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G = M4(2)⋊14D4order 128 = 27

1st semidirect product of M4(2) and D4 acting via D4/C22=C2

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

Aliases: M4(2)⋊14D4, C24.111D4, C86(C2×D4), C8⋊D41C2, C82D42C2, C87D425C2, C88D411C2, C4.Q83C22, (C2×D8)⋊18C22, C4⋊C4.24C23, C2.D814C22, C4⋊D455C22, C221(C8⋊C22), (C2×C8).251C23, (C2×C4).259C24, (C22×C8)⋊20C22, (C2×D4).62C23, C23.384(C2×D4), (C22×C4).429D4, C4.153(C22×D4), C22⋊Q867C22, (C2×Q8).50C23, C4.172(C4⋊D4), D4⋊C450C22, C22.19C248C2, Q8⋊C451C22, (C2×SD16)⋊11C22, (C22×M4(2))⋊4C2, M4(2)⋊C412C2, C23.36D440C2, C23.37D433C2, C22.84(C4⋊D4), (C2×M4(2))⋊52C22, (C22×C4).981C23, (C23×C4).551C22, C22.519(C22×D4), C2.15(D8⋊C22), (C22×D4).348C22, C42⋊C2.108C22, C4.26(C2×C4○D4), (C2×C4⋊D4)⋊48C2, (C2×C8⋊C22)⋊18C2, (C2×C4).475(C2×D4), C2.77(C2×C4⋊D4), C2.19(C2×C8⋊C22), (C2×C4).477(C4○D4), (C2×C4⋊C4).591C22, (C2×C4○D4).125C22, SmallGroup(128,1787)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — M4(2)⋊14D4
C1C2C22C2×C4C22×C4C23×C4C22×M4(2) — M4(2)⋊14D4
C1C2C2×C4 — M4(2)⋊14D4
C1C22C23×C4 — M4(2)⋊14D4
C1C2C2C2×C4 — M4(2)⋊14D4

Generators and relations for M4(2)⋊14D4
 G = < a,b,c,d | a8=b2=c4=d2=1, bab=a5, cac-1=dad=a-1, bc=cb, bd=db, dcd=c-1 >

Subgroups: 588 in 274 conjugacy classes, 102 normal (44 characteristic)
C1, C2 [×3], C2 [×8], C4 [×2], C4 [×2], C4 [×7], C22, C22 [×4], C22 [×24], C8 [×4], C8 [×2], C2×C4 [×4], C2×C4 [×4], C2×C4 [×19], D4 [×22], Q8 [×2], C23 [×3], C23 [×14], C42, C22⋊C4 [×9], C4⋊C4 [×2], C4⋊C4 [×2], C4⋊C4 [×3], C2×C8 [×4], C2×C8 [×4], M4(2) [×4], M4(2) [×6], D8 [×4], SD16 [×4], C22×C4 [×6], C22×C4 [×6], C2×D4, C2×D4 [×2], C2×D4 [×16], C2×Q8, C4○D4 [×4], C24, C24, D4⋊C4 [×6], Q8⋊C4 [×2], C4.Q8 [×2], C2.D8 [×2], C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C4×D4 [×2], C22≀C2, C4⋊D4 [×2], C4⋊D4 [×4], C4⋊D4 [×2], C22⋊Q8 [×2], C22.D4, C22×C8 [×2], C2×M4(2) [×2], C2×M4(2) [×2], C2×M4(2) [×4], C2×D8 [×2], C2×SD16 [×2], C8⋊C22 [×4], C23×C4, C22×D4, C22×D4, C2×C4○D4, C23.36D4, C23.37D4, M4(2)⋊C4, C88D4 [×2], C87D4 [×2], C8⋊D4 [×2], C82D4 [×2], C2×C4⋊D4, C22.19C24, C22×M4(2), C2×C8⋊C22, M4(2)⋊14D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×8], C23 [×15], C2×D4 [×12], C4○D4 [×2], C24, C4⋊D4 [×4], C8⋊C22 [×2], C22×D4 [×2], C2×C4○D4, C2×C4⋊D4, C2×C8⋊C22, D8⋊C22, M4(2)⋊14D4

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

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

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

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

32 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J2K4A···4F4G4H···4L8A···8H
order1222222222224···444···48···8
size1111222248882···248···84···4

32 irreducible representations

dim111111111111222244
type++++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2C2D4D4D4C4○D4C8⋊C22D8⋊C22
kernelM4(2)⋊14D4C23.36D4C23.37D4M4(2)⋊C4C88D4C87D4C8⋊D4C82D4C2×C4⋊D4C22.19C24C22×M4(2)C2×C8⋊C22M4(2)C22×C4C24C2×C4C22C2
# reps111122221111431422

Matrix representation of M4(2)⋊14D4 in GL6(𝔽17)

1600000
0160000
0011600
00215160
0001116
0021600
,
100000
010000
00160161
00016152
000010
000001
,
1150000
1160000
0040013
008131313
000040
0000013
,
100000
1160000
0011600
0001600
000001
000010

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,2,0,2,0,0,16,15,1,16,0,0,0,16,1,0,0,0,0,0,16,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,16,15,1,0,0,0,1,2,0,1],[1,1,0,0,0,0,15,16,0,0,0,0,0,0,4,8,0,0,0,0,0,13,0,0,0,0,0,13,4,0,0,0,13,13,0,13],[1,1,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,16,16,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

M4(2)⋊14D4 in GAP, Magma, Sage, TeX

M_4(2)\rtimes_{14}D_4
% in TeX

G:=Group("M4(2):14D4");
// GroupNames label

G:=SmallGroup(128,1787);
// by ID

G=gap.SmallGroup(128,1787);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,120,758,1018,2804,172]);
// Polycyclic

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

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