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G = C4≀C2⋊C4  order 128 = 27

1st semidirect product of C4≀C2 and C4 acting via C4/C2=C2

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

Aliases: C4≀C2⋊1C4, D4⋊3(C4⋊C4), Q8⋊3(C4⋊C4), C42⋊8(C2×C4), C4○D4.1Q8, C4⋊C4.302D4, C4.140(C4×D4), C42⋊9C4⋊2C2, (C2×D4).271D4, M4(2)⋊2(C2×C4), (C2×Q8).212D4, C22.30(C4×D4), (C22×C4).20D4, C42⋊6C4⋊16C2, C4.4(C22⋊Q8), C2.3(D4⋊4D4), C23.554(C2×D4), M4(2)⋊C4⋊2C2, C22.C42⋊1C2, C22.82C22≀C2, C2.3(D4.10D4), (C2×C42).265C22, (C22×C4).675C23, C23.36D4.7C2, C4.4(C22.D4), C22.49(C22⋊Q8), C42⋊C2.10C22, C2.15(C23.8Q8), (C2×M4(2)).173C22, C23.33C23.1C2, C4.9(C2×C4⋊C4), (C2×C4≀C2).1C2, (C2×C4).9(C2×Q8), C4○D4.4(C2×C4), (C2×C4).985(C2×D4), (C2×C4).50(C4○D4), (C2×C4⋊C4).56C22, (C2×C4).181(C22×C4), (C2×C4○D4).11C22, SmallGroup(128,591)

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

Derived series
C1 — C2×C4 — C4≀C2⋊C4
Chief series
C1 — C2 — C22 — C2×C4 — C22×C4 — C2×C4○D4 — C23.33C23 — C4≀C2⋊C4
Lower central
C1 — C2 — C2×C4 — C4≀C2⋊C4
Upper central
C1 — C22 — C22×C4 — C4≀C2⋊C4
Jennings
C1 — C2 — C2 — C22×C4 — C4≀C2⋊C4

Generators and relations for C4≀C2⋊C4
 G = < a,b,c,d | a4=b2=c4=d4=1, bab=a-1, ac=ca, ad=da, cbc-1=a-1b, bd=db, dcd-1=a-1c-1 >

Subgroups: 300 in 151 conjugacy classes, 56 normal (38 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, M4(2), M4(2), C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C4○D4, D4⋊C4, Q8⋊C4, C4≀C2, C4.Q8, C2.D8, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C42⋊C2, C42⋊C2, C4×D4, C4×Q8, C2×M4(2), C2×C4○D4, C42⋊6C4, C22.C42, C42⋊9C4, C23.36D4, C2×C4≀C2, M4(2)⋊C4, C23.33C23, C4≀C2⋊C4
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C4⋊C4, C22×C4, C2×D4, C2×Q8, C4○D4, C2×C4⋊C4, C4×D4, C22≀C2, C22⋊Q8, C22.D4, C23.8Q8, D4⋊4D4, D4.10D4, C4≀C2⋊C4

Smallest permutation representation of C4≀C2⋊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 20)(2 19)(3 18)(4 17)(5 21)(6 24)(7 23)(8 22)(9 16)(10 15)(11 14)(12 13)(25 32)(26 31)(27 30)(28 29)

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

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

32 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E···4R4S4T8A8B8C8D
order1222222244444···4448888
size1111224422224···4888888

32 irreducible representations

dim11111111122222244
type++++++++++++-+-
imageC1C2C2C2C2C2C2C2C4D4D4D4D4Q8C4○D4D4⋊4D4D4.10D4
kernelC4≀C2⋊C4C42⋊6C4C22.C42C42⋊9C4C23.36D4C2×C4≀C2M4(2)⋊C4C23.33C23C4≀C2C4⋊C4C22×C4C2×D4C2×Q8C4○D4C2×C4C2C2
# reps11111111822112422

Matrix representation of C4≀C2⋊C4 ►in GL6(𝔽17)

100000
010000
000100
0016000
0000016
000010
,
1600000
0160000
0000016
000010
000100
0016000
,
0160000
100000
001000
000100
000001
0000160
,
0130000
1300000
0000160
000001
0016000
000100

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

C4≀C2⋊C4 in GAP, Magma, Sage, TeX 

C_4\wr C_2\rtimes C_4
 
% in TeX
 
G:=Group("C4wrC2:C4");
 
// GroupNames label
 
G:=SmallGroup(128,591);
 
// by ID
 
G=gap.SmallGroup(128,591);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,232,422,2019,1018,248,2804,718,172]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^4=b^2=c^4=d^4=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=a^-1*b,b*d=d*b,d*c*d^-1=a^-1*c^-1>;
 
// generators/relations
 

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