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G = C24.9Q8  order 128 = 27

8th non-split extension by C24 of Q8 acting via Q8/C2=C22

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

Aliases: C24.9Q8, (C2×C8).101D4, C8.1(C22⋊C4), C23.80(C2×Q8), (C22×C4).44Q8, C23.27(C4⋊C4), (C22×C4).272D4, C4.183(C4⋊D4), C4.C42⋊19C2, (C2×M4(2)).12C4, C4.56(C42⋊C2), C24.4C4.17C2, (C22×C8).213C22, (C23×C4).238C22, (C22×M4(2)).2C2, C22.23(C22⋊Q8), C2.8(M4(2).C4), (C22×C4).1333C23, C2.20(C23.7Q8), (C2×M4(2)).155C22, (C2×C8.C4)⋊2C2, (C2×C4).43(C4⋊C4), (C2×C8).143(C2×C4), C4.90(C2×C22⋊C4), C22.94(C2×C4⋊C4), (C2×C4).1321(C2×D4), (C2×C4).555(C4○D4), (C2×C4).531(C22×C4), (C22×C4).265(C2×C4), SmallGroup(128,543)

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

Derived series
C1 — C2×C4 — C24.9Q8
Chief series
C1 — C2 — C4 — C2×C4 — C22×C4 — C23×C4 — C22×M4(2) — C24.9Q8
Lower central
C1 — C2 — C2×C4 — C24.9Q8
Upper central
C1 — C2×C4 — C23×C4 — C24.9Q8
Jennings
C1 — C2 — C2 — C22×C4 — C24.9Q8

Generators and relations for C24.9Q8
 G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e4=d, f2=bde2, ab=ba, faf-1=ac=ca, eae-1=ad=da, bc=cb, fbf-1=bd=db, be=eb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e3 >

Subgroups: 236 in 136 conjugacy classes, 60 normal (16 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C8, C8, C2×C4, C2×C4, C2×C4, C23, C23, C23, C2×C8, C2×C8, M4(2), C22×C4, C22×C4, C22×C4, C24, C22⋊C8, C8.C4, C22×C8, C2×M4(2), C2×M4(2), C23×C4, C4.C42, C24.4C4, C2×C8.C4, C22×M4(2), C24.9Q8
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, C4○D4, C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C4⋊D4, C22⋊Q8, C23.7Q8, M4(2).C4, C24.9Q8

Smallest permutation representation of C24.9Q8
►On 32 points
Generators in S32
(2 6)(4 8)(9 31)(10 28)(11 25)(12 30)(13 27)(14 32)(15 29)(16 26)(17 21)(19 23)

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

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

(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)(25 29)(26 30)(27 31)(28 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 28 24 12 5 32 20 16)(2 31 17 15 6 27 21 11)(3 26 18 10 7 30 22 14)(4 29 19 13 8 25 23 9)
 
G:=sub<Sym(32)| (2,6)(4,8)(9,31)(10,28)(11,25)(12,30)(13,27)(14,32)(15,29)(16,26)(17,21)(19,23), (1,18)(2,19)(3,20)(4,21)(5,22)(6,23)(7,24)(8,17)(9,31)(10,32)(11,25)(12,26)(13,27)(14,28)(15,29)(16,30), (1,18)(2,19)(3,20)(4,21)(5,22)(6,23)(7,24)(8,17)(9,27)(10,28)(11,29)(12,30)(13,31)(14,32)(15,25)(16,26), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32), (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), (1,28,24,12,5,32,20,16)(2,31,17,15,6,27,21,11)(3,26,18,10,7,30,22,14)(4,29,19,13,8,25,23,9)>;
 
G:=Group( (2,6)(4,8)(9,31)(10,28)(11,25)(12,30)(13,27)(14,32)(15,29)(16,26)(17,21)(19,23), (1,18)(2,19)(3,20)(4,21)(5,22)(6,23)(7,24)(8,17)(9,31)(10,32)(11,25)(12,26)(13,27)(14,28)(15,29)(16,30), (1,18)(2,19)(3,20)(4,21)(5,22)(6,23)(7,24)(8,17)(9,27)(10,28)(11,29)(12,30)(13,31)(14,32)(15,25)(16,26), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32), (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), (1,28,24,12,5,32,20,16)(2,31,17,15,6,27,21,11)(3,26,18,10,7,30,22,14)(4,29,19,13,8,25,23,9) );
 
G=PermutationGroup([[(2,6),(4,8),(9,31),(10,28),(11,25),(12,30),(13,27),(14,32),(15,29),(16,26),(17,21),(19,23)], [(1,18),(2,19),(3,20),(4,21),(5,22),(6,23),(7,24),(8,17),(9,31),(10,32),(11,25),(12,26),(13,27),(14,28),(15,29),(16,30)], [(1,18),(2,19),(3,20),(4,21),(5,22),(6,23),(7,24),(8,17),(9,27),(10,28),(11,29),(12,30),(13,31),(14,32),(15,25),(16,26)], [(1,5),(2,6),(3,7),(4,8),(9,13),(10,14),(11,15),(12,16),(17,21),(18,22),(19,23),(20,24),(25,29),(26,30),(27,31),(28,32)], [(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)], [(1,28,24,12,5,32,20,16),(2,31,17,15,6,27,21,11),(3,26,18,10,7,30,22,14),(4,29,19,13,8,25,23,9)]])
 

32 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E4F4G4H8A···8H8I···8P
order12222222444444448···88···8
size11112244111122444···48···8

32 irreducible representations

dim111111222224
type+++++++--
imageC1C2C2C2C2C4D4D4Q8Q8C4○D4M4(2).C4
kernelC24.9Q8C4.C42C24.4C4C2×C8.C4C22×M4(2)C2×M4(2)C2×C8C22×C4C22×C4C24C2×C4C2
# reps122218421144

Matrix representation of C24.9Q8 ►in GL6(𝔽17)

1600000
210000
001000
0001600
000010
0000016
,
100000
010000
001000
000100
0000160
0000016
,
1600000
0160000
001000
000100
000010
000001
,
100000
010000
0016000
0001600
0000160
0000016
,
1300000
840000
000100
0013000
0000013
0000160
,
13130000
040000
000010
000001
004000
000400

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

C24.9Q8 in GAP, Magma, Sage, TeX 

C_2^4._9Q_8
 
% in TeX
 
G:=Group("C2^4.9Q8");
 
// GroupNames label
 
G:=SmallGroup(128,543);
 
// by ID
 
G=gap.SmallGroup(128,543);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,64,422,723,2019,248,124]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^4=d,f^2=b*d*e^2,a*b=b*a,f*a*f^-1=a*c=c*a,e*a*e^-1=a*d=d*a,b*c=c*b,f*b*f^-1=b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=e^3>;
 
// generators/relations
 

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