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G = (C23×C4).C4  order 128 = 27

20th non-split extension by C23×C4 of C4 acting faithfully

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

Aliases: C4.51C22≀C2, (C2×D4).262D4, C24.28(C2×C4), (C23×C4).20C4, (C2×Q8).205D4, (C22×D4).28C4, (C22×C4).261D4, C24.4C4⋊24C2, C2.10(C24⋊3C4), C23.30(C22⋊C4), C23.184(C22×C4), (C23×C4).230C22, (C22×C4).655C23, (C22×D4).451C22, (C22×Q8).379C22, (C2×M4(2)).148C22, C2.22(M4(2).8C22), (C2×C4).226(C2×D4), (C2×C4.D4)⋊14C2, (C22×C4○D4).3C2, (C2×C4.10D4)⋊14C2, (C2×C4).43(C22⋊C4), (C22×C4).441(C2×C4), C22.32(C2×C22⋊C4), SmallGroup(128,517)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series
C1 — C23 — (C23×C4).C4
Chief series
C1 — C2 — C4 — C2×C4 — C22×C4 — C23×C4 — C22×C4○D4 — (C23×C4).C4
Lower central
C1 — C2 — C23 — (C23×C4).C4
Upper central
C1 — C22 — C23×C4 — (C23×C4).C4
Jennings
C1 — C2 — C2 — C22×C4 — (C23×C4).C4

Generators and relations for (C23×C4).C4
 G = < a,b,c,d,e | a2=b2=c2=d4=1, e4=d2, ab=ba, ac=ca, ad=da, eae-1=ad2, bc=cb, ede-1=bd=db, be=eb, cd=dc, ece-1=acd2 >

Subgroups: 580 in 292 conjugacy classes, 68 normal (14 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, D4, Q8, C23, C23, C23, C2×C8, M4(2), C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4, C24, C24, C22⋊C8, C4.D4, C4.10D4, C2×M4(2), C23×C4, C23×C4, C22×D4, C22×D4, C22×Q8, C2×C4○D4, C24.4C4, C2×C4.D4, C2×C4.10D4, C22×C4○D4, (C23×C4).C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C2×C22⋊C4, C22≀C2, C24⋊3C4, M4(2).8C22, (C23×C4).C4

Smallest permutation representation of (C23×C4).C4
►On 32 points
Generators in S32
(2 6)(4 8)(9 13)(11 15)(18 22)(20 24)(26 30)(28 32)

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

(2 6)(3 7)(11 15)(12 16)(17 21)(20 24)(25 29)(28 32)

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

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

32 conjugacy classes

class 1 2A2B2C2D2E2F···2K4A···4H4I4J4K4L8A···8H
order1222222···24···444448···8
size1111224···42···244448···8

32 irreducible representations

dim11111112224
type++++++++
imageC1C2C2C2C2C4C4D4D4D4M4(2).8C22
kernel(C23×C4).C4C24.4C4C2×C4.D4C2×C4.10D4C22×C4○D4C23×C4C22×D4C22×C4C2×D4C2×Q8C2
# reps12221444444

Matrix representation of (C23×C4).C4 ►in GL6(𝔽17)

1600000
0160000
001000
000100
0000160
0000016
,
1600000
0160000
0016000
0001600
0000160
0000016
,
010000
100000
001000
0001600
000010
0000016
,
0160000
1600000
004000
000400
0000130
0000013
,
400000
0130000
000010
000001
000100
0016000

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16],[0,16,0,0,0,0,16,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,13,0,0,0,0,0,0,13],[4,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0] >;
 

(C23×C4).C4 in GAP, Magma, Sage, TeX 

(C_2^3\times C_4).C_4
 
% in TeX
 
G:=Group("(C2^3xC4).C4");
 
// GroupNames label
 
G:=SmallGroup(128,517);
 
// by ID
 
G=gap.SmallGroup(128,517);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,422,352,2019,2028,124]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^4=1,e^4=d^2,a*b=b*a,a*c=c*a,a*d=d*a,e*a*e^-1=a*d^2,b*c=c*b,e*d*e^-1=b*d=d*b,b*e=e*b,c*d=d*c,e*c*e^-1=a*c*d^2>;
 
// generators/relations
 

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