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G = C4≀C2⋊C4order 128 = 27

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

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

Aliases: C4≀C21C4, D43(C4⋊C4), Q83(C4⋊C4), C428(C2×C4), C4○D4.1Q8, C4⋊C4.302D4, C4.140(C4×D4), C429C42C2, (C2×D4).271D4, M4(2)⋊2(C2×C4), (C2×Q8).212D4, C22.30(C4×D4), (C22×C4).20D4, C426C416C2, C4.4(C22⋊Q8), C2.3(D44D4), C23.554(C2×D4), M4(2)⋊C42C2, C22.C421C2, 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

C1C2×C4 — C4≀C2⋊C4
C1C2C22C2×C4C22×C4C2×C4○D4C23.33C23 — C4≀C2⋊C4
C1C2C2×C4 — C4≀C2⋊C4
C1C22C22×C4 — C4≀C2⋊C4
C1C2C2C22×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 [×3], C2 [×4], C4 [×4], C4 [×11], C22 [×3], C22 [×6], C8 [×3], C2×C4 [×6], C2×C4 [×22], D4 [×2], D4 [×5], Q8 [×2], Q8, C23, C23, C42 [×2], C42 [×4], C22⋊C4 [×3], C4⋊C4 [×2], C4⋊C4 [×10], C2×C8 [×2], M4(2) [×2], M4(2) [×3], C22×C4, C22×C4 [×2], C22×C4 [×5], C2×D4, C2×D4, C2×Q8, C4○D4 [×4], C4○D4 [×2], D4⋊C4, Q8⋊C4, C4≀C2 [×4], C4.Q8, C2.D8, C2×C42, C2×C4⋊C4 [×2], C2×C4⋊C4 [×3], C42⋊C2, C42⋊C2, C4×D4 [×3], C4×Q8, C2×M4(2) [×2], C2×C4○D4, C426C4, C22.C42, C429C4, C23.36D4, C2×C4≀C2, M4(2)⋊C4, C23.33C23, C4≀C2⋊C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×6], Q8 [×2], C23, C4⋊C4 [×4], C22×C4, C2×D4 [×3], C2×Q8, C4○D4 [×2], C2×C4⋊C4, C4×D4 [×2], C22≀C2, C22⋊Q8 [×2], C22.D4, C23.8Q8, D44D4, 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 23)(6 22)(7 21)(8 24)(9 14)(10 13)(11 16)(12 15)(25 30)(26 29)(27 32)(28 31)
(1 10)(2 11)(3 12)(4 9)(5 32)(6 29)(7 30)(8 31)(13 19 15 17)(14 20 16 18)(21 28 23 26)(22 25 24 27)
(1 26 10 22)(2 27 11 23)(3 28 12 24)(4 25 9 21)(5 19 32 16)(6 20 29 13)(7 17 30 14)(8 18 31 15)

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

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

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

32 conjugacy classes

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

32 irreducible representations

dim11111111122222244
type++++++++++++-+-
imageC1C2C2C2C2C2C2C2C4D4D4D4D4Q8C4○D4D44D4D4.10D4
kernelC4≀C2⋊C4C426C4C22.C42C429C4C23.36D4C2×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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