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G = C2×C4.9C42  order 128 = 27

Direct product of C2 and C4.9C42

direct product, p-group, metabelian, nilpotent (class 2), monomial

Aliases: C2×C4.9C42, C24.62D4, (C22×C8)⋊6C4, (C2×C42)⋊4C4, C42⋊3(C2×C4), C4○(C4.9C42), (C2×C4).87C42, C4.40(C2×C42), C42⋊C2⋊12C4, (C22×C4).33Q8, C23.57(C4⋊C4), (C2×M4(2))⋊11C4, C23.121(C2×D4), (C22×C4).252D4, (C23×C4).215C22, (C22×C4).644C23, (C22×M4(2)).8C2, C23.189(C22⋊C4), C4.13(C2.C42), C42⋊C2.255C22, (C2×M4(2)).293C22, C22.26(C2.C42), (C2×C8)⋊2(C2×C4), C4.24(C2×C4⋊C4), C22.6(C2×C4⋊C4), (C2×C4).224(C2×D4), C4.78(C2×C22⋊C4), (C2×C4).111(C2×Q8), (C2×C4).119(C4⋊C4), (C2×C42⋊C2).4C2, (C22×C4).477(C2×C4), (C2×C4).514(C22×C4), C22.25(C2×C22⋊C4), C2.7(C2×C2.C42), (C2×C4).111(C22⋊C4), SmallGroup(128,462)

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

Derived series
C1 — C4 — C2×C4.9C42
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C23×C4 — C2×C42⋊C2 — C2×C4.9C42
Lower central
C1 — C4 — C2×C4.9C42
Upper central
C1 — C2×C4 — C2×C4.9C42
Jennings
C1 — C2 — C2 — C22×C4 — C2×C4.9C42

Generators and relations for C2×C4.9C42
 G = < a,b,c,d | a2=b4=c4=d4=1, ab=ba, ac=ca, ad=da, dcd-1=bc=cb, bd=db >

Subgroups: 340 in 202 conjugacy classes, 108 normal (30 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, C23, C23, C23, C42, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), C22×C4, C22×C4, C22×C4, C24, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C42⋊C2, C22×C8, C2×M4(2), C2×M4(2), C23×C4, C4.9C42, C2×C42⋊C2, C22×M4(2), C2×C4.9C42
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, C2.C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C4.9C42, C2×C2.C42, C2×C4.9C42

Smallest permutation representation of C2×C4.9C42
►On 32 points
Generators in S32
(1 16)(2 13)(3 14)(4 15)(5 25)(6 26)(7 27)(8 28)(9 19)(10 20)(11 17)(12 18)(21 31)(22 32)(23 29)(24 30)

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

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

44 conjugacy classes

class 1 2A2B2C2D···2I4A4B4C4D4E···4J4K···4Z8A···8H
order12222···244444···44···48···8
size11112···211112···24···44···4

44 irreducible representations

dim111111112224
type+++++-+
imageC1C2C2C2C4C4C4C4D4Q8D4C4.9C42
kernelC2×C4.9C42C4.9C42C2×C42⋊C2C22×M4(2)C2×C42C42⋊C2C22×C8C2×M4(2)C22×C4C22×C4C24C2
# reps142188445214

Matrix representation of C2×C4.9C42 ►in GL6(𝔽17)

1600000
0160000
0016000
0001600
0000160
0000016
,
1600000
0160000
004000
000400
000040
000004
,
410000
0130000
000010
000001
000100
001000
,
1300000
1540000
001000
0001600
000040
0000013

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

C2×C4.9C42 in GAP, Magma, Sage, TeX 

C_2\times C_4._9C_4^2
 
% in TeX
 
G:=Group("C2xC4.9C4^2");
 
// GroupNames label
 
G:=SmallGroup(128,462);
 
// by ID
 
G=gap.SmallGroup(128,462);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,2,-2,112,141,232,248,1411,4037]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^2=b^4=c^4=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,d*c*d^-1=b*c=c*b,b*d=d*b>;
 
// generators/relations
 

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