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G = C42.398D4order 128 = 27

31st non-split extension by C42 of D4 acting via D4/C22=C2

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

Aliases: C42.398D4, C42.599C23, D4⋊C82C2, (C2×D4)⋊6C8, (C4×C8)⋊1C22, D4.5(C2×C8), C4⋊C855C22, (C4×D4).14C4, C4.5(C22×C8), C42.52(C2×C4), C4.5(C2×M4(2)), C4.11(C22⋊C8), (C22×D4).23C4, (C22×C4).655D4, (C2×C4).16M4(2), C4.130(C8⋊C22), C42.12C47C2, (C4×D4).263C22, C22.27(C22⋊C8), (C2×C42).155C22, C23.167(C22⋊C4), C2.2(C42⋊C22), C2.1(C23.37D4), (C2×C4×D4).6C2, (C2×C4⋊C4).38C4, (C2×C4).17(C2×C8), C4⋊C4.177(C2×C4), C2.14(C2×C22⋊C8), (C2×D4).191(C2×C4), (C2×C4).1442(C2×D4), (C2×C4).304(C22×C4), (C22×C4).177(C2×C4), C22.98(C2×C22⋊C4), (C2×C4).237(C22⋊C4), SmallGroup(128,210)

Series: Derived Chief Lower central Upper central Jennings

C1C4 — C42.398D4
C1C2C22C2×C4C42C2×C42C2×C4×D4 — C42.398D4
C1C2C4 — C42.398D4
C1C2×C4C2×C42 — C42.398D4
C1C22C22C42 — C42.398D4

Generators and relations for C42.398D4
 G = < a,b,c,d | a4=b4=1, c4=a2b2, d2=b, ab=ba, cac-1=a-1, ad=da, bc=cb, bd=db, dcd-1=a2b-1c3 >

Subgroups: 332 in 156 conjugacy classes, 62 normal (28 characteristic)
C1, C2 [×3], C2 [×6], C4 [×2], C4 [×4], C4 [×5], C22, C22 [×2], C22 [×18], C8 [×4], C2×C4 [×6], C2×C4 [×4], C2×C4 [×15], D4 [×4], D4 [×6], C23, C23 [×10], C42 [×4], C22⋊C4 [×4], C4⋊C4 [×2], C4⋊C4, C2×C8 [×4], C22×C4 [×3], C22×C4 [×9], C2×D4 [×6], C2×D4 [×3], C24, C4×C8 [×2], C4×C8, C22⋊C8 [×2], C4⋊C8 [×2], C4⋊C8, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C4×D4 [×4], C4×D4 [×2], C23×C4, C22×D4, D4⋊C8 [×4], C42.12C4 [×2], C2×C4×D4, C42.398D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C8 [×4], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], C2×C8 [×6], M4(2) [×2], C22×C4, C2×D4 [×2], C22⋊C8 [×4], C2×C22⋊C4, C22×C8, C2×M4(2), C8⋊C22 [×2], C2×C22⋊C8, C23.37D4, C42⋊C22, C42.398D4

Smallest permutation representation of C42.398D4
On 32 points
Generators in S32
(1 23 27 13)(2 14 28 24)(3 17 29 15)(4 16 30 18)(5 19 31 9)(6 10 32 20)(7 21 25 11)(8 12 26 22)
(1 11 31 17)(2 12 32 18)(3 13 25 19)(4 14 26 20)(5 15 27 21)(6 16 28 22)(7 9 29 23)(8 10 30 24)
(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 10 11 30 31 24 17 8)(2 29 12 23 32 7 18 9)(3 22 13 6 25 16 19 28)(4 5 14 15 26 27 20 21)

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

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

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

44 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I4A4B4C4D4E···4N4O4P4Q4R8A···8P
order122222222244444···444448···8
size111122444411112···244444···4

44 irreducible representations

dim1111111122244
type+++++++
imageC1C2C2C2C4C4C4C8D4D4M4(2)C8⋊C22C42⋊C22
kernelC42.398D4D4⋊C8C42.12C4C2×C4×D4C2×C4⋊C4C4×D4C22×D4C2×D4C42C22×C4C2×C4C4C2
# reps14212421622422

Matrix representation of C42.398D4 in GL6(𝔽17)

100000
010000
0016200
0016100
0000162
0000161
,
1300000
0130000
004000
000400
000040
000004
,
4150000
6130000
000010
0000116
0013800
000400
,
4150000
10130000
000010
000001
004000
000400

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,16,0,0,0,0,2,1,0,0,0,0,0,0,16,16,0,0,0,0,2,1],[13,0,0,0,0,0,0,13,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[4,6,0,0,0,0,15,13,0,0,0,0,0,0,0,0,13,0,0,0,0,0,8,4,0,0,1,1,0,0,0,0,0,16,0,0],[4,10,0,0,0,0,15,13,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,1,0,0,0,0,0,0,1,0,0] >;

C42.398D4 in GAP, Magma, Sage, TeX

C_4^2._{398}D_4
% in TeX

G:=Group("C4^2.398D4");
// GroupNames label

G:=SmallGroup(128,210);
// by ID

G=gap.SmallGroup(128,210);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,387,184,1123,570,136,172]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^4=1,c^4=a^2*b^2,d^2=b,a*b=b*a,c*a*c^-1=a^-1,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=a^2*b^-1*c^3>;
// generators/relations

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