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G = M4(2).27D4order 128 = 27

8th non-split extension by M4(2) of D4 acting via D4/C4=C2

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

Aliases: M4(2).27D4, C8.9(C4⋊C4), (C2×C8).17Q8, C8.C49C4, (C2×C8).357D4, C4.149(C4×D4), C22⋊C4.5Q8, C23.5(C2×Q8), C22.7(C4⋊Q8), C22.10(C4×Q8), C4.202(C4⋊D4), C82M4(2).4C2, M4(2).12(C2×C4), C4.18(C42.C2), M4(2).C4.1C2, M4(2)⋊4C4.1C2, (C22×C8).224C22, (C22×C4).698C23, C22.27(C22⋊Q8), C42⋊C2.277C22, (C2×M4(2)).205C22, C2.20(C23.65C23), C4.47(C2×C4⋊C4), (C2×C8).70(C2×C4), (C2×C4).118(C2×Q8), (C2×C4).63(C4○D4), (C2×C4).1018(C2×D4), (C2×C8.C4).18C2, (C2×C4).199(C22×C4), SmallGroup(128,685)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — M4(2).27D4
C1C2C4C2×C4C22×C4C22×C8C82M4(2) — M4(2).27D4
C1C2C2×C4 — M4(2).27D4
C1C4C22×C4 — M4(2).27D4
C1C2C2C22×C4 — M4(2).27D4

Generators and relations for M4(2).27D4
 G = < a,b,c,d | a8=b2=1, c4=a4, d2=a2, bab=a5, ac=ca, dad-1=a-1b, bc=cb, dbd-1=a4b, dcd-1=c3 >

Subgroups: 140 in 92 conjugacy classes, 54 normal (34 characteristic)
C1, C2, C2 [×3], C4 [×4], C4 [×2], C22 [×3], C22, C8 [×4], C8 [×8], C2×C4 [×6], C2×C4 [×2], C23, C42, C22⋊C4 [×2], C4⋊C4, C2×C8 [×6], C2×C8 [×5], M4(2) [×6], M4(2) [×7], C22×C4, C4×C8, C8⋊C4, C8.C4 [×4], C8.C4 [×6], C42⋊C2, C22×C8, C2×M4(2), C2×M4(2) [×4], M4(2)⋊4C4 [×2], C82M4(2), C2×C8.C4 [×2], M4(2).C4 [×2], M4(2).27D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], Q8 [×4], C23, C4⋊C4 [×4], C22×C4, C2×D4 [×2], C2×Q8 [×2], C4○D4 [×2], C2×C4⋊C4, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C42.C2, C4⋊Q8, C23.65C23, M4(2).27D4

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

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

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

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

32 conjugacy classes

class 1 2A2B2C2D4A4B4C4D4E4F4G4H4I8A8B8C8D8E···8J8K···8R
order1222244444444488888···88···8
size1122211222444422224···48···8

32 irreducible representations

dim111111222224
type+++++-+-+
imageC1C2C2C2C2C4Q8D4Q8D4C4○D4M4(2).27D4
kernelM4(2).27D4M4(2)⋊4C4C82M4(2)C2×C8.C4M4(2).C4C8.C4C22⋊C4C2×C8C2×C8M4(2)C2×C4C1
# reps121228222244

Matrix representation of M4(2).27D4 in GL4(𝔽17) generated by

9000
16800
0088
0009
,
131300
8400
0044
00913
,
9000
0900
00150
00015
,
00150
00015
2000
0200
G:=sub<GL(4,GF(17))| [9,16,0,0,0,8,0,0,0,0,8,0,0,0,8,9],[13,8,0,0,13,4,0,0,0,0,4,9,0,0,4,13],[9,0,0,0,0,9,0,0,0,0,15,0,0,0,0,15],[0,0,2,0,0,0,0,2,15,0,0,0,0,15,0,0] >;

M4(2).27D4 in GAP, Magma, Sage, TeX

M_4(2)._{27}D_4
% in TeX

G:=Group("M4(2).27D4");
// GroupNames label

G:=SmallGroup(128,685);
// by ID

G=gap.SmallGroup(128,685);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,232,422,100,2019,1018,248,2804,172,124]);
// Polycyclic

G:=Group<a,b,c,d|a^8=b^2=1,c^4=a^4,d^2=a^2,b*a*b=a^5,a*c=c*a,d*a*d^-1=a^-1*b,b*c=c*b,d*b*d^-1=a^4*b,d*c*d^-1=c^3>;
// generators/relations

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