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G = 2- 1+4⋊4C4order 128 = 27

3rd semidirect product of 2- 1+4 and C4 acting via C4/C2=C2

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C4 — 2- 1+4⋊4C4
 Chief series C1 — C2 — C22 — C2×C4 — C22×C4 — C2×C4○D4 — C2×2- 1+4 — 2- 1+4⋊4C4
 Lower central C1 — C2 — C4 — 2- 1+4⋊4C4
 Upper central C1 — C22 — C2×C4○D4 — 2- 1+4⋊4C4
 Jennings C1 — C2 — C2 — C2×C4 — 2- 1+4⋊4C4

Generators and relations for 2- 1+44C4
G = < a,b,c,d,e | a4=b2=e4=1, c2=d2=a2, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=a2c, ece-1=cd, ede-1=a2d >

Subgroups: 588 in 363 conjugacy classes, 172 normal (18 characteristic)
C1, C2 [×3], C2 [×8], C4 [×2], C4 [×6], C4 [×10], C22, C22 [×6], C22 [×10], C8 [×4], C2×C4, C2×C4 [×15], C2×C4 [×34], D4 [×14], D4 [×13], Q8 [×10], Q8 [×15], C23 [×3], C23, C42 [×3], C22⋊C4 [×3], C4⋊C4, C4⋊C4 [×3], C4⋊C4 [×3], C2×C8, C2×C8 [×3], C2×C8 [×6], M4(2) [×6], C22×C4 [×3], C22×C4 [×9], C2×D4, C2×D4 [×3], C2×D4 [×3], C2×Q8, C2×Q8 [×15], C2×Q8 [×17], C4○D4 [×20], C4○D4 [×30], D4⋊C4, D4⋊C4 [×3], Q8⋊C4 [×12], C2×C4⋊C4 [×3], C42⋊C2 [×3], C4×D4 [×3], C4×Q8, C22×C8 [×3], C2×M4(2) [×3], C8○D4 [×4], C22×Q8 [×3], C22×Q8, C2×C4○D4, C2×C4○D4 [×3], C2×C4○D4 [×3], 2- 1+4 [×8], 2- 1+4 [×4], C2×Q8⋊C4 [×3], C23.24D4 [×3], C23.36D4 [×3], C23.38D4 [×3], C23.33C23, C2×C8○D4, C2×2- 1+4, 2- 1+44C4
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], D4 [×8], C23 [×15], C22⋊C4 [×16], C22×C4 [×14], C2×D4 [×12], C24, C2×C22⋊C4 [×12], C23×C4, C22×D4 [×2], C22×C22⋊C4, D4○SD16, Q8○D8, 2- 1+44C4

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

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

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

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

44 conjugacy classes

 class 1 2A 2B 2C 2D ··· 2I 2J 2K 4A ··· 4H 4I ··· 4V 8A 8B 8C 8D 8E ··· 8J order 1 2 2 2 2 ··· 2 2 2 4 ··· 4 4 ··· 4 8 8 8 8 8 ··· 8 size 1 1 1 1 2 ··· 2 4 4 2 ··· 2 4 ··· 4 2 2 2 2 4 ··· 4

44 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 4 4 type + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 C2 C4 D4 D4 D4 D4○SD16 Q8○D8 kernel 2- 1+4⋊4C4 C2×Q8⋊C4 C23.24D4 C23.36D4 C23.38D4 C23.33C23 C2×C8○D4 C2×2- 1+4 2- 1+4 C2×D4 C2×Q8 C4○D4 C2 C2 # reps 1 3 3 3 3 1 1 1 16 3 1 4 2 2

Matrix representation of 2- 1+44C4 in GL6(𝔽17)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 8 16 16 2 0 0 0 7 1 0 0 0 0 1 10 0 0 0 10 8 1 9
,
 16 0 0 0 0 0 0 16 0 0 0 0 0 0 0 1 10 0 0 0 6 10 10 14 0 0 8 16 16 2 0 0 7 9 16 8
,
 16 0 0 0 0 0 0 1 0 0 0 0 0 0 9 1 1 15 0 0 0 7 1 0 0 0 0 1 10 0 0 0 7 0 10 8
,
 16 0 0 0 0 0 0 16 0 0 0 0 0 0 0 1 0 0 0 0 16 0 0 0 0 0 16 1 1 15 0 0 16 0 1 16
,
 0 1 0 0 0 0 16 0 0 0 0 0 0 0 10 7 6 11 0 0 7 1 0 6 0 0 16 7 0 9 0 0 7 0 10 6

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

2- 1+44C4 in GAP, Magma, Sage, TeX

`2_-^{1+4}\rtimes_4C_4`
`% in TeX`

`G:=Group("ES-(2,2):4C4");`
`// GroupNames label`

`G:=SmallGroup(128,1630);`
`// by ID`

`G=gap.SmallGroup(128,1630);`
`# by ID`

`G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,456,521,2804,1411,172]);`
`// Polycyclic`

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

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