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G = C4×C4≀C2  order 128 = 27

Direct product of C4 and C4≀C2

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

Aliases: C4×C4≀C2, C43⋊6C2, D4⋊1C42, Q8⋊1C42, C42.458D4, (C4×D4)⋊9C4, (C4×Q8)⋊9C4, C42⋊36(C2×C4), C4.161(C4×D4), C4.1(C2×C42), C22.26(C4×D4), C4○2(C42⋊6C4), C42⋊6C4⋊34C2, (C4×M4(2))⋊21C2, M4(2)⋊12(C2×C4), (C22×C4).670D4, C23.541(C2×D4), C4.1(C42⋊C2), C42○(C42⋊6C4), (C2×C42).1045C22, (C22×C4).1305C23, C42⋊C2.261C22, (C2×M4(2)).306C22, C2.5(C2×C4≀C2), C42○(C2×C4≀C2), (C4×C4○D4).5C2, (C2×C4≀C2).16C2, C4⋊C4.187(C2×C4), C4○D4.13(C2×C4), C2.16(C4×C22⋊C4), (C2×D4).196(C2×C4), (C2×C4).1499(C2×D4), (C2×Q8).179(C2×C4), (C2×C4).536(C4○D4), (C2×C4).348(C22×C4), (C2×C4).399(C22⋊C4), (C2×C4○D4).250C22, C22.117(C2×C22⋊C4), 2-Sylow(GL(3,5)), SmallGroup(128,490)

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

Derived series
C1 — C4 — C4×C4≀C2
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C43 — C4×C4≀C2
Lower central
C1 — C2 — C4 — C4×C4≀C2
Upper central
C1 — C42 — C2×C42 — C4×C4≀C2
Jennings
C1 — C2 — C2 — C22×C4 — C4×C4≀C2

Generators and relations for C4×C4≀C2
 G = < a,b,c,d | a4=b4=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=b-1, bd=db, dcd-1=b-1c >

Subgroups: 300 in 180 conjugacy classes, 80 normal (20 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, C42, 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, C4×C8, C8⋊C4, C4≀C2, C2×C42, C2×C42, C2×C42, C42⋊C2, C42⋊C2, C4×D4, C4×D4, C4×Q8, C2×M4(2), C2×C4○D4, C42⋊6C4, C43, C4×M4(2), C2×C4≀C2, C4×C4○D4, C4×C4≀C2
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C42, C22⋊C4, C22×C4, C2×D4, C4○D4, C4≀C2, C2×C42, C2×C22⋊C4, C42⋊C2, C4×D4, C4×C22⋊C4, C2×C4≀C2, C4×C4≀C2

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

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

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

56 conjugacy classes

class 1 2A2B2C2D2E2F2G4A···4L4M···4AH4AI···4AN8A···8H
order122222224···44···44···48···8
size111122441···12···24···44···4

56 irreducible representations

dim1111111112222
type++++++++
imageC1C2C2C2C2C2C4C4C4D4D4C4○D4C4≀C2
kernelC4×C4≀C2C42⋊6C4C43C4×M4(2)C2×C4≀C2C4×C4○D4C4≀C2C4×D4C4×Q8C42C22×C4C2×C4C4
# reps121121164422416

Matrix representation of C4×C4≀C2 ►in GL3(𝔽17) generated by

400
040
004
,
100
040
0013
,
1600
001
010
,
1600
040
0016
G:=sub<GL(3,GF(17))| [4,0,0,0,4,0,0,0,4],[1,0,0,0,4,0,0,0,13],[16,0,0,0,0,1,0,1,0],[16,0,0,0,4,0,0,0,16] >;
 

C4×C4≀C2 in GAP, Magma, Sage, TeX 

C_4\times C_4\wr C_2
 
% in TeX
 
G:=Group("C4xC4wrC2");
 
// GroupNames label
 
G:=SmallGroup(128,490);
 
// by ID
 
G=gap.SmallGroup(128,490);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,2,-2,112,141,232,100,2019,248,4037]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^4=b^4=c^2=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,b*d=d*b,d*c*d^-1=b^-1*c>;
 
// generators/relations
 

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