C2^4.88D4 - GroupNames

G = C₂⁴.₈₈D₄ order 128 = 2⁷

43^rd non-split extension by C₂⁴ of D₄ acting via D₄/C₂=C₂²

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

Aliases: C₂⁴.₈₈D₄, C₂³.₉Q₁₆, (C₂×C₈).₅₄D₄, C₂.₁₆(C₈⋊₂D₄), (C₂²×C₄).₁₆₁D₄, C₂³.₉₃₁(C₂×D₄), C₂².₆₁(C₂×Q₁₆), C₂².₄Q₁₆⋊₃₀C₂, C₄.₅₅(C₄.₄D₄), C₄.₁₉(C₄²⋊₂C₂), C₂.₁₆(C₈.₁₈D₄), C₂².₁₂₃(C₄○D₈), (C₂³×C₄).₂₇₆C₂², (C₂²×C₈).₁₁₆C₂², C₂³.₇Q₈.₂₁C₂, C₂².₂₅₂(C₄⋊D₄), C₂².₁₅₁(C₈⋊C₂²), (C₂²×C₄).₁₄₆₅C₂³, C₂.₈(C₂³.₄₈D₄), C₂.₁₁(C₂³.₁₁D₄), C₂.₁₁(C₂³.₁₉D₄), C₄.₁₁₁(C₂².D₄), C₂².₁₂₁(C₂².D₄), (C₂×C₂.D₈)⋊₁₁C₂, (C₂×C₄).₁₃₇₄(C₂×D₄), (C₂×C₂²⋊C₈).₂₉C₂, (C₂×C₄).₆₂₇(C₄○D₄), (C₂×C₄⋊C₄).₁₅₀C₂², SmallGroup(128,808)

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

Derived series

C₁ — C₂²×C₄ — C₂⁴.₈₈D₄

Chief series

C₁ — C₂ — C₂² — C₂×C₄ — C₂²×C₄ — C₂×C₄⋊C₄ — C₂³.₇Q₈ — C₂⁴.₈₈D₄

Lower central

C₁ — C₂ — C₂²×C₄ — C₂⁴.₈₈D₄

Upper central

C₁ — C₂³ — C₂³×C₄ — C₂⁴.₈₈D₄

Jennings

C₁ — C₂ — C₂ — C₂²×C₄ — C₂⁴.₈₈D₄

Generators and relations for C₂⁴.₈₈D₄
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=de³ >

Subgroups: 288 in 128 conjugacy classes, 48 normal (22 characteristic)
C₁, C₂, C₂, C₂, C₄, C₄, C₄, C₂², C₂², C₂², C₈, C₂×C₄, C₂×C₄, C₂×C₄, C₂³, C₂³, C₂³, C₂²⋊C₄, C₄⋊C₄, C₂×C₈, C₂×C₈, C₂²×C₄, C₂²×C₄, C₂⁴, C₂.C₄², C₂²⋊C₈, C₂.D₈, C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₂²×C₈, C₂³×C₄, C₂².₄Q₁₆, C₂².₄Q₁₆, C₂³.₇Q₈, C₂×C₂²⋊C₈, C₂×C₂.D₈, C₂⁴.₈₈D₄
Quotients: C₁, C₂, C₂², D₄, C₂³, Q₁₆, C₂×D₄, C₄○D₄, C₄⋊D₄, C₂².D₄, C₄.₄D₄, C₄²⋊₂C₂, C₂×Q₁₆, C₄○D₈, C₈⋊C₂², C₂³.₁₁D₄, C₈.₁₈D₄, C₈⋊₂D₄, C₂³.₁₉D₄, C₂³.₄₈D₄, C₂⁴.₈₈D₄

Smallest permutation representation of C₂⁴.₈₈D₄
►On 64 points

Generators in S₆₄

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

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

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

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

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

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

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

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

type

image

C₁

C₂

D₄

C₄○D₄

Q₁₆

C₄○D₈

C₈⋊C₂²

kernel

C₂⁴.₈₈D₄

C₂².₄Q₁₆

C₂³.₇Q₈

C₂×C₂²⋊C₈

C₂×C₂.D₈

C₂×C₈

C₂²×C₄

C₂⁴

C₂×C₄

C₂³

C₂²

# reps

Matrix representation of C₂⁴.₈₈D₄ ►in GL₆(𝔽₁₇)

1	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	15	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

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

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

15	0	0	0	0	0
0	8	0	0	0	0
0	0	2	0	0	0
0	0	0	9	0	0
0	0	0	0	2	2
0	0	0	0	6	15

0	8	0	0	0	0
2	0	0	0	0	0
0	0	0	9	0	0
0	0	15	0	0	0
0	0	0	0	8	8
0	0	0	0	11	9

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

C₂⁴.₈₈D₄ in GAP, Magma, Sage, TeX

C_2^4._{88}D_4

% in TeX

G:=Group("C2^4.88D4");

// GroupNames label

G:=SmallGroup(128,808);

// by ID

G=gap.SmallGroup(128,808);

# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,112,141,422,387,394,718,172]);

// 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=d*e^3>;

// generators/relations

G = C24.88D4 order 128 = 27

43rd non-split extension by C24 of D4 acting via D4/C2=C22

G = C₂⁴.₈₈D₄ order 128 = 2⁷

43^rd non-split extension by C₂⁴ of D₄ acting via D₄/C₂=C₂²