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G = M4(2).29C23order 128 = 27

11st non-split extension by M4(2) of C23 acting via C23/C22=C2

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

Aliases: M4(2).29C23, C8○D4.1C4, C4○D4.59D4, D4(C8.C4), (C2×Q8).24Q8, (C2×D4).33Q8, Q8(C8.C4), D4.11(C4⋊C4), Q8.11(C4⋊C4), C8.17(C22×C4), C4.54(C23×C4), C23.48(C2×Q8), (C2×C8).583C23, (C2×C4).192C24, C4.190(C22×D4), Q8○M4(2).7C2, C22.3(C22×Q8), M4(2).26(C2×C4), M4(2).C411C2, C8.C4.24C22, (C22×C4).909C23, (C22×C8).247C22, (C2×M4(2)).243C22, C4.23(C2×C4⋊C4), (C2×C8).95(C2×C4), (C2×C8○D4).8C2, C22.5(C2×C4⋊C4), (C2×C4).98(C2×Q8), C4○D4.34(C2×C4), C4○D4(C8.C4), C2.31(C22×C4⋊C4), (C2×C8.C4)⋊22C2, (C2×C4).1083(C2×D4), (C2×C4).254(C22×C4), (C2×C4○D4).279C22, SmallGroup(128,1648)

Series: Derived Chief Lower central Upper central Jennings

C1C4 — M4(2).29C23
C1C2C4C2×C4C22×C4C2×C4○D4C2×C8○D4 — M4(2).29C23
C1C2C4 — M4(2).29C23
C1C4C2×C4○D4 — M4(2).29C23
C1C2C2C2×C4 — M4(2).29C23

Generators and relations for M4(2).29C23
 G = < a,b,c,d,e | a8=b2=d2=e2=1, c2=a2, bab=a5, cac-1=a-1b, ad=da, ae=ea, cbc-1=a4b, bd=db, be=eb, cd=dc, ce=ec, ede=a4d >

Subgroups: 276 in 220 conjugacy classes, 170 normal (13 characteristic)
C1, C2, C2 [×7], C4 [×2], C4 [×6], C22, C22 [×6], C22 [×3], C8 [×2], C8 [×6], C8 [×8], C2×C4, C2×C4 [×15], D4 [×12], Q8 [×4], C23 [×3], C2×C8, C2×C8 [×15], C2×C8 [×12], M4(2) [×20], M4(2) [×12], C22×C4 [×3], C2×D4 [×3], C2×Q8, C4○D4 [×8], C8.C4 [×16], C22×C8 [×3], C2×M4(2) [×15], C8○D4 [×8], C8○D4 [×8], C2×C4○D4, C2×C8.C4 [×6], M4(2).C4 [×6], C2×C8○D4, Q8○M4(2) [×2], M4(2).29C23
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], D4 [×4], Q8 [×4], C23 [×15], C4⋊C4 [×16], C22×C4 [×14], C2×D4 [×6], C2×Q8 [×6], C24, C2×C4⋊C4 [×12], C23×C4, C22×D4, C22×Q8, C22×C4⋊C4, M4(2).29C23

Smallest permutation representation of M4(2).29C23
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 5)(3 7)(10 14)(12 16)(17 21)(19 23)(26 30)(28 32)
(1 11 3 13 5 15 7 9)(2 14 4 16 6 10 8 12)(17 31 19 25 21 27 23 29)(18 26 20 28 22 30 24 32)
(1 30)(2 31)(3 32)(4 25)(5 26)(6 27)(7 28)(8 29)(9 22)(10 23)(11 24)(12 17)(13 18)(14 19)(15 20)(16 21)
(17 21)(18 22)(19 23)(20 24)(25 29)(26 30)(27 31)(28 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), (1,5)(3,7)(10,14)(12,16)(17,21)(19,23)(26,30)(28,32), (1,11,3,13,5,15,7,9)(2,14,4,16,6,10,8,12)(17,31,19,25,21,27,23,29)(18,26,20,28,22,30,24,32), (1,30)(2,31)(3,32)(4,25)(5,26)(6,27)(7,28)(8,29)(9,22)(10,23)(11,24)(12,17)(13,18)(14,19)(15,20)(16,21), (17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32)>;

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

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

44 conjugacy classes

class 1 2A2B···2H4A4B4C···4I8A8B8C8D8E···8Z
order122···2444···488888···8
size112···2112···222224···4

44 irreducible representations

dim1111112224
type+++++--+
imageC1C2C2C2C2C4Q8Q8D4M4(2).29C23
kernelM4(2).29C23C2×C8.C4M4(2).C4C2×C8○D4Q8○M4(2)C8○D4C2×D4C2×Q8C4○D4C1
# reps16612163144

Matrix representation of M4(2).29C23 in GL4(𝔽17) generated by

0080
0008
9000
0900
,
16000
01600
0010
0001
,
0010
0001
4000
0400
,
0100
1000
0001
0010
,
1000
01600
0010
00016
G:=sub<GL(4,GF(17))| [0,0,9,0,0,0,0,9,8,0,0,0,0,8,0,0],[16,0,0,0,0,16,0,0,0,0,1,0,0,0,0,1],[0,0,4,0,0,0,0,4,1,0,0,0,0,1,0,0],[0,1,0,0,1,0,0,0,0,0,0,1,0,0,1,0],[1,0,0,0,0,16,0,0,0,0,1,0,0,0,0,16] >;

M4(2).29C23 in GAP, Magma, Sage, TeX

M_4(2)._{29}C_2^3
% in TeX

G:=Group("M4(2).29C2^3");
// GroupNames label

G:=SmallGroup(128,1648);
// by ID

G=gap.SmallGroup(128,1648);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,120,521,2804,172,124]);
// Polycyclic

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

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