(1,30),(2,31),(3,32),(4,29),(5,18),(6,19),(7,20),(8,17),(9,22),(10,23),(11,24),(12,21),(13,26),(14,27),(15,28),(16,25)], [(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)]])

44 conjugacy classes

 class 1 2A ··· 2G 2H ··· 2S 2T 2U 2V 2W 4A ··· 4T order 1 2 ··· 2 2 ··· 2 2 2 2 2 4 ··· 4 size 1 1 ··· 1 2 ··· 2 4 4 4 4 4 ··· 4

44 irreducible representations

 dim 1 1 1 1 1 1 1 2 4 type + + + + + + image C1 C2 C2 C2 C4 C4 C4 D4 C23⋊C4 kernel C22×C23⋊C4 C2×C23⋊C4 C22×C22⋊C4 D4×C23 C23×C4 C22×D4 C25 C24 C22 # reps 1 12 2 1 2 12 2 8 4

Matrix representation of C22×C23⋊C4 in GL8(𝔽5)

 4 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 4
,
 4 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 4 0 0 0 0 0 0 0 4 1 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 1 0 0 0 0 0 2 0 1 4
,
 4 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 4 0 0 0 0 0 2 3 0 4
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 4
,
 3 4 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 2 3 1 3 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 2

G:=sub<GL(8,GF(5))| [4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[4,4,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,2,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,2,0,0,0,0,0,1,0,3,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[3,0,0,0,0,0,0,0,4,2,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,3,1,0,0,0,0,0,1,1,0,0,0,0,0,0,0,3,0,2] >;

C22×C23⋊C4 in GAP, Magma, Sage, TeX

C_2^2\times C_2^3\rtimes C_4
% in TeX

G:=Group("C2^2xC2^3:C4");
// GroupNames label

G:=SmallGroup(128,1613);
// by ID

G=gap.SmallGroup(128,1613);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,2804,2028]);
// Polycyclic

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

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