C2^4.89D4 - GroupNames

G = C₂⁴.₈₉D₄ order 128 = 2⁷

44^th non-split extension by C₂⁴ of D₄ acting via D₄/C₂=C₂²

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

Aliases: C₂⁴.₈₉D₄, C₂³.₁₉SD₁₆, (C₂×C₈).₁₆₄D₄, C₂.₂₂(C₈⋊₈D₄), C₂.₂₂(C₈⋊D₄), C₂³.₉₃₂(C₂×D₄), (C₂²×C₄).₁₆₂D₄, C₂².₄Q₁₆⋊₃₁C₂, C₄.₅₆(C₄.₄D₄), C₄.₂₀(C₄²⋊₂C₂), C₂².₁₂₄(C₄○D₈), (C₂³×C₄).₂₇₇C₂², (C₂²×C₈).₃₂₅C₂², C₂².₁₀₅(C₂×SD₁₆), C₂³.₇Q₈.₂₂C₂, C₂².₂₅₃(C₄⋊D₄), C₂².₁₅₂(C₈⋊C₂²), (C₂²×C₄).₁₄₆₆C₂³, C₂.₈(C₂³.₄₆D₄), C₂.₈(C₂³.₄₇D₄), C₄.₁₁₂(C₂².D₄), C₂.₁₂(C₂³.₁₉D₄), C₂.₁₂(C₂³.₂₀D₄), C₂.₁₂(C₂³.₁₁D₄), C₂².₁₄₁(C₈.C₂²), C₂².₁₂₂(C₂².D₄), (C₂×C₄.Q₈)⋊₂₃C₂, (C₂×C₄).₁₃₇₅(C₂×D₄), (C₂×C₂²⋊C₈).₃₈C₂, (C₂×C₄).₆₂₈(C₄○D₄), (C₂×C₄⋊C₄).₁₅₁C₂², SmallGroup(128,809)

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

Derived series

C₁ — C₂²×C₄ — C₂⁴.₈₉D₄

Chief series

C₁ — C₂ — C₂² — C₂×C₄ — C₂²×C₄ — C₂×C₄⋊C₄ — C₂³.₇Q₈ — C₂⁴.₈₉D₄

Lower central

C₁ — C₂ — C₂²×C₄ — C₂⁴.₈₉D₄

Upper central

C₁ — C₂³ — C₂³×C₄ — C₂⁴.₈₉D₄

Jennings

C₁ — C₂ — C₂ — C₂²×C₄ — C₂⁴.₈₉D₄

Generators and relations for C₂⁴.₈₉D₄
G = < a,b,c,d,e,f | a²=b²=c²=d²=1, e⁴=d, f²=db=bd, eae^-1=ab=ba, ac=ca, ad=da, faf^-1=abc, bc=cb, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef^-1=e³ >

Subgroups: 288 in 128 conjugacy classes, 48 normal (44 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₈, C₂×C₄, C₂×C₄, C₂³, C₂³, C₂³, C₂²⋊C₄, C₄⋊C₄, C₂×C₈, C₂×C₈, C₂²×C₄, C₂²×C₄, C₂⁴, C₂.C₄², C₂²⋊C₈, C₄.Q₈, C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₂²×C₈, C₂³×C₄, C₂².₄Q₁₆, C₂³.₇Q₈, C₂×C₂²⋊C₈, C₂×C₄.Q₈, C₂⁴.₈₉D₄
Quotients: C₁, C₂, C₂², D₄, C₂³, SD₁₆, C₂×D₄, C₄○D₄, C₄⋊D₄, C₂².D₄, C₄.₄D₄, C₄²⋊₂C₂, C₂×SD₁₆, C₄○D₈, C₈⋊C₂², C₈.C₂², C₂³.₁₁D₄, C₈⋊₈D₄, C₈⋊D₄, C₂³.₄₆D₄, C₂³.₁₉D₄, C₂³.₄₇D₄, C₂³.₂₀D₄, C₂⁴.₈₉D₄

Smallest permutation representation of C₂⁴.₈₉D₄
►On 64 points

Generators in S₆₄

(2 20)(4 22)(6 24)(8 18)(9 58)(10 40)(11 60)(12 34)(13 62)(14 36)(15 64)(16 38)(26 50)(28 52)(30 54)(32 56)(33 48)(35 42)(37 44)(39 46)(41 61)(43 63)(45 57)(47 59)

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

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

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

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

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

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

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	2	2	2	2	2	2	4	4
type	+	+	+	+	+	+	+	+				+	-
image	C₁	C₂	C₂	C₂	C₂	D₄	D₄	D₄	C₄○D₄	SD₁₆	C₄○D₈	C₈⋊C₂²	C₈.C₂²
kernel	C₂⁴.₈₉D₄	C₂².₄Q₁₆	C₂³.₇Q₈	C₂×C₂²⋊C₈	C₂×C₄.Q₈	C₂×C₈	C₂²×C₄	C₂⁴	C₂×C₄	C₂³	C₂²	C₂²	C₂²
# reps	1	3	2	1	1	2	1	1	10	4	4	1	1

Matrix representation of C₂⁴.₈₉D₄ ►in GL₆(𝔽₁₇)

1	0	0	0	0	0
4	16	0	0	0	0
0	0	1	0	0	0
0	0	1	16	0	0
0	0	0	0	1	0
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	1	0
0	0	0	0	0	1

1	0	0	0	0	0
0	1	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	16

1	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	16	0
0	0	0	0	0	16

13	2	0	0	0	0
0	4	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	2	0
0	0	0	0	0	8

1	8	0	0	0	0
4	16	0	0	0	0
0	0	1	15	0	0
0	0	0	16	0	0
0	0	0	0	0	8
0	0	0	0	2	0

G:=sub<GL(6,GF(17))| [1,4,0,0,0,0,0,16,0,0,0,0,0,0,1,1,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[13,0,0,0,0,0,2,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,2,0,0,0,0,0,0,8],[1,4,0,0,0,0,8,16,0,0,0,0,0,0,1,0,0,0,0,0,15,16,0,0,0,0,0,0,0,2,0,0,0,0,8,0] >;

C₂⁴.₈₉D₄ in GAP, Magma, Sage, TeX

C_2^4._{89}D_4

% in TeX

G:=Group("C2^4.89D4");

// GroupNames label

G:=SmallGroup(128,809);

// by ID

G=gap.SmallGroup(128,809);

# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,560,141,422,387,58,718,172]);

// Polycyclic

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

// generators/relations

G = C24.89D4 order 128 = 27

44th non-split extension by C24 of D4 acting via D4/C2=C22

G = C₂⁴.₈₉D₄ order 128 = 2⁷

44^th non-split extension by C₂⁴ of D₄ acting via D₄/C₂=C₂²