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G = C4×C23⋊C4order 128 = 27

Direct product of C4 and C23⋊C4

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

Aliases: C4×C23⋊C4, C231C42, C24.163C23, (C23×C4)⋊3C4, (C2×C4)⋊1C42, (C2×C42)⋊6C4, C24.25(C2×C4), C22.24(C4×D4), C23.2(C4○D4), C23.540(C2×D4), (C22×C4).669D4, C22.9(C2×C42), C23.1(C22×C4), C42(C23.9D4), C23.9D416C2, (C23×C4).18C22, (C22×D4).444C22, C22.20(C42⋊C2), C2.5(C23.C23), (C2×C4⋊C4)⋊25C4, (C2×C4×D4).9C2, (C4×C22⋊C4)⋊1C2, C2.5(C2×C23⋊C4), C22⋊C423(C2×C4), (C2×C22⋊C4)⋊14C4, C2.12(C4×C22⋊C4), (C2×D4).156(C2×C4), (C2×C23⋊C4).11C2, (C22×C4).435(C2×C4), (C2×C4)(C23.9D4), (C2×C4).396(C22⋊C4), C22.114(C2×C22⋊C4), (C2×C22⋊C4).408C22, (C2×C4)(C2×C23⋊C4), SmallGroup(128,486)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C4×C23⋊C4
C1C2C22C23C24C23×C4C4×C22⋊C4 — C4×C23⋊C4
C1C2C22 — C4×C23⋊C4
C1C2×C4C23×C4 — C4×C23⋊C4
C1C2C24 — C4×C23⋊C4

Generators and relations for C4×C23⋊C4
 G = < a,b,c,d,e | a4=b2=c2=d2=e4=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, ebe-1=bcd, ece-1=cd=dc, de=ed >

Subgroups: 436 in 206 conjugacy classes, 78 normal (26 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, C2×C4, D4, C23, C23, C23, C42, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C24, C2.C42, C23⋊C4, C2×C42, C2×C42, C2×C22⋊C4, C2×C22⋊C4, C2×C4⋊C4, C4×D4, C23×C4, C22×D4, C23.9D4, C4×C22⋊C4, C2×C23⋊C4, C2×C4×D4, C4×C23⋊C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C42, C22⋊C4, C22×C4, C2×D4, C4○D4, C23⋊C4, C2×C42, C2×C22⋊C4, C42⋊C2, C4×D4, C4×C22⋊C4, C2×C23⋊C4, C23.C23, C4×C23⋊C4

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

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

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

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

44 conjugacy classes

class 1 2A2B2C2D···2I2J2K4A4B4C4D4E···4J4K···4AF
order12222···22244444···44···4
size11112···24411112···24···4

44 irreducible representations

dim11111111112244
type+++++++
imageC1C2C2C2C2C4C4C4C4C4D4C4○D4C23⋊C4C23.C23
kernelC4×C23⋊C4C23.9D4C4×C22⋊C4C2×C23⋊C4C2×C4×D4C23⋊C4C2×C42C2×C22⋊C4C2×C4⋊C4C23×C4C22×C4C23C4C2
# reps122211622224422

Matrix representation of C4×C23⋊C4 in GL6(𝔽5)

200000
020000
001000
000100
000010
000001
,
040000
400000
000010
000004
001000
000400
,
400000
040000
000100
001000
000004
000040
,
100000
010000
004000
000400
000040
000004
,
030000
200000
000004
000040
004000
000400

G:=sub<GL(6,GF(5))| [2,0,0,0,0,0,0,2,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,4,0,0,0,0,4,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,1,0,0,0,0,0,0,4,0,0],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,4,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[0,2,0,0,0,0,3,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,4,0,0,0,0,4,0,0,0] >;

C4×C23⋊C4 in GAP, Magma, Sage, TeX

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

G:=Group("C4xC2^3:C4");
// GroupNames label

G:=SmallGroup(128,486);
// by ID

G=gap.SmallGroup(128,486);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,112,141,232,100,1018,4037]);
// Polycyclic

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

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