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G = C22.M5(2)  order 128 = 27

2nd non-split extension by C22 of M5(2) acting via M5(2)/C2×C8=C2

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

Aliases: C22.3M5(2), (C2×C4)⋊C16, (C2×C8).292D4, (C2×C42).7C4, (C22×C8).7C4, (C22×C4).3C8, C22⋊C16.1C2, C23.27(C2×C8), C22.3(C2×C16), C4.39(C23⋊C4), C2.4(C22⋊C16), C2.2(C23.C8), (C2×C4).55M4(2), (C22×C8).3C22, C4.20(C4.10D4), C22.34(C22⋊C8), C2.2(C22.M4(2)), (C2×C4⋊C8).8C2, (C22×C4).428(C2×C4), (C2×C4).378(C22⋊C4), SmallGroup(128,54)

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

Derived series
C1 — C22 — C22.M5(2)
Chief series
C1 — C2 — C4 — C2×C4 — C2×C8 — C22×C8 — C2×C4⋊C8 — C22.M5(2)
Lower central
C1 — C2 — C22 — C22.M5(2)
Upper central
C1 — C2×C4 — C22×C8 — C22.M5(2)
Jennings
C1 — C2 — C2 — C2 — C2 — C4 — C2×C4 — C22×C8 — C22.M5(2)

Generators and relations for C22.M5(2)
 G = < a,b,c,d | a2=b2=c16=1, d2=b, cac-1=ab=ba, ad=da, bc=cb, bd=db, dcd-1=abc9 >

C1
C2
C2
C2
2C2
2C2
C22
C22
C4
C4
C22
2C4
2C4
2C4
2C22
2C22
4C4
C2×C4
C2×C4
C23
C2×C4
C2×C4
2C2×C4
2C2×C4
2C2×C4
2C2×C4
2C8
2C8
2C2×C4
4C8
4C2×C4
C22×C4
C2×C8
C22×C4
C2×C8
C22×C4
2C42
2C2×C8
2C42
4C16
4C2×C8
4C16
4C2×C8
C22×C8
C2×C42
C22×C8
2C2×C16
2C4⋊C8
2C4⋊C8
2C2×C16
C22⋊C16
C2×C4⋊C8
C22⋊C16
C22.M5(2)

Smallest permutation representation of C22.M5(2)
►On 64 points
Generators in S64
(1 9)(2 48)(3 11)(4 34)(5 13)(6 36)(7 15)(8 38)(10 40)(12 42)(14 44)(16 46)(17 25)(18 62)(19 27)(20 64)(21 29)(22 50)(23 31)(24 52)(26 54)(28 56)(30 58)(32 60)(33 41)(35 43)(37 45)(39 47)(49 57)(51 59)(53 61)(55 63)

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

(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)

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

44 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G4H4I4J8A···8H8I8J8K8L16A···16P
order12222244444444448···8888816···16
size11112211112244442···244444···4

44 irreducible representations

dim1111111222444
type+++++-
imageC1C2C2C4C4C8C16D4M4(2)M5(2)C23⋊C4C4.10D4C23.C8
kernelC22.M5(2)C22⋊C16C2×C4⋊C8C2×C42C22×C8C22×C4C2×C4C2×C8C2×C4C22C4C4C2
# reps12122816224112

Matrix representation of C22.M5(2) ►in GL6(𝔽17)

100000
010000
001000
000100
00312160
00315016
,
100000
010000
0016000
0001600
0000160
0000016
,
770000
3100000
00111040
007699
0016493
00165148
,
1600000
210000
0013000
000400
00014130
0010384

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,3,3,0,0,0,1,12,15,0,0,0,0,16,0,0,0,0,0,0,16],[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],[7,3,0,0,0,0,7,10,0,0,0,0,0,0,11,7,16,16,0,0,10,6,4,5,0,0,4,9,9,14,0,0,0,9,3,8],[16,2,0,0,0,0,0,1,0,0,0,0,0,0,13,0,0,10,0,0,0,4,14,3,0,0,0,0,13,8,0,0,0,0,0,4] >;
 

C22.M5(2) in GAP, Magma, Sage, TeX 

C_2^2.M_5(2)
 
% in TeX
 
G:=Group("C2^2.M5(2)");
 
// GroupNames label
 
G:=SmallGroup(128,54);
 
// by ID
 
G=gap.SmallGroup(128,54);
 
# by ID
 
G:=PCGroup([7,-2,2,-2,2,-2,2,-2,56,85,120,422,346,136,124]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^2=b^2=c^16=1,d^2=b,c*a*c^-1=a*b=b*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=a*b*c^9>;
 
// generators/relations
 

Export

Subgroup lattice of C22.M5(2) in TeX

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