C2^4.85D4 - GroupNames

G = C₂⁴.₈₅D₄ order 128 = 2⁷

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

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

Aliases: C₂⁴.₈₅D₄, C₂³.₁₈SD₁₆, C₄⋊C₄.₉₀D₄, (C₂×C₈).₁₅₆D₄, (C₂×Q₈).₉₀D₄, C₂.₁₈(C₈⋊₈D₄), (C₂²×C₄).₁₄₇D₄, C₂³.₉₁₁(C₂×D₄), C₄.₁₄₄(C₄⋊D₄), C₂².₄Q₁₆⋊₂₃C₂, C₂.₂₁(Q₈⋊D₄), C₂.₁₃(C₈.D₄), C₄.₃₈(C₄.₄D₄), (C₂²×C₈).₇₁C₂², C₂².₉₆(C₂×SD₁₆), C₂².₂₁₇C₂²≀C₂, C₂.₃₂(D₄.₇D₄), C₂².₁₀₉(C₄○D₈), (C₂³×C₄).₂₇₃C₂², C₂³.₇Q₈.₁₈C₂, (C₂²×Q₈).₆₃C₂², C₂².₂₂₈(C₄⋊D₄), (C₂²×C₄).₁₄₄₅C₂³, C₂.₈(C₂³.₂₀D₄), C₂.₆(C₂³.₄₇D₄), C₄.₁₉(C₂².D₄), C₂.₈(C₂³.₁₀D₄), C₂².₁₂₅(C₈.C₂²), C₂².₁₁₄(C₂².D₄), (C₂×C₄.Q₈)⋊₂₁C₂, (C₂×Q₈⋊C₄)⋊₁₃C₂, (C₂×C₄).₁₀₃₇(C₂×D₄), (C₂×C₂²⋊C₈).₃₇C₂, (C₂×C₂²⋊Q₈).₁₃C₂, (C₂×C₄).₇₇₂(C₄○D₄), (C₂×C₄⋊C₄).₁₂₀C₂², SmallGroup(128,767)

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

Derived series

C₁ — C₂²×C₄ — C₂⁴.₈₅D₄

Chief series

C₁ — C₂ — C₂² — C₂×C₄ — C₂²×C₄ — C₂×C₄⋊C₄ — C₂³.₇Q₈ — C₂⁴.₈₅D₄

Lower central

C₁ — C₂ — C₂²×C₄ — C₂⁴.₈₅D₄

Upper central

C₁ — C₂³ — C₂³×C₄ — C₂⁴.₈₅D₄

Jennings

C₁ — C₂ — C₂ — C₂²×C₄ — C₂⁴.₈₅D₄

Generators and relations for C₂⁴.₈₅D₄
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=be³ >

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, C₄.Q₈, 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×C₄.Q₈, C₂×C₂²⋊Q₈, C₂⁴.₈₅D₄
Quotients: C₁, C₂, C₂², D₄, C₂³, SD₁₆, C₂×D₄, C₄○D₄, C₂²≀C₂, C₄⋊D₄, C₂².D₄, C₄.₄D₄, C₂×SD₁₆, C₄○D₈, C₈.C₂², C₂³.₁₀D₄, Q₈⋊D₄, D₄.₇D₄, C₈⋊₈D₄, C₈.D₄, C₂³.₄₇D₄, C₂³.₂₀D₄, C₂⁴.₈₅D₄

Smallest permutation representation of C₂⁴.₈₅D₄
►On 64 points

Generators in S₆₄

(2 14)(4 16)(6 10)(8 12)(17 45)(18 39)(19 47)(20 33)(21 41)(22 35)(23 43)(24 37)(25 62)(27 64)(29 58)(31 60)(34 51)(36 53)(38 55)(40 49)(42 52)(44 54)(46 56)(48 50)

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

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

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

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

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

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

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
type	+	+	+	+	+	+	+	+	+	+	+	+				-
image	C₁	C₂	C₂	C₂	C₂	C₂	C₂	D₄	D₄	D₄	D₄	D₄	C₄○D₄	SD₁₆	C₄○D₈	C₈.C₂²
kernel	C₂⁴.₈₅D₄	C₂².₄Q₁₆	C₂³.₇Q₈	C₂×C₂²⋊C₈	C₂×Q₈⋊C₄	C₂×C₄.Q₈	C₂×C₂²⋊Q₈	C₄⋊C₄	C₂×C₈	C₂²×C₄	C₂×Q₈	C₂⁴	C₂×C₄	C₂³	C₂²	C₂²
# reps	1	1	1	1	2	1	1	2	2	1	2	1	6	4	4	2

Matrix representation of C₂⁴.₈₅D₄ ►in GL₆(𝔽₁₇)

1	0	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	0
0	0	0	0	3	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	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	16	0	0	0
0	0	0	16	0	0
0	0	0	0	1	0
0	0	0	0	0	1

13	0	0	0	0	0
0	13	0	0	0	0
0	0	2	0	0	0
0	0	0	9	0	0
0	0	0	0	3	15
0	0	0	0	5	14

0	4	0	0	0	0
13	0	0	0	0	0
0	0	0	8	0	0
0	0	2	0	0	0
0	0	0	0	1	0
0	0	0	0	3	16

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,3,0,0,0,0,0,16],[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],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[13,0,0,0,0,0,0,13,0,0,0,0,0,0,2,0,0,0,0,0,0,9,0,0,0,0,0,0,3,5,0,0,0,0,15,14],[0,13,0,0,0,0,4,0,0,0,0,0,0,0,0,2,0,0,0,0,8,0,0,0,0,0,0,0,1,3,0,0,0,0,0,16] >;

C₂⁴.₈₅D₄ in GAP, Magma, Sage, TeX

C_2^4._{85}D_4

% in TeX

G:=Group("C2^4.85D4");

// GroupNames label

G:=SmallGroup(128,767);

// by ID

G=gap.SmallGroup(128,767);

# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,448,141,422,387,58,2804,718,172]);

// 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*e^3>;

// generators/relations

G = C24.85D4 order 128 = 27

40th non-split extension by C24 of D4 acting via D4/C2=C22

G = C₂⁴.₈₅D₄ order 128 = 2⁷

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