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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

C1C22 — C22.M5(2)
C1C2C4C2×C4C2×C8C22×C8C2×C4⋊C8 — C22.M5(2)
C1C2C22 — C22.M5(2)
C1C2×C4C22×C8 — C22.M5(2)
C1C2C2C2C2C4C2×C4C22×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 >

2C2
2C2
2C4
2C4
2C4
2C22
2C22
4C4
2C2×C4
2C2×C4
2C2×C4
2C2×C4
2C8
2C8
2C2×C4
4C8
4C2×C4
2C42
2C2×C8
2C42
4C16
4C2×C8
4C16
4C2×C8
2C2×C16
2C4⋊C8
2C4⋊C8
2C2×C16

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

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

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

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

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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