C2^4.411C2^3 - GroupNames

G = C₂⁴.₄₁₁C₂³ order 128 = 2⁷

251^st non-split extension by C₂⁴ of C₂³ acting via C₂³/C₂=C₂²

p-group, metabelian, nilpotent (class 2), monomial

Aliases: C₂⁴.₄₁₁C₂³, C₂³.₆₀₉C₂⁴, C₂².₃₈₃2₊¹⁺⁴, (C₂×D₄).₁₄₃D₄, C₂³.₇₀(C₂×D₄), C₂.₁₁₄(D₄⋊₅D₄), C₂³.₁₀D₄⋊₉₂C₂, C₂³.₂₃D₄⋊₉₆C₂, C₂.₄₉(C₂³⋊₃D₄), (C₂³×C₄).₁₅₃C₂², (C₂²×C₄).₁₈₇C₂³, (C₂×C₄²).₆₆₀C₂², C₂².₄₁₈(C₂²×D₄), (C₂²×D₄).₂₄₄C₂², C₂³.₆₃C₂³⋊₁₃₉C₂, C₂.₁₉(C₂².₅₄C₂⁴), C₂.C₄².₃₁₅C₂², C₂.₄₆(C₂².₃₄C₂⁴), C₂.₃₃(C₂².₅₃C₂⁴), (C₂×C₄).₄₂₃(C₂×D₄), (C₂×C₄⋊₁D₄).₁₉C₂, (C₂×C₄).₁₉₆(C₄○D₄), (C₂×C₄⋊C₄).₄₂₂C₂², C₂².₄₇₁(C₂×C₄○D₄), (C₂×C₂².D₄)⋊₄₂C₂, (C₂×C₂²⋊C₄).₂₇₅C₂², SmallGroup(128,1441)

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

Derived series

C₁ — C₂³ — C₂⁴.₄₁₁C₂³

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂⁴ — C₂³×C₄ — C₂³.₂₃D₄ — C₂⁴.₄₁₁C₂³

Lower central

C₁ — C₂³ — C₂⁴.₄₁₁C₂³

Upper central

C₁ — C₂³ — C₂⁴.₄₁₁C₂³

Jennings

C₁ — C₂³ — C₂⁴.₄₁₁C₂³

Generators and relations for C₂⁴.₄₁₁C₂³
G = < a,b,c,d,e,f,g | a²=b²=c²=d²=e²=1, f²=b, g²=cb=bc, faf^-1=ab=ba, ac=ca, ad=da, eae=abc, ag=ga, bd=db, fef^-1=be=eb, bf=fb, bg=gb, cd=dc, geg^-1=ce=ec, cf=fc, cg=gc, de=ed, gfg^-1=df=fd, dg=gd >

Subgroups: 708 in 306 conjugacy classes, 96 normal (22 characteristic)
C₁, C₂, C₂, C₂, C₄, C₂², C₂², C₂², C₂×C₄, C₂×C₄, D₄, C₂³, C₂³, C₂³, C₄², C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂⁴, C₂⁴, C₂.C₄², C₂×C₄², C₂×C₂²⋊C₄, C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₂².D₄, C₄⋊₁D₄, C₂³×C₄, C₂²×D₄, C₂²×D₄, C₂³.₂₃D₄, C₂³.₂₃D₄, C₂³.₆₃C₂³, C₂³.₁₀D₄, C₂×C₂².D₄, C₂×C₄⋊₁D₄, C₂⁴.₄₁₁C₂³
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₄○D₄, C₂⁴, C₂²×D₄, C₂×C₄○D₄, 2₊¹⁺⁴, C₂³⋊₃D₄, C₂².₃₄C₂⁴, D₄⋊₅D₄, C₂².₅₃C₂⁴, C₂².₅₄C₂⁴, C₂⁴.₄₁₁C₂³

Smallest permutation representation of C₂⁴.₄₁₁C₂³
►On 64 points

Generators in S₆₄

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

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

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

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

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

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

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

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

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

32 conjugacy classes

class	1	2A	···	2G	2H	2I	2J	2K	2L	2M	4A	···	4N	4O	4P	4Q	4R
order	1	2	···	2	2	2	2	2	2	2	4	···	4	4	4	4	4
size	1	1	···	1	4	4	4	4	8	8	4	···	4	8	8	8	8

32 irreducible representations

dim	1	1	1	1	1	1	2	2	4
type	+	+	+	+	+	+	+		+
image	C₁	C₂	C₂	C₂	C₂	C₂	D₄	C₄○D₄	2₊¹⁺⁴
kernel	C₂⁴.₄₁₁C₂³	C₂³.₂₃D₄	C₂³.₆₃C₂³	C₂³.₁₀D₄	C₂×C₂².D₄	C₂×C₄⋊₁D₄	C₂×D₄	C₂×C₄	C₂²
# reps	1	5	2	6	1	1	4	8	4

Matrix representation of C₂⁴.₄₁₁C₂³ ►in GL₆(𝔽₅)

4	0	0	0	0	0
0	1	0	0	0	0
0	0	0	1	0	0
0	0	1	0	0	0
0	0	0	0	1	0
0	0	0	0	0	1

4	0	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	1	0
0	0	0	0	0	1

1	0	0	0	0	0
0	1	0	0	0	0
0	0	4	0	0	0
0	0	0	4	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	1	0	0	0
0	0	0	1	0	0
0	0	0	0	4	0
0	0	0	0	0	4

0	1	0	0	0	0
1	0	0	0	0	0
0	0	4	0	0	0
0	0	0	1	0	0
0	0	0	0	4	0
0	0	0	0	0	4

0	1	0	0	0	0
4	0	0	0	0	0
0	0	4	0	0	0
0	0	0	4	0	0
0	0	0	0	4	0
0	0	0	0	3	1

2	0	0	0	0	0
0	2	0	0	0	0
0	0	0	2	0	0
0	0	2	0	0	0
0	0	0	0	2	3
0	0	0	0	4	3

G:=sub<GL(6,GF(5))| [4,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[0,4,0,0,0,0,1,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,3,0,0,0,0,0,1],[2,0,0,0,0,0,0,2,0,0,0,0,0,0,0,2,0,0,0,0,2,0,0,0,0,0,0,0,2,4,0,0,0,0,3,3] >;

C₂⁴.₄₁₁C₂³ in GAP, Magma, Sage, TeX

C_2^4._{411}C_2^3

% in TeX

G:=Group("C2^4.411C2^3");

// GroupNames label

G:=SmallGroup(128,1441);

// by ID

G=gap.SmallGroup(128,1441);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,253,120,758,723,268,1571,346]);

// Polycyclic

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

// generators/relations

G = C24.411C23 order 128 = 27

251st non-split extension by C24 of C23 acting via C23/C2=C22

G = C₂⁴.₄₁₁C₂³ order 128 = 2⁷

251^st non-split extension by C₂⁴ of C₂³ acting via C₂³/C₂=C₂²