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G = M4(2):12D4order 128 = 27

6th semidirect product of M4(2) and D4 acting via D4/C4=C2

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

Aliases: M4(2):12D4, C42.113D4, C4:1(C4.D4), C24.11(C2xC4), C22.58(C4xD4), C4.17(C4:1D4), C4.78(C4:D4), (C4xM4(2)):24C2, (C22xD4).14C4, C4:M4(2):28C2, C4.63(C4.4D4), C23.197(C22xC4), (C22xC4).699C23, (C2xC42).319C22, (C22xD4).46C22, (C2xM4(2)).208C22, C2.17(C24.3C22), (C2xC4:1D4).7C2, (C2xC4).64(C4oD4), (C2xC4).1351(C2xD4), (C2xC4.D4):22C2, C2.28(C2xC4.D4), (C22xC4).283(C2xC4), (C2xC4).257(C22:C4), C22.287(C2xC22:C4), SmallGroup(128,697)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — M4(2):12D4
C1C2C4C2xC4C22xC4C2xC42C4xM4(2) — M4(2):12D4
C1C2C23 — M4(2):12D4
C1C22C2xC42 — M4(2):12D4
C1C2C2C22xC4 — M4(2):12D4

Generators and relations for M4(2):12D4
 G = < a,b,c,d | a8=b2=c4=d2=1, bab=a5, ac=ca, dad=ab, bc=cb, bd=db, dcd=c-1 >

Subgroups: 516 in 198 conjugacy classes, 60 normal (16 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2xC4, C2xC4, C2xC4, D4, C23, C23, C42, C42, C2xC8, M4(2), M4(2), C22xC4, C22xC4, C2xD4, C24, C4xC8, C8:C4, C4.D4, C4:C8, C2xC42, C4:1D4, C2xM4(2), C22xD4, C4xM4(2), C2xC4.D4, C4:M4(2), C2xC4:1D4, M4(2):12D4
Quotients: C1, C2, C4, C22, C2xC4, D4, C23, C22:C4, C22xC4, C2xD4, C4oD4, C4.D4, C2xC22:C4, C4xD4, C4:D4, C4.4D4, C4:1D4, C24.3C22, C2xC4.D4, M4(2):12D4

Smallest permutation representation of M4(2):12D4
On 32 points
Generators in S32
(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 19)(10 24)(11 21)(12 18)(13 23)(14 20)(15 17)(16 22)
(1 19 31 13)(2 20 32 14)(3 21 25 15)(4 22 26 16)(5 23 27 9)(6 24 28 10)(7 17 29 11)(8 18 30 12)
(1 31)(3 29)(4 8)(5 27)(7 25)(10 24)(11 15)(12 22)(14 20)(16 18)(17 21)(26 30)

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

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

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

32 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I4A···4H4I4J8A···8H8I8J8K8L
order12222222224···4448···88888
size11112288882···2444···48888

32 irreducible representations

dim1111112224
type++++++++
imageC1C2C2C2C2C4D4D4C4oD4C4.D4
kernelM4(2):12D4C4xM4(2)C2xC4.D4C4:M4(2)C2xC4:1D4C22xD4C42M4(2)C2xC4C4
# reps1141184444

Matrix representation of M4(2):12D4 in GL6(Z)

-100000
0-10000
000010
000001
000100
00-1000
,
100000
010000
001000
000100
0000-10
00000-1
,
-1-20000
110000
000100
00-1000
000001
0000-10
,
-1-20000
010000
00-1000
000100
0000-10
000001

G:=sub<GL(6,Integers())| [-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1],[-1,1,0,0,0,0,-2,1,0,0,0,0,0,0,0,-1,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,1,0],[-1,0,0,0,0,0,-2,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1] >;

M4(2):12D4 in GAP, Magma, Sage, TeX

M_4(2)\rtimes_{12}D_4
% in TeX

G:=Group("M4(2):12D4");
// GroupNames label

G:=SmallGroup(128,697);
// by ID

G=gap.SmallGroup(128,697);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,288,422,100,2019,1018,2028,124]);
// Polycyclic

G:=Group<a,b,c,d|a^8=b^2=c^4=d^2=1,b*a*b=a^5,a*c=c*a,d*a*d=a*b,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

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