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## G = C2×M4(2).C4order 128 = 27

### Direct product of C2 and M4(2).C4

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

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 2A 2B 2C 2D ··· 2I 4A 4B 4C 4D 4E ··· 4J 8A ··· 8X order 1 2 2 2 2 ··· 2 4 4 4 4 4 ··· 4 8 ··· 8 size 1 1 1 1 2 ··· 2 1 1 1 1 2 ··· 2 4 ··· 4

44 irreducible representations

 dim 1 1 1 1 1 2 2 2 4 type + + + + + - - image C1 C2 C2 C2 C4 D4 Q8 Q8 M4(2).C4 kernel C2×M4(2).C4 C2×C8.C4 M4(2).C4 C22×M4(2) C2×M4(2) C22×C4 C22×C4 C24 C2 # reps 1 4 8 3 16 4 3 1 4

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

 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 16 0 0 0 0 1 0 0 0 0 0 0 0 4 9 0 0 0 0 10 13 0 0 0 0 7 4 0 1 0 0 13 4 13 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 16 0 0 0 0 0 16 1 0 0 0 0 1 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 2 0 0 0 0 0 1 16 0 0 10 0 16 0 0 0 10 13 16 0

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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