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G = M4(2).5Q8  order 128 = 27

3rd non-split extension by M4(2) of Q8 acting via Q8/C4=C2

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

Aliases: M4(2).5Q8, C8.7(C4⋊C4), (C2×C8).15Q8, C8.C4⋊7C4, C4⋊C4.220D4, C4.147(C4×D4), (C2×C8).369D4, C4.19(C4⋊Q8), C22.8(C4×Q8), C4.78(C22⋊Q8), C2.6(D4.3D4), C8○2M4(2).2C2, M4(2).10(C2×C4), C23.267(C4○D4), C22.C42.1C2, M4(2)⋊C4.3C2, (C22×C4).696C23, (C22×C8).222C22, C22.9(C42.C2), C22.141(C4⋊D4), C42⋊C2.275C22, (C2×M4(2)).323C22, C2.18(C23.65C23), C4.45(C2×C4⋊C4), (C2×C8).68(C2×C4), (C2×C4.Q8).7C2, (C2×C4).116(C2×Q8), (C2×C4).61(C4○D4), (C2×C4).1016(C2×D4), (C2×C4⋊C4).81C22, (C2×C8.C4).16C2, (C2×C4).197(C22×C4), SmallGroup(128,683)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series
C1 — C2×C4 — M4(2).5Q8
Chief series
C1 — C2 — C22 — C2×C4 — C22×C4 — C22×C8 — C8○2M4(2) — M4(2).5Q8
Lower central
C1 — C2 — C2×C4 — M4(2).5Q8
Upper central
C1 — C22 — C22×C4 — M4(2).5Q8
Jennings
C1 — C2 — C2 — C22×C4 — M4(2).5Q8

Generators and relations for M4(2).5Q8
 G = < a,b,c,d | a8=b2=c4=1, d2=c2, bab=cac-1=a5, dad-1=ab, cbc-1=a4b, bd=db, dcd-1=a6bc-1 >

Subgroups: 172 in 98 conjugacy classes, 56 normal (26 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, C23, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C2×C8, M4(2), M4(2), C22×C4, C22×C4, C4×C8, C8⋊C4, C4.Q8, C2.D8, C8.C4, C2×C4⋊C4, C42⋊C2, C22×C8, C2×M4(2), C2×M4(2), C22.C42, C8○2M4(2), C2×C4.Q8, M4(2)⋊C4, C2×C8.C4, M4(2).5Q8
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C4⋊C4, C22×C4, C2×D4, C2×Q8, C4○D4, C2×C4⋊C4, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C42.C2, C4⋊Q8, C23.65C23, D4.3D4, M4(2).5Q8

Smallest permutation representation of M4(2).5Q8
►On 64 points
Generators in S64
(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)(33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56)(57 58 59 60 61 62 63 64)

(2 6)(4 8)(9 13)(11 15)(18 22)(20 24)(25 29)(27 31)(33 37)(35 39)(42 46)(44 48)(49 53)(51 55)(58 62)(60 64)

(1 18 41 11)(2 23 42 16)(3 20 43 13)(4 17 44 10)(5 22 45 15)(6 19 46 12)(7 24 47 9)(8 21 48 14)(25 40 55 63)(26 37 56 60)(27 34 49 57)(28 39 50 62)(29 36 51 59)(30 33 52 64)(31 38 53 61)(32 35 54 58)

(1 52 41 30)(2 49 42 27)(3 50 43 28)(4 55 44 25)(5 56 45 26)(6 53 46 31)(7 54 47 32)(8 51 48 29)(9 64 24 33)(10 57 17 34)(11 62 18 39)(12 63 19 40)(13 60 20 37)(14 61 21 38)(15 58 22 35)(16 59 23 36)
 
G:=sub<Sym(64)| (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)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64), (2,6)(4,8)(9,13)(11,15)(18,22)(20,24)(25,29)(27,31)(33,37)(35,39)(42,46)(44,48)(49,53)(51,55)(58,62)(60,64), (1,18,41,11)(2,23,42,16)(3,20,43,13)(4,17,44,10)(5,22,45,15)(6,19,46,12)(7,24,47,9)(8,21,48,14)(25,40,55,63)(26,37,56,60)(27,34,49,57)(28,39,50,62)(29,36,51,59)(30,33,52,64)(31,38,53,61)(32,35,54,58), (1,52,41,30)(2,49,42,27)(3,50,43,28)(4,55,44,25)(5,56,45,26)(6,53,46,31)(7,54,47,32)(8,51,48,29)(9,64,24,33)(10,57,17,34)(11,62,18,39)(12,63,19,40)(13,60,20,37)(14,61,21,38)(15,58,22,35)(16,59,23,36)>;
 
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)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64), (2,6)(4,8)(9,13)(11,15)(18,22)(20,24)(25,29)(27,31)(33,37)(35,39)(42,46)(44,48)(49,53)(51,55)(58,62)(60,64), (1,18,41,11)(2,23,42,16)(3,20,43,13)(4,17,44,10)(5,22,45,15)(6,19,46,12)(7,24,47,9)(8,21,48,14)(25,40,55,63)(26,37,56,60)(27,34,49,57)(28,39,50,62)(29,36,51,59)(30,33,52,64)(31,38,53,61)(32,35,54,58), (1,52,41,30)(2,49,42,27)(3,50,43,28)(4,55,44,25)(5,56,45,26)(6,53,46,31)(7,54,47,32)(8,51,48,29)(9,64,24,33)(10,57,17,34)(11,62,18,39)(12,63,19,40)(13,60,20,37)(14,61,21,38)(15,58,22,35)(16,59,23,36) );
 
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),(33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56),(57,58,59,60,61,62,63,64)], [(2,6),(4,8),(9,13),(11,15),(18,22),(20,24),(25,29),(27,31),(33,37),(35,39),(42,46),(44,48),(49,53),(51,55),(58,62),(60,64)], [(1,18,41,11),(2,23,42,16),(3,20,43,13),(4,17,44,10),(5,22,45,15),(6,19,46,12),(7,24,47,9),(8,21,48,14),(25,40,55,63),(26,37,56,60),(27,34,49,57),(28,39,50,62),(29,36,51,59),(30,33,52,64),(31,38,53,61),(32,35,54,58)], [(1,52,41,30),(2,49,42,27),(3,50,43,28),(4,55,44,25),(5,56,45,26),(6,53,46,31),(7,54,47,32),(8,51,48,29),(9,64,24,33),(10,57,17,34),(11,62,18,39),(12,63,19,40),(13,60,20,37),(14,61,21,38),(15,58,22,35),(16,59,23,36)]])
 

32 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G4H4I4J4K4L8A8B8C8D8E···8J8K8L8M8N
order12222244444444444488888···88888
size11112222224444888822224···48888

32 irreducible representations

dim11111112222224
type++++++++--
imageC1C2C2C2C2C2C4D4D4Q8Q8C4○D4C4○D4D4.3D4
kernelM4(2).5Q8C22.C42C8○2M4(2)C2×C4.Q8M4(2)⋊C4C2×C8.C4C8.C4C4⋊C4C2×C8C2×C8M4(2)C2×C4C23C2
# reps12112182222224

Matrix representation of M4(2).5Q8 ►in GL6(𝔽17)

100000
0160000
0000010
00001210
0001000
00121000
,
1600000
0160000
001000
000100
0000160
0000016
,
400000
040000
0000160
0000016
001000
000100
,
0150000
900000
00111500
0010600
000083
000019

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,12,0,0,0,0,10,10,0,0,0,12,0,0,0,0,10,10,0,0],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,16,0,0,0,0,0,0,16,0,0],[0,9,0,0,0,0,15,0,0,0,0,0,0,0,11,10,0,0,0,0,15,6,0,0,0,0,0,0,8,1,0,0,0,0,3,9] >;
 

M4(2).5Q8 in GAP, Magma, Sage, TeX 

M_4(2)._5Q_8
 
% in TeX
 
G:=Group("M4(2).5Q8");
 
// GroupNames label
 
G:=SmallGroup(128,683);
 
// by ID
 
G=gap.SmallGroup(128,683);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,232,422,100,2019,1018,248,2804,172]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^8=b^2=c^4=1,d^2=c^2,b*a*b=c*a*c^-1=a^5,d*a*d^-1=a*b,c*b*c^-1=a^4*b,b*d=d*b,d*c*d^-1=a^6*b*c^-1>;
 
// generators/relations
 

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