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G = D4.C42order 128 = 27

1st non-split extension by D4 of C42 acting via C42/C2×C4=C2

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

Aliases: D4.1C42, Q8.1C42, C42.380D4, C4≀C24C4, (C4×D4)⋊10C4, C427(C2×C4), (C4×Q8)⋊10C4, C4.162(C4×D4), C4.2(C2×C42), C424C46C2, C22.27(C4×D4), C426C411C2, M4(2)⋊13(C2×C4), (C4×M4(2))⋊22C2, (C22×C4).129D4, C23.542(C2×D4), C4.2(C42⋊C2), (C2×C42).234C22, (C22×C4).1306C23, C2.5(C42⋊C22), C42⋊C2.262C22, (C2×M4(2)).307C22, (C2×C4≀C2).5C2, (C4×C4○D4).6C2, C4⋊C4.188(C2×C4), C4○D4.14(C2×C4), C2.17(C4×C22⋊C4), (C2×D4).197(C2×C4), (C2×C4).1500(C2×D4), (C2×Q8).180(C2×C4), (C2×C4).537(C4○D4), (C2×C4).349(C22×C4), (C2×C4).326(C22⋊C4), (C2×C4○D4).251C22, C22.118(C2×C22⋊C4), SmallGroup(128,491)

Series: Derived Chief Lower central Upper central Jennings

C1C4 — D4.C42
C1C2C22C23C22×C4C2×C42C424C4 — D4.C42
C1C2C4 — D4.C42
C1C2×C4C2×C42 — D4.C42
C1C2C2C22×C4 — D4.C42

Generators and relations for D4.C42
 G = < a,b,c,d | a4=b2=c4=d4=1, bab=dad-1=a-1, dcd-1=ac=ca, cbc-1=ab, dbd-1=a2b >

Subgroups: 284 in 160 conjugacy classes, 76 normal (20 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×2], C4 [×2], C4 [×12], C22, C22 [×2], C22 [×6], C8 [×4], C2×C4 [×2], C2×C4 [×8], C2×C4 [×21], D4 [×2], D4 [×5], Q8 [×2], Q8, C23, C23, C42 [×2], C42 [×6], C42 [×5], C22⋊C4 [×3], C4⋊C4 [×2], C4⋊C4 [×2], C2×C8 [×2], M4(2) [×4], M4(2) [×2], C22×C4, C22×C4 [×2], C22×C4 [×5], C2×D4, C2×D4, C2×Q8, C4○D4 [×4], C4○D4 [×2], C2.C42 [×2], C4×C8, C8⋊C4, C4≀C2 [×8], C2×C42, C2×C42 [×2], C2×C42, C42⋊C2, C42⋊C2, C4×D4 [×2], C4×D4 [×2], C4×Q8 [×2], C2×M4(2) [×2], C2×C4○D4, C426C4 [×2], C424C4, C4×M4(2), C2×C4≀C2 [×2], C4×C4○D4, D4.C42
Quotients: C1, C2 [×7], C4 [×12], C22 [×7], C2×C4 [×18], D4 [×4], C23, C42 [×4], C22⋊C4 [×4], C22×C4 [×3], C2×D4 [×2], C4○D4 [×2], C2×C42, C2×C22⋊C4, C42⋊C2, C4×D4 [×4], C4×C22⋊C4, C42⋊C22 [×2], D4.C42

Smallest permutation representation of D4.C42
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 29)(2 32)(3 31)(4 30)(5 12)(6 11)(7 10)(8 9)(13 26)(14 25)(15 28)(16 27)(17 24)(18 23)(19 22)(20 21)
(1 14 11 18)(2 15 12 19)(3 16 9 20)(4 13 10 17)(5 23 30 27)(6 24 31 28)(7 21 32 25)(8 22 29 26)
(1 22 9 26)(2 21 10 25)(3 24 11 28)(4 23 12 27)(5 14 30 20)(6 13 31 19)(7 16 32 18)(8 15 29 17)

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

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

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

44 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E···4N4O···4AB8A···8H
order1222222244444···44···48···8
size1111224411112···24···44···4

44 irreducible representations

dim1111111112224
type++++++++
imageC1C2C2C2C2C2C4C4C4D4D4C4○D4C42⋊C22
kernelD4.C42C426C4C424C4C4×M4(2)C2×C4≀C2C4×C4○D4C4≀C2C4×D4C4×Q8C42C22×C4C2×C4C2
# reps12112116442244

Matrix representation of D4.C42 in GL6(𝔽17)

1600000
0160000
004000
000400
0000130
0000013
,
1620000
010000
000018
0000016
001800
0001600
,
100000
1160000
0041500
00161300
0000169
0000131
,
1380000
040000
0000160
0000016
001000
000100

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,13,0,0,0,0,0,0,13],[16,0,0,0,0,0,2,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,8,16,0,0,1,0,0,0,0,0,8,16,0,0],[1,1,0,0,0,0,0,16,0,0,0,0,0,0,4,16,0,0,0,0,15,13,0,0,0,0,0,0,16,13,0,0,0,0,9,1],[13,0,0,0,0,0,8,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,16,0,0,0,0,0,0,16,0,0] >;

D4.C42 in GAP, Magma, Sage, TeX

D_4.C_4^2
% in TeX

G:=Group("D4.C4^2");
// GroupNames label

G:=SmallGroup(128,491);
// by ID

G=gap.SmallGroup(128,491);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,112,141,232,723,100,2019,248,4037]);
// Polycyclic

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

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