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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, C41(C4.D4), C24.11(C2×C4), C22.58(C4×D4), C4.17(C41D4), C4.78(C4⋊D4), (C4×M4(2))⋊24C2, (C22×D4).14C4, C4⋊M4(2)⋊28C2, C4.63(C4.4D4), C23.197(C22×C4), (C22×C4).699C23, (C2×C42).319C22, (C22×D4).46C22, (C2×M4(2)).208C22, C2.17(C24.3C22), (C2×C41D4).7C2, (C2×C4).64(C4○D4), (C2×C4).1351(C2×D4), (C2×C4.D4)⋊22C2, C2.28(C2×C4.D4), (C22×C4).283(C2×C4), (C2×C4).257(C22⋊C4), C22.287(C2×C22⋊C4), SmallGroup(128,697)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — M4(2)⋊12D4
C1C2C4C2×C4C22×C4C2×C42C4×M4(2) — M4(2)⋊12D4
C1C2C23 — M4(2)⋊12D4
C1C22C2×C42 — M4(2)⋊12D4
C1C2C2C22×C4 — 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 [×2], C2 [×6], C4 [×2], C4 [×6], C4 [×2], C22, C22 [×2], C22 [×22], C8 [×6], C2×C4 [×2], C2×C4 [×8], C2×C4 [×4], D4 [×24], C23, C23 [×20], C42 [×2], C42 [×2], C2×C8 [×4], M4(2) [×4], M4(2) [×6], C22×C4, C22×C4 [×2], C2×D4 [×28], C24 [×4], C4×C8, C8⋊C4, C4.D4 [×8], C4⋊C8 [×2], C2×C42, C41D4 [×4], C2×M4(2) [×4], C22×D4 [×6], C4×M4(2), C2×C4.D4 [×4], C4⋊M4(2), C2×C41D4, M4(2)⋊12D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×8], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×4], C4○D4 [×2], C4.D4 [×4], C2×C22⋊C4, C4×D4 [×2], C4⋊D4 [×2], C4.4D4, C41D4, C24.3C22, C2×C4.D4 [×2], 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++++++++
imageC1C2C2C2C2C4D4D4C4○D4C4.D4
kernelM4(2)⋊12D4C4×M4(2)C2×C4.D4C4⋊M4(2)C2×C41D4C22×D4C42M4(2)C2×C4C4
# 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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