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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

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 2A 2B 2C 2D ··· 2I 4A 4B 4C 4D 4E ··· 4N 8A ··· 8H order 1 2 2 2 2 ··· 2 4 4 4 4 4 ··· 4 8 ··· 8 size 1 1 1 1 2 ··· 2 2 2 2 2 4 ··· 4 8 ··· 8

32 irreducible representations

 dim 1 1 1 1 1 1 2 2 4 4 type + + + + + + - image C1 C2 C2 C2 C4 C4 D4 M4(2) C23⋊C4 C4.10D4 kernel (C2×C4)⋊M4(2) C22.M4(2) C24.4C4 C22×C4⋊C4 C2×C4⋊C4 C23×C4 C22×C4 C2×C4 C22 C22 # reps 1 4 2 1 4 4 4 8 2 2

Matrix representation of (C2×C4)⋊M4(2) 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 9 0 16 0 0 0 14 0 0 16
,
 16 0 0 0 0 0 0 1 0 0 0 0 0 0 16 2 0 0 0 0 16 1 0 0 0 0 0 9 4 15 0 0 4 14 0 13
,
 0 16 0 0 0 0 4 0 0 0 0 0 0 0 4 0 1 0 0 0 0 0 11 16 0 0 0 0 13 0 0 0 1 16 7 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

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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