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G = M5(2)⋊C4order 128 = 27

5th semidirect product of M5(2) and C4 acting via C4/C2=C2

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

Aliases: M5(2)⋊5C4, (C2×C8).2C8, C4.9(C4⋊C8), C8.34(C4⋊C4), (C2×C8).27Q8, (C2×C8).374D4, (C22×C4).4C8, C22.3(C4×C8), C4.3(C8⋊C4), C22.4(C4⋊C8), (C22×C8).15C4, C23.28(C2×C8), (C2×C4).54C42, (C2×C42).12C4, C4.31(C22⋊C8), C8.50(C22⋊C4), C2.3(C23.C8), (C2×M5(2)).8C2, (C2×C4).70M4(2), C22.38(C22⋊C8), (C22×C8).368C22, C4.27(C2.C42), C2.15(C22.7C42), (C2×C4).75(C2×C8), (C2×C8).116(C2×C4), (C2×C8⋊C4).18C2, (C2×C4).105(C4⋊C4), (C22×C4).468(C2×C4), (C2×C4).385(C22⋊C4), SmallGroup(128,109)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — M5(2)⋊C4
C1C2C4C2×C4C2×C8C22×C8C2×C8⋊C4 — M5(2)⋊C4
C1C2C22 — M5(2)⋊C4
C1C2×C4C22×C8 — M5(2)⋊C4
C1C2C2C2C2C4C4C22×C8 — M5(2)⋊C4

Generators and relations for M5(2)⋊C4
 G = < a,b,c | a16=b2=c4=1, bab=a9, cac-1=ab, bc=cb >

Subgroups: 104 in 70 conjugacy classes, 44 normal (24 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×4], C4 [×2], C22 [×3], C22 [×2], C8 [×4], C8 [×2], C2×C4 [×6], C2×C4 [×4], C23, C16 [×4], C42 [×2], C2×C8 [×2], C2×C8 [×6], C2×C8 [×2], C22×C4, C22×C4 [×2], C8⋊C4 [×2], C2×C16 [×2], M5(2) [×4], M5(2) [×2], C2×C42, C22×C8 [×2], C2×C8⋊C4, C2×M5(2) [×2], M5(2)⋊C4
Quotients: C1, C2 [×3], C4 [×6], C22, C8 [×4], C2×C4 [×3], D4 [×3], Q8, C42, C22⋊C4 [×3], C4⋊C4 [×3], C2×C8 [×2], M4(2) [×2], C2.C42, C4×C8, C8⋊C4, C22⋊C8 [×2], C4⋊C8 [×2], C22.7C42, C23.C8 [×2], M5(2)⋊C4

Smallest permutation representation of M5(2)⋊C4
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)
(1 43)(2 36)(3 45)(4 38)(5 47)(6 40)(7 33)(8 42)(9 35)(10 44)(11 37)(12 46)(13 39)(14 48)(15 41)(16 34)(17 53)(18 62)(19 55)(20 64)(21 57)(22 50)(23 59)(24 52)(25 61)(26 54)(27 63)(28 56)(29 49)(30 58)(31 51)(32 60)
(1 29 43 49)(2 58 44 22)(3 23 45 59)(4 52 46 32)(5 17 47 53)(6 62 48 26)(7 27 33 63)(8 56 34 20)(9 21 35 57)(10 50 36 30)(11 31 37 51)(12 60 38 24)(13 25 39 61)(14 54 40 18)(15 19 41 55)(16 64 42 28)

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

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

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

44 conjugacy classes

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

44 irreducible representations

dim111111112224
type++++-
imageC1C2C2C4C4C4C8C8D4Q8M4(2)C23.C8
kernelM5(2)⋊C4C2×C8⋊C4C2×M5(2)M5(2)C2×C42C22×C8C2×C8C22×C4C2×C8C2×C8C2×C4C2
# reps112822883144

Matrix representation of M5(2)⋊C4 in GL6(𝔽17)

13110000
1140000
0091012
0015832
0043812
0097149
,
1600000
0160000
0016065
00016158
000010
000001
,
010000
1600000
00163154
0051710
00001010
0000127

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

M5(2)⋊C4 in GAP, Magma, Sage, TeX

M_5(2)\rtimes C_4
% in TeX

G:=Group("M5(2):C4");
// GroupNames label

G:=SmallGroup(128,109);
// by ID

G=gap.SmallGroup(128,109);
# by ID

G:=PCGroup([7,-2,2,-2,2,2,-2,-2,56,85,120,1430,1018,136,124]);
// Polycyclic

G:=Group<a,b,c|a^16=b^2=c^4=1,b*a*b=a^9,c*a*c^-1=a*b,b*c=c*b>;
// generators/relations

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