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G = C42.371D4  order 128 = 27

4th non-split extension by C42 of D4 acting via D4/C22=C2

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

Aliases: C42.371D4, C4○(C23⋊C8), (C22×C4)⋊4C8, C23⋊C8.11C2, C24.19(C2×C4), C23.16(C2×C8), (C23×C4).14C4, (C2×C42).14C4, C4.22(C22⋊C8), (C2×C4).72M4(2), C22.8(C22×C8), C42.12C4⋊1C2, C22⋊C8.153C22, C23.159(C22×C4), (C2×C42).145C22, (C22×C4).422C23, C22.11(C2×M4(2)), C4○(C22.M4(2)), C22.M4(2)⋊18C2, C2.1(C23.C23), C2.1(M4(2).8C22), (C2×C4⋊C4).29C4, (C2×C4).37(C2×C8), C2.6(C2×C22⋊C8), (C2×C4).1119(C2×D4), (C2×C22⋊C4).16C4, (C2×C4⋊C4).732C22, (C22×C4).470(C2×C4), (C2×C42⋊C2).1C2, C22.90(C2×C22⋊C4), (C2×C4).388(C22⋊C4), (C2×C22⋊C4).399C22, SmallGroup(128,190)

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

Derived series
C1 — C22 — C42.371D4
Chief series
C1 — C2 — C22 — C2×C4 — C22×C4 — C2×C42 — C2×C42⋊C2 — C42.371D4
Lower central
C1 — C2 — C22 — C42.371D4
Upper central
C1 — C2×C4 — C2×C42 — C42.371D4
Jennings
C1 — C2 — C22 — C22×C4 — C42.371D4

Generators and relations for C42.371D4
 G = < a,b,c,d | a4=b4=1, c4=b2, d2=b, ab=ba, ac=ca, ad=da, cbc-1=a2b, bd=db, dcd-1=b-1c3 >

Subgroups: 244 in 132 conjugacy classes, 60 normal (26 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, C23, C23, C23, C42, C42, C22⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C22×C4, C24, C4×C8, C22⋊C8, C4⋊C8, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C23×C4, C23⋊C8, C22.M4(2), C42.12C4, C2×C42⋊C2, C42.371D4
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, C23, C22⋊C4, C2×C8, M4(2), C22×C4, C2×D4, C22⋊C8, C2×C22⋊C4, C22×C8, C2×M4(2), C2×C22⋊C8, C23.C23, M4(2).8C22, C42.371D4

Smallest permutation representation of C42.371D4
►On 32 points
Generators in S32
(1 15 25 23)(2 16 26 24)(3 9 27 17)(4 10 28 18)(5 11 29 19)(6 12 30 20)(7 13 31 21)(8 14 32 22)

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

44 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E···4N4O···4T8A···8P
order1222222244444···44···48···8
size1111224411112···24···44···4

44 irreducible representations

dim11111111112244
type++++++
imageC1C2C2C2C2C4C4C4C4C8D4M4(2)C23.C23M4(2).8C22
kernelC42.371D4C23⋊C8C22.M4(2)C42.12C4C2×C42⋊C2C2×C42C2×C22⋊C4C2×C4⋊C4C23×C4C22×C4C42C2×C4C2C2
# reps122212222164422

Matrix representation of C42.371D4 ►in GL6(𝔽17)

100000
010000
004000
000400
000040
000004
,
1300000
0130000
000100
0016000
0011001
0006160
,
1420000
630000
0071120
00610015
00281011
0081567
,
3150000
15140000
00106150
00610015
0000711
0010117

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[13,0,0,0,0,0,0,13,0,0,0,0,0,0,0,16,11,0,0,0,1,0,0,6,0,0,0,0,0,16,0,0,0,0,1,0],[14,6,0,0,0,0,2,3,0,0,0,0,0,0,7,6,2,8,0,0,11,10,8,15,0,0,2,0,10,6,0,0,0,15,11,7],[3,15,0,0,0,0,15,14,0,0,0,0,0,0,10,6,0,1,0,0,6,10,0,0,0,0,15,0,7,11,0,0,0,15,11,7] >;
 

C42.371D4 in GAP, Magma, Sage, TeX 

C_4^2._{371}D_4
 
% in TeX
 
G:=Group("C4^2.371D4");
 
// GroupNames label
 
G:=SmallGroup(128,190);
 
// by ID
 
G=gap.SmallGroup(128,190);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,184,1123,851,172]);
 
// 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,b*d=d*b,d*c*d^-1=b^-1*c^3>;
 
// generators/relations
 

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