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G = (C2×C4)⋊M4(2)  order 128 = 27

The semidirect product of C2×C4 and M4(2) acting via M4(2)/C22=C4

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

Aliases: (C2×C4)⋊6M4(2), (C23×C4).15C4, C24.106(C2×C4), (C22×C4).199D4, C22⋊C8.119C22, C22.26(C23⋊C4), C24.4C4.11C2, C2.7(C24.4C4), C23.164(C22×C4), (C22×C4).427C23, (C23×C4).198C22, C22.16(C2×M4(2)), C23.166(C22⋊C4), C22.9(C4.10D4), C22.M4(2)⋊12C2, (C2×C4⋊C4).32C4, C2.7(C2×C23⋊C4), (C22×C4⋊C4).8C2, (C2×C4).1124(C2×D4), (C22×C4).69(C2×C4), C2.5(C2×C4.10D4), (C2×C4⋊C4).735C22, (C2×C4).69(C22⋊C4), C22.145(C2×C22⋊C4), SmallGroup(128,195)

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

Derived series
C1 — C23 — (C2×C4)⋊M4(2)
Chief series
C1 — C2 — C22 — C2×C4 — C22×C4 — C23×C4 — C22×C4⋊C4 — (C2×C4)⋊M4(2)
Lower central
C1 — C2 — C23 — (C2×C4)⋊M4(2)
Upper central
C1 — C22 — C23×C4 — (C2×C4)⋊M4(2)
Jennings
C1 — C2 — C22 — C22×C4 — (C2×C4)⋊M4(2)

Generators and relations for (C2×C4)⋊M4(2)
 G = < a,b,c,d | a2=b4=c8=d2=1, ab=ba, cac-1=ab2, ad=da, cbc-1=ab-1, bd=db, dcd=c5 >

Subgroups: 308 in 156 conjugacy classes, 52 normal (16 characteristic)
C1, C2, C2, C4, C22, C22, C22, C8, C2×C4, C2×C4, C23, C23, C4⋊C4, C2×C8, M4(2), C22×C4, C22×C4, C22×C4, C24, C22⋊C8, C22⋊C8, C2×C4⋊C4, C2×C4⋊C4, C2×M4(2), C23×C4, C22.M4(2), C24.4C4, C22×C4⋊C4, (C2×C4)⋊M4(2)
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, M4(2), C22×C4, C2×D4, C23⋊C4, C4.10D4, C2×C22⋊C4, C2×M4(2), C24.4C4, C2×C23⋊C4, C2×C4.10D4, (C2×C4)⋊M4(2)

Smallest permutation representation of (C2×C4)⋊M4(2)
►On 32 points
Generators in S32
(2 32)(4 26)(6 28)(8 30)(10 18)(12 20)(14 22)(16 24)

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

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

32 conjugacy classes

class 1 2A2B2C2D···2I4A4B4C4D4E···4N8A···8H
order12222···244444···48···8
size11112···222224···48···8

32 irreducible representations

dim1111112244
type++++++-
imageC1C2C2C2C4C4D4M4(2)C23⋊C4C4.10D4
kernel(C2×C4)⋊M4(2)C22.M4(2)C24.4C4C22×C4⋊C4C2×C4⋊C4C23×C4C22×C4C2×C4C22C22
# reps1421444822

Matrix representation of (C2×C4)⋊M4(2) ►in GL6(𝔽17)

1600000
0160000
001000
000100
0090160
00140016
,
1600000
010000
0016200
0016100
0009415
00414013
,
0160000
400000
004010
00001116
0000130
0011670
,
100000
0160000
0016000
0001600
0000160
0000016

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,9,14,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[16,0,0,0,0,0,0,1,0,0,0,0,0,0,16,16,0,4,0,0,2,1,9,14,0,0,0,0,4,0,0,0,0,0,15,13],[0,4,0,0,0,0,16,0,0,0,0,0,0,0,4,0,0,1,0,0,0,0,0,16,0,0,1,11,13,7,0,0,0,16,0,0],[1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16] >;
 

(C2×C4)⋊M4(2) in GAP, Magma, Sage, TeX 

(C_2\times C_4)\rtimes M_4(2)
 
% in TeX
 
G:=Group("(C2xC4):M4(2)");
 
// GroupNames label
 
G:=SmallGroup(128,195);
 
// by ID
 
G=gap.SmallGroup(128,195);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,232,1430,1123,851,172]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^2=b^4=c^8=d^2=1,a*b=b*a,c*a*c^-1=a*b^2,a*d=d*a,c*b*c^-1=a*b^-1,b*d=d*b,d*c*d=c^5>;
 
// generators/relations
 

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