C2^4.71D4 - GroupNames

G = C₂⁴.₇₁D₄ order 128 = 2⁷

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

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

Aliases: C₂⁴.₇₁D₄, C₂²⋊C₈⋊₈C₄, C₄⋊C₄.₂₉₉D₄, C₄.₁₃₈(C₄×D₄), C₄.₃(C₂²⋊Q₈), C₂.₂(D₄⋊D₄), (C₂²×C₄).₅₀Q₈, C₂³.₂₉(C₄⋊C₄), (C₂²×C₄).₆₈₁D₄, C₂³.₇₅₉(C₂×D₄), C₂².₄Q₁₆⋊₃₄C₂, C₂.₂(D₄.₇D₄), C₂².₇₉C₂²≀C₂, C₂².₄₈(C₄○D₈), C₂².₆₉(C₈⋊C₂²), (C₂²×C₈).₃₁₂C₂², (C₂³×C₄).₂₅₀C₂², C₂³.₇Q₈.₁₄C₂, (C₂²×C₄).₁₃₅₁C₂³, C₂.₂(C₂³.₂₀D₄), C₂.₂(C₂³.₁₉D₄), C₂².₅₈(C₈.C₂²), C₂.₁₀(C₂³.₈Q₈), C₂.₁₂(M₄(2)⋊C₄), C₂.₁₁(C₂³.₂₅D₄), C₂².₈₃(C₂².D₄), (C₂×C₂.D₈)⋊₃C₂, (C₂×C₄.Q₈)⋊₁₆C₂, (C₂×C₄).₉₁(C₄⋊C₄), (C₂×C₈).₁₀₆(C₂×C₄), (C₂×C₄).₉₈₁(C₂×D₄), (C₂×C₄).₂₀₁(C₂×Q₈), (C₂×C₄⋊C₄).₅₃C₂², (C₂×C₂²⋊C₈).₃₃C₂, C₂².₁₁₁(C₂×C₄⋊C₄), (C₂×C₄).₇₄₇(C₄○D₄), (C₂²×C₄).₂₇₃(C₂×C₄), (C₂×C₄).₅₅₀(C₂²×C₄), (C₂×C₄²⋊C₂).₁₉C₂, SmallGroup(128,586)

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

Derived series

C₁ — C₂×C₄ — C₂⁴.₇₁D₄

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂³×C₄ — C₂×C₄²⋊C₂ — C₂⁴.₇₁D₄

Lower central

C₁ — C₂ — C₂×C₄ — C₂⁴.₇₁D₄

Upper central

C₁ — C₂³ — C₂³×C₄ — C₂⁴.₇₁D₄

Jennings

C₁ — C₂ — C₂ — C₂²×C₄ — C₂⁴.₇₁D₄

Generators and relations for C₂⁴.₇₁D₄
G = < a,b,c,d,e,f | a²=b²=c²=d²=1, e⁴=d, f²=c, eae^-1=ab=ba, ac=ca, ad=da, faf^-1=abd, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef^-1=de³ >

Subgroups: 308 in 158 conjugacy classes, 64 normal (44 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₈, C₂×C₄, C₂×C₄, C₂×C₄, C₂³, C₂³, C₂³, C₄², C₂²⋊C₄, C₄⋊C₄, C₄⋊C₄, C₂×C₈, C₂×C₈, C₂²×C₄, C₂²×C₄, C₂²×C₄, C₂⁴, C₂.C₄², C₂²⋊C₈, C₄.Q₈, C₂.D₈, C₂×C₄², C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₄²⋊C₂, C₂²×C₈, C₂³×C₄, C₂².₄Q₁₆, C₂³.₇Q₈, C₂×C₂²⋊C₈, C₂×C₄.Q₈, C₂×C₂.D₈, C₂×C₄²⋊C₂, C₂⁴.₇₁D₄
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, D₄, Q₈, C₂³, C₄⋊C₄, C₂²×C₄, C₂×D₄, C₂×Q₈, C₄○D₄, C₂×C₄⋊C₄, C₄×D₄, C₂²≀C₂, C₂²⋊Q₈, C₂².D₄, C₄○D₈, C₈⋊C₂², C₈.C₂², C₂³.₈Q₈, C₂³.₂₅D₄, M₄(2)⋊C₄, D₄⋊D₄, D₄.₇D₄, C₂³.₁₉D₄, C₂³.₂₀D₄, C₂⁴.₇₁D₄

Smallest permutation representation of C₂⁴.₇₁D₄
►On 64 points

Generators in S₆₄

(2 24)(4 18)(6 20)(8 22)(9 63)(11 57)(13 59)(15 61)(25 46)(26 30)(27 48)(28 32)(29 42)(31 44)(33 37)(34 50)(35 39)(36 52)(38 54)(40 56)(41 45)(43 47)(49 53)(51 55)

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

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

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

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

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

38 conjugacy classes

class	1	2A	···	2G	2H	2I	4A	···	4H	4I	···	4P	4Q	4R	4S	4T	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	4	···	4	8	8	8	8	4	···	4

38 irreducible representations

dim	1	1	1	1	1	1	1	1	2	2	2	2	2	2	4	4
type	+	+	+	+	+	+	+		+	+	-	+			+	-
image	C₁	C₂	C₂	C₂	C₂	C₂	C₂	C₄	D₄	D₄	Q₈	D₄	C₄○D₄	C₄○D₈	C₈⋊C₂²	C₈.C₂²
kernel	C₂⁴.₇₁D₄	C₂².₄Q₁₆	C₂³.₇Q₈	C₂×C₂²⋊C₈	C₂×C₄.Q₈	C₂×C₂.D₈	C₂×C₄²⋊C₂	C₂²⋊C₈	C₄⋊C₄	C₂²×C₄	C₂²×C₄	C₂⁴	C₂×C₄	C₂²	C₂²	C₂²
# reps	1	2	1	1	1	1	1	8	4	1	2	1	4	8	1	1

Matrix representation of C₂⁴.₇₁D₄ ►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	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	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

9	0	0	0	0	0
0	2	0	0	0	0
0	0	2	9	0	0
0	0	7	15	0	0
0	0	0	0	0	16
0	0	0	0	16	0

0	15	0	0	0	0
9	0	0	0	0	0
0	0	8	2	0	0
0	0	10	9	0	0
0	0	0	0	0	1
0	0	0	0	1	0

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16],[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],[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],[9,0,0,0,0,0,0,2,0,0,0,0,0,0,2,7,0,0,0,0,9,15,0,0,0,0,0,0,0,16,0,0,0,0,16,0],[0,9,0,0,0,0,15,0,0,0,0,0,0,0,8,10,0,0,0,0,2,9,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

C₂⁴.₇₁D₄ in GAP, Magma, Sage, TeX

C_2^4._{71}D_4

% in TeX

G:=Group("C2^4.71D4");

// GroupNames label

G:=SmallGroup(128,586);

// by ID

G=gap.SmallGroup(128,586);

# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,232,422,352,2019,1018,248]);

// Polycyclic

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

// generators/relations

G = C24.71D4 order 128 = 27

26th non-split extension by C24 of D4 acting via D4/C2=C22

G = C₂⁴.₇₁D₄ order 128 = 2⁷

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