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G = C2×M4(2).C4  order 128 = 27

Direct product of C2 and M4(2).C4

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

Aliases: C2×M4(2).C4, C24.16Q8, M4(2).28C23, C8.16(C22×C4), C4.53(C23×C4), (C22×C4).66Q8, C23.33(C4⋊C4), (C2×C4).191C24, (C2×C8).245C23, C4.189(C22×D4), (C22×C4).786D4, C4○(M4(2).C4), C23.108(C2×Q8), C8.C4⋊10C22, C22.2(C22×Q8), (C2×M4(2)).16C4, M4(2).25(C2×C4), (C22×C8).246C22, (C23×C4).520C22, (C22×C4).1508C23, (C22×M4(2)).31C2, (C2×M4(2)).340C22, C4.68(C2×C4⋊C4), (C2×C8).94(C2×C4), C2.30(C22×C4⋊C4), C22.39(C2×C4⋊C4), (C2×C4).241(C2×Q8), (C2×C8.C4)⋊21C2, (C2×C4).153(C4⋊C4), (C2×C4).1410(C2×D4), (C22×C4).332(C2×C4), (C2×C4).253(C22×C4), SmallGroup(128,1647)

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

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

Generators and relations for C2×M4(2).C4
 G = < a,b,c,d | a2=b8=c2=1, d4=b4, ab=ba, ac=ca, ad=da, cbc=b5, dbd-1=b-1, dcd-1=b4c >

Subgroups: 300 in 230 conjugacy classes, 172 normal (22 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C8, C8, C2×C4, C2×C4, C23, C23, C23, C2×C8, C2×C8, M4(2), M4(2), C22×C4, C22×C4, C24, C8.C4, C22×C8, C22×C8, C2×M4(2), C2×M4(2), C23×C4, C2×C8.C4, M4(2).C4, C22×M4(2), C22×M4(2), C2×M4(2).C4
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C4⋊C4, C22×C4, C2×D4, C2×Q8, C24, C2×C4⋊C4, C23×C4, C22×D4, C22×Q8, M4(2).C4, C22×C4⋊C4, C2×M4(2).C4

Smallest permutation representation of C2×M4(2).C4
►On 32 points
Generators in S32
(1 23)(2 24)(3 17)(4 18)(5 19)(6 20)(7 21)(8 22)(9 27)(10 28)(11 29)(12 30)(13 31)(14 32)(15 25)(16 26)

(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 5)(3 7)(9 13)(11 15)(17 21)(19 23)(25 29)(27 31)

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

44 conjugacy classes

class 1 2A2B2C2D···2I4A4B4C4D4E···4J8A···8X
order12222···244444···48···8
size11112···211112···24···4

44 irreducible representations

dim111112224
type+++++--
imageC1C2C2C2C4D4Q8Q8M4(2).C4
kernelC2×M4(2).C4C2×C8.C4M4(2).C4C22×M4(2)C2×M4(2)C22×C4C22×C4C24C2
# reps1483164314

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

1600000
0160000
001000
000100
000010
000001
,
0160000
100000
004900
00101300
007401
00134130
,
100000
010000
0016000
0016100
001010
0000016
,
10160000
1670000
001020
0000116
00100160
001013160

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,16,0,0,0,0,0,0,0,4,10,7,13,0,0,9,13,4,4,0,0,0,0,0,13,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,16,1,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16],[10,16,0,0,0,0,16,7,0,0,0,0,0,0,1,0,10,10,0,0,0,0,0,13,0,0,2,1,16,16,0,0,0,16,0,0] >;
 

C2×M4(2).C4 in GAP, Magma, Sage, TeX 

C_2\times M_4(2).C_4
 
% in TeX
 
G:=Group("C2xM4(2).C4");
 
// GroupNames label
 
G:=SmallGroup(128,1647);
 
// by ID
 
G=gap.SmallGroup(128,1647);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,120,723,2804,172,124]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^2=b^8=c^2=1,d^4=b^4,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^5,d*b*d^-1=b^-1,d*c*d^-1=b^4*c>;
 
// generators/relations
 

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