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## G = M5(2)⋊1C4order 128 = 27

### 1st semidirect product of M5(2) and C4 acting via C4/C2=C2

p-group, metabelian, nilpotent (class 4), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C8 — M5(2)⋊1C4
 Chief series C1 — C2 — C4 — C2×C4 — C2×C8 — C22×C8 — C2×M5(2) — M5(2)⋊1C4
 Lower central C1 — C2 — C4 — C8 — M5(2)⋊1C4
 Upper central C1 — C22 — C22×C4 — C22×C8 — M5(2)⋊1C4
 Jennings C1 — C2 — C2 — C2 — C2 — C4 — C4 — C2×C8 — M5(2)⋊1C4

Generators and relations for M5(2)⋊1C4
G = < a,b,c | a16=b2=c4=1, bab=a9, cac-1=a7, bc=cb >

Subgroups: 156 in 78 conjugacy classes, 52 normal (22 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C8, C8, C2×C4, C2×C4, C2×C4, C23, C16, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C22×C4, C4.Q8, C2.D8, C2.D8, C2.D8, C2×C16, M5(2), C2×C4⋊C4, C42⋊C2, C22×C8, C163C4, C164C4, C2×C2.D8, C23.25D4, C2×M5(2), M5(2)⋊1C4
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C4⋊C4, D8, Q16, C22×C4, C2×D4, C2×Q8, C2.D8, C2×C4⋊C4, C2×D8, C2×Q16, C2×C2.D8, C16⋊C22, Q32⋊C2, M5(2)⋊1C4

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

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

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

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

32 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E ··· 4L 8A 8B 8C 8D 8E 8F 16A ··· 16H order 1 2 2 2 2 2 4 4 4 4 4 ··· 4 8 8 8 8 8 8 16 ··· 16 size 1 1 1 1 2 2 2 2 2 2 8 ··· 8 2 2 2 2 4 4 4 ··· 4

32 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 4 4 type + + + + + + + - + + - + + - image C1 C2 C2 C2 C2 C2 C4 D4 Q8 D4 D8 Q16 D8 C16⋊C22 Q32⋊C2 kernel M5(2)⋊1C4 C16⋊3C4 C16⋊4C4 C2×C2.D8 C23.25D4 C2×M5(2) M5(2) C2×C8 C2×C8 C22×C4 C2×C4 C2×C4 C23 C2 C2 # reps 1 2 2 1 1 1 8 1 2 1 2 4 2 2 2

Matrix representation of M5(2)⋊1C4 in GL6(𝔽17)

 15 6 0 0 0 0 4 13 0 0 0 0 0 0 6 1 0 0 0 0 6 11 0 0 0 0 1 5 0 2 0 0 4 6 16 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 5 16 0 0 0 0 0 0 1 0 0 0 12 0 0 16
,
 8 3 0 0 0 0 1 9 0 0 0 0 0 0 13 0 4 0 0 0 14 8 10 4 0 0 9 0 4 0 0 0 6 14 7 9

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

M5(2)⋊1C4 in GAP, Magma, Sage, TeX

M_5(2)\rtimes_1C_4
% in TeX

G:=Group("M5(2):1C4");
// GroupNames label

G:=SmallGroup(128,891);
// by ID

G=gap.SmallGroup(128,891);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,112,141,288,723,1123,360,4037,124]);
// Polycyclic

G:=Group<a,b,c|a^16=b^2=c^4=1,b*a*b=a^9,c*a*c^-1=a^7,b*c=c*b>;
// generators/relations

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