ES-(2,2):4C4 - GroupNames

G = 2_-¹⁺⁴⋊₄C₄ order 128 = 2⁷

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

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

Aliases: 2_-¹⁺⁴⋊₄C₄, Q₈ ○(D₄⋊C₄), D₄ ○(Q₈⋊C₄), C₄○D₄.₅₁D₄, C₄.₉(C₂³×C₄), C₂.₁(Q₈○D₈), (C₂×D₄).₃₄₄D₄, (C₂×Q₈).₂₆₃D₄, C₄⋊C₄.₃₄₅C₂³, (C₂×C₄).₁₇₉C₂⁴, (C₂×C₈).₄₆₉C₂³, C₂.₂(D₄○SD₁₆), D₄.₂₁(C₂²×C₄), C₄.₁₄₄(C₂²×D₄), C₂³.₄₃₂(C₂×D₄), Q₈.₂₁(C₂²×C₄), D₄.₂₀(C₂²⋊C₄), (C₂×D₄).₃₆₃C₂³, Q₈.₂₀(C₂²⋊C₄), (C₂×Q₈).₃₃₆C₂³, C₂³.₃₆D₄⋊₃₇C₂, C₂³.₃₈D₄⋊₃₁C₂, C₂³.₂₄D₄⋊₃₆C₂, (C₂²×C₈).₄₂₅C₂², (C₂²×C₄).₉₀₃C₂³, C₂².₁₂₉(C₂²×D₄), (C₂×2_-¹⁺⁴).₅C₂, D₄⋊C₄.₁₉₃C₂², C₄²⋊C₂.₇₇C₂², C₂³.₃₃C₂³⋊₃C₂, Q₈⋊C₄.₁₉₃C₂², (C₂²×Q₈).₂₅₇C₂², (C₂×M₄(2)).₃₃₄C₂², C₄○D₄⋊₂(C₂×C₄), (C₂×C₈○D₄)⋊₁₈C₂, (C₂×Q₈)⋊₁₈(C₂×C₄), (C₂×Q₈)○(D₄⋊C₄), (C₂×C₄).₄₄₇(C₂×D₄), C₄.₃₄(C₂×C₂²⋊C₄), (C₂×Q₈⋊C₄)⋊₅₁C₂, (C₂×C₄).₆₃(C₂²×C₄), C₂².₇(C₂×C₂²⋊C₄), (C₂×C₄⋊C₄).₅₇₁C₂², (C₂×C₄○D₄).₈₅C₂², C₂.₄₁(C₂²×C₂²⋊C₄), SmallGroup(128,1630)

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⁺⁴⋊₄C₄
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, C₂, C₄, C₄, C₄, C₂², C₂², C₂², C₈, C₂×C₄, C₂×C₄, C₂×C₄, D₄, D₄, Q₈, Q₈, C₂³, C₂³, C₄², C₂²⋊C₄, C₄⋊C₄, C₄⋊C₄, C₄⋊C₄, C₂×C₈, C₂×C₈, C₂×C₈, M₄(2), C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂×D₄, C₂×Q₈, C₂×Q₈, C₂×Q₈, C₄○D₄, C₄○D₄, D₄⋊C₄, D₄⋊C₄, Q₈⋊C₄, C₂×C₄⋊C₄, C₄²⋊C₂, C₄×D₄, C₄×Q₈, C₂²×C₈, C₂×M₄(2), C₈○D₄, C₂²×Q₈, C₂²×Q₈, C₂×C₄○D₄, C₂×C₄○D₄, C₂×C₄○D₄, 2_-¹⁺⁴, 2_-¹⁺⁴, C₂×Q₈⋊C₄, C₂³.₂₄D₄, C₂³.₃₆D₄, C₂³.₃₈D₄, C₂³.₃₃C₂³, C₂×C₈○D₄, C₂×2_-¹⁺⁴, 2_-¹⁺⁴⋊₄C₄
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, D₄, C₂³, C₂²⋊C₄, C₂²×C₄, C₂×D₄, C₂⁴, C₂×C₂²⋊C₄, C₂³×C₄, C₂²×D₄, C₂²×C₂²⋊C₄, D₄○SD₁₆, Q₈○D₈, 2_-¹⁺⁴⋊₄C₄

Smallest permutation representation of 2_-¹⁺⁴⋊₄C₄
►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

type

image

C₁

C₂

C₄

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

# reps

Matrix representation of 2_-¹⁺⁴⋊₄C₄ ►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⁺⁴⋊₄C₄ 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

G = 2- 1+4⋊4C4 order 128 = 27

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

G = 2_-¹⁺⁴⋊₄C₄ order 128 = 2⁷

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