C2^4.86D4 - GroupNames

G = C₂⁴.₈₆D₄ order 128 = 2⁷

41^st non-split extension by C₂⁴ of D₄ acting via D₄/C₂=C₂²

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

Aliases: C₂⁴.₈₆D₄, C₂³.₈Q₁₆, C₄⋊C₄.₉₁D₄, (C₂×C₈).₄₅D₄, (C₂×Q₈).₉₁D₄, C₂.₁₈(C₈⋊D₄), (C₂²×C₄).₁₄₈D₄, C₂³.₉₁₂(C₂×D₄), C₂.₃₂(D₄⋊D₄), C₂².₅₅(C₂×Q₁₆), C₄.₁₄₅(C₄⋊D₄), C₂².₄Q₁₆⋊₂₄C₂, C₄.₃₉(C₄.₄D₄), (C₂²×C₈).₇₂C₂², C₂².₂₁₈C₂²≀C₂, C₂.₁₃(C₈.₁₈D₄), C₂².₁₁₀(C₄○D₈), (C₂³×C₄).₂₇₄C₂², C₂.₂₁(C₂²⋊Q₁₆), C₂³.₇Q₈.₁₉C₂, (C₂²×Q₈).₆₄C₂², C₂².₂₂₉(C₄⋊D₄), C₂².₁₃₇(C₈⋊C₂²), (C₂²×C₄).₁₄₄₆C₂³, C₄.₂₀(C₂².D₄), C₂.₉(C₂³.₂₀D₄), C₂.₆(C₂³.₄₈D₄), C₂.₉(C₂³.₁₀D₄), C₂².₁₂₆(C₈.C₂²), C₂².₁₁₅(C₂².D₄), (C₂×C₂.D₈)⋊₈C₂, (C₂×Q₈⋊C₄)⋊₁₄C₂, (C₂×C₄).₁₀₃₈(C₂×D₄), (C₂×C₂²⋊C₈).₂₇C₂, (C₂×C₂²⋊Q₈).₁₄C₂, (C₂×C₄).₇₇₃(C₄○D₄), (C₂×C₄⋊C₄).₁₂₁C₂², SmallGroup(128,768)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series

C₁ — C₂²×C₄ — C₂⁴.₈₆D₄

Chief series

C₁ — C₂ — C₂² — C₂×C₄ — C₂²×C₄ — C₂×C₄⋊C₄ — C₂³.₇Q₈ — C₂⁴.₈₆D₄

Lower central

C₁ — C₂ — C₂²×C₄ — C₂⁴.₈₆D₄

Upper central

C₁ — C₂³ — C₂³×C₄ — C₂⁴.₈₆D₄

Jennings

C₁ — C₂ — C₂ — C₂²×C₄ — C₂⁴.₈₆D₄

Generators and relations for C₂⁴.₈₆D₄
G = < a,b,c,d,e,f | a²=b²=c²=d²=1, e⁴=f²=d, faf^-1=ab=ba, ac=ca, ad=da, eae^-1=abc, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef^-1=bde³ >

Subgroups: 336 in 158 conjugacy classes, 52 normal (44 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₈, C₂×C₄, C₂×C₄, Q₈, C₂³, C₂³, C₂³, C₂²⋊C₄, C₄⋊C₄, C₄⋊C₄, C₂×C₈, C₂×C₈, C₂²×C₄, C₂²×C₄, C₂×Q₈, C₂×Q₈, C₂⁴, C₂.C₄², C₂²⋊C₈, Q₈⋊C₄, C₂.D₈, C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₂×C₄⋊C₄, C₂²⋊Q₈, C₂²×C₈, C₂³×C₄, C₂²×Q₈, C₂².₄Q₁₆, C₂³.₇Q₈, C₂×C₂²⋊C₈, C₂×Q₈⋊C₄, C₂×C₂.D₈, C₂×C₂²⋊Q₈, C₂⁴.₈₆D₄
Quotients: C₁, C₂, C₂², D₄, C₂³, Q₁₆, C₂×D₄, C₄○D₄, C₂²≀C₂, C₄⋊D₄, C₂².D₄, C₄.₄D₄, C₂×Q₁₆, C₄○D₈, C₈⋊C₂², C₈.C₂², C₂³.₁₀D₄, D₄⋊D₄, C₂²⋊Q₁₆, C₈.₁₈D₄, C₈⋊D₄, C₂³.₄₈D₄, C₂³.₂₀D₄, C₂⁴.₈₆D₄

Smallest permutation representation of C₂⁴.₈₆D₄
►On 64 points

Generators in S₆₄

(2 9)(4 11)(6 13)(8 15)(17 40)(18 47)(19 34)(20 41)(21 36)(22 43)(23 38)(24 45)(26 59)(28 61)(30 63)(32 57)(33 49)(35 51)(37 53)(39 55)(42 52)(44 54)(46 56)(48 50)

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

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

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

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

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

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

32 conjugacy classes

class	1	2A	···	2G	2H	2I	4A	4B	4C	4D	4E	4F	4G	···	4N	8A	···	8H
order	1	2	···	2	2	2	4	4	4	4	4	4	4	···	4	8	···	8
size	1	1	···	1	4	4	2	2	2	2	4	4	8	···	8	4	···	4

32 irreducible representations

dim	1	1	1	1	1	1	1	2	2	2	2	2	2	2	2	4	4
type	+	+	+	+	+	+	+	+	+	+	+	+		-		+	-
image	C₁	C₂	C₂	C₂	C₂	C₂	C₂	D₄	D₄	D₄	D₄	D₄	C₄○D₄	Q₁₆	C₄○D₈	C₈⋊C₂²	C₈.C₂²
kernel	C₂⁴.₈₆D₄	C₂².₄Q₁₆	C₂³.₇Q₈	C₂×C₂²⋊C₈	C₂×Q₈⋊C₄	C₂×C₂.D₈	C₂×C₂²⋊Q₈	C₄⋊C₄	C₂×C₈	C₂²×C₄	C₂×Q₈	C₂⁴	C₂×C₄	C₂³	C₂²	C₂²	C₂²
# reps	1	1	1	1	2	1	1	2	2	1	2	1	6	4	4	1	1

Matrix representation of C₂⁴.₈₆D₄ ►in GL₆(𝔽₁₇)

1	0	0	0	0	0
16	16	0	0	0	0
0	0	1	0	0	0
0	0	0	16	0	0
0	0	0	0	1	0
0	0	0	0	11	16

16	0	0	0	0	0
0	16	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	16	0
0	0	0	0	0	16

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	0	0	1	0
0	0	0	0	0	1

16	0	0	0	0	0
0	16	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	1	0
0	0	0	0	0	1

15	0	0	0	0	0
14	9	0	0	0	0
0	0	0	1	0	0
0	0	16	0	0	0
0	0	0	0	14	16
0	0	0	0	10	3

8	16	0	0	0	0
14	9	0	0	0	0
0	0	16	0	0	0
0	0	0	1	0	0
0	0	0	0	5	13
0	0	0	0	6	12

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

C₂⁴.₈₆D₄ in GAP, Magma, Sage, TeX

C_2^4._{86}D_4

% in TeX

G:=Group("C2^4.86D4");

// GroupNames label

G:=SmallGroup(128,768);

// by ID

G=gap.SmallGroup(128,768);

# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,448,141,422,387,394,2804,718,172]);

// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^4=f^2=d,f*a*f^-1=a*b=b*a,a*c=c*a,a*d=d*a,e*a*e^-1=a*b*c,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=b*d*e^3>;

// generators/relations

G = C24.86D4 order 128 = 27

41st non-split extension by C24 of D4 acting via D4/C2=C22

G = C₂⁴.₈₆D₄ order 128 = 2⁷

41^st non-split extension by C₂⁴ of D₄ acting via D₄/C₂=C₂²