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G = C42.393D4  order 128 = 27

26th non-split extension by C42 of D4 acting via D4/C22=C2

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

Aliases: C42.393D4, (C2×D4)⋊4C8, C4○3(C23⋊C8), C23⋊C8⋊19C2, C23.3(C2×C8), (C23×C4).7C4, C4.7(C22⋊C8), C24.20(C2×C4), (C22×D4).19C4, (C2×C4).13M4(2), C42.12C4⋊2C2, C4.22(C4.D4), C22.10(C22×C8), C22⋊C8.155C22, (C2×C42).146C22, C23.161(C22×C4), (C22×C4).424C23, C22.13(C2×M4(2)), C2.2(C23.C23), (C2×C4×D4).2C2, (C2×C4⋊C4).30C4, (C2×C4).15(C2×C8), C2.8(C2×C22⋊C8), C2.2(C2×C4.D4), (C2×C4).1121(C2×D4), (C22×C4).41(C2×C4), C22.92(C2×C22⋊C4), (C2×C4).310(C22⋊C4), (C2×C22⋊C4).401C22, SmallGroup(128,192)

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

Derived series
C1 — C22 — C42.393D4
Chief series
C1 — C2 — C22 — C2×C4 — C22×C4 — C2×C42 — C2×C4×D4 — C42.393D4
Lower central
C1 — C2 — C22 — C42.393D4
Upper central
C1 — C2×C4 — C2×C42 — C42.393D4
Jennings
C1 — C2 — C22 — C22×C4 — C42.393D4

Generators and relations for C42.393D4
 G = < a,b,c,d | a4=b4=1, c4=b2, d2=b, ab=ba, ac=ca, ad=da, cbc-1=a2b-1, bd=db, dcd-1=b-1c3 >

Subgroups: 332 in 156 conjugacy classes, 62 normal (18 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, C23, C23, C23, C42, C22⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C24, C4×C8, C22⋊C8, C4⋊C8, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C4×D4, C23×C4, C22×D4, C23⋊C8, C42.12C4, C2×C4×D4, C42.393D4
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, C23, C22⋊C4, C2×C8, M4(2), C22×C4, C2×D4, C22⋊C8, C4.D4, C2×C22⋊C4, C22×C8, C2×M4(2), C2×C22⋊C8, C23.C23, C2×C4.D4, C42.393D4

Smallest permutation representation of C42.393D4
►On 32 points
Generators in S32
(1 15 29 17)(2 16 30 18)(3 9 31 19)(4 10 32 20)(5 11 25 21)(6 12 26 22)(7 13 27 23)(8 14 28 24)

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

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

44 conjugacy classes

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

44 irreducible representations

dim111111112244
type++++++
imageC1C2C2C2C4C4C4C8D4M4(2)C4.D4C23.C23
kernelC42.393D4C23⋊C8C42.12C4C2×C4×D4C2×C4⋊C4C23×C4C22×D4C2×D4C42C2×C4C4C2
# reps1421242164422

Matrix representation of C42.393D4 ►in GL6(𝔽17)

400000
040000
0013000
0001300
0000130
0000013
,
400000
040000
000100
001000
000001
000010
,
1620000
610000
0000013
000040
0013000
000400
,
1150000
7160000
000010
000001
000100
001000

G:=sub<GL(6,GF(17))| [4,0,0,0,0,0,0,4,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,13],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[16,6,0,0,0,0,2,1,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,4,0,0,0,4,0,0,0,0,13,0,0,0],[1,7,0,0,0,0,15,16,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0] >;
 

C42.393D4 in GAP, Magma, Sage, TeX 

C_4^2._{393}D_4
 
% in TeX
 
G:=Group("C4^2.393D4");
 
// GroupNames label
 
G:=SmallGroup(128,192);
 
// by ID
 
G=gap.SmallGroup(128,192);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,520,1123,851,172]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^4=b^4=1,c^4=b^2,d^2=b,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=a^2*b^-1,b*d=d*b,d*c*d^-1=b^-1*c^3>;
 
// generators/relations
 

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