C2^3.624C2^4 - GroupNames

G = C₂³.₆₂₄C₂⁴ order 128 = 2⁷

341^st central stem extension by C₂³ of C₂⁴

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

Aliases: C₂³.₆₂₄C₂⁴, C₂⁴.₄₁₉C₂³, C₂².₂₉₉2_-¹⁺⁴, C₂².₃₉₇2₊¹⁺⁴, (C₂×D₄).₁₄₅D₄, C₂³.₂₂₁(C₂×D₄), C₂.₇₅(D₄⋊₆D₄), C₂³.₁₁D₄⋊₉₉C₂, C₂.₅₁(C₂³⋊₃D₄), (C₂³×C₄).₁₅₇C₂², (C₂²×C₄).₈₈₇C₂³, (C₂×C₄²).₆₇₅C₂², C₂³.₈Q₈⋊₁₂₀C₂, C₂².₄₃₃(C₂²×D₄), (C₂²×D₄).₂₅₃C₂², C₂³.₈₁C₂³⋊₉₉C₂, C₂⁴.₃C₂².₆₄C₂, C₂³.₆₅C₂³⋊₁₃₄C₂, C₂.C₄².₃₃₀C₂², C₂.₃₆(C₂².₅₃C₂⁴), C₂.₂₆(C₂².₅₇C₂⁴), C₂.₇₆(C₂².₃₃C₂⁴), (C₂×C₄).₁₁₉(C₂×D₄), (C₂×C₄).₂₀₅(C₄○D₄), (C₂×C₄⋊C₄).₄₃₇C₂², C₂².₄₈₆(C₂×C₄○D₄), (C₂×C₂²⋊C₄).₂₈₈C₂², (C₂×C₂².D₄).₃₀C₂, SmallGroup(128,1456)

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

Derived series

C₁ — C₂³ — C₂³.₆₂₄C₂⁴

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂⁴ — C₂³×C₄ — C₂³.₈Q₈ — C₂³.₆₂₄C₂⁴

Lower central

C₁ — C₂³ — C₂³.₆₂₄C₂⁴

Upper central

C₁ — C₂³ — C₂³.₆₂₄C₂⁴

Jennings

C₁ — C₂³ — C₂³.₆₂₄C₂⁴

Generators and relations for C₂³.₆₂₄C₂⁴
G = < a,b,c,d,e,f,g | a²=b²=c²=d²=1, e²=f²=g²=cb=bc, ab=ba, faf^-1=ac=ca, ad=da, ae=ea, gag^-1=abc, bd=db, geg^-1=be=eb, bf=fb, bg=gb, cd=dc, ce=ec, gfg^-1=cf=fc, cg=gc, fef^-1=de=ed, df=fd, dg=gd >

Subgroups: 500 in 252 conjugacy classes, 96 normal (14 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, C₂.C₄², C₂×C₄², C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₂×C₄⋊C₄, C₂².D₄, C₂³×C₄, C₂²×D₄, C₂³.₈Q₈, C₂³.₆₅C₂³, C₂⁴.₃C₂², C₂³.₁₁D₄, C₂³.₈₁C₂³, C₂×C₂².D₄, C₂³.₆₂₄C₂⁴
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₄○D₄, C₂⁴, C₂²×D₄, C₂×C₄○D₄, 2₊¹⁺⁴, 2_-¹⁺⁴, C₂³⋊₃D₄, C₂².₃₃C₂⁴, D₄⋊₆D₄, C₂².₅₃C₂⁴, C₂².₅₇C₂⁴, C₂³.₆₂₄C₂⁴

Smallest permutation representation of C₂³.₆₂₄C₂⁴
►On 64 points

Generators in S₆₄

(5 37)(6 38)(7 39)(8 40)(13 15)(14 16)(17 51)(18 52)(19 49)(20 50)(21 43)(22 44)(23 41)(24 42)(29 31)(30 32)(33 62)(34 63)(35 64)(36 61)(45 47)(46 48)(57 59)(58 60)

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

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

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

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

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

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

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

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

32 conjugacy classes

class	1	2A	···	2G	2H	2I	2J	2K	4A	···	4N	4O	···	4T
order	1	2	···	2	2	2	2	2	4	···	4	4	···	4
size	1	1	···	1	4	4	4	4	4	···	4	8	···	8

32 irreducible representations

dim	1	1	1	1	1	1	1	2	2	4	4
type	+	+	+	+	+	+	+	+		+	-
image	C₁	C₂	C₂	C₂	C₂	C₂	C₂	D₄	C₄○D₄	2₊¹⁺⁴	2_-¹⁺⁴
kernel	C₂³.₆₂₄C₂⁴	C₂³.₈Q₈	C₂³.₆₅C₂³	C₂⁴.₃C₂²	C₂³.₁₁D₄	C₂³.₈₁C₂³	C₂×C₂².D₄	C₂×D₄	C₂×C₄	C₂²	C₂²
# reps	1	4	2	1	4	2	2	4	8	2	2

Matrix representation of C₂³.₆₂₄C₂⁴ ►in GL₆(𝔽₅)

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

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

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

0	2	0	0	0	0
3	0	0	0	0	0
0	0	2	0	0	0
0	0	3	3	0	0
0	0	0	0	3	0
0	0	0	0	0	3

0	1	0	0	0	0
1	0	0	0	0	0
0	0	3	0	0	0
0	0	0	3	0	0
0	0	0	0	2	4
0	0	0	0	0	3

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

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

C₂³.₆₂₄C₂⁴ in GAP, Magma, Sage, TeX

C_2^3._{624}C_2^4

% in TeX

G:=Group("C2^3.624C2^4");

// GroupNames label

G:=SmallGroup(128,1456);

// by ID

G=gap.SmallGroup(128,1456);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,336,253,344,758,723,268,1571,346]);

// Polycyclic

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

// generators/relations

G = C23.624C24 order 128 = 27

341st central stem extension by C23 of C24

G = C₂³.₆₂₄C₂⁴ order 128 = 2⁷

341^st central stem extension by C₂³ of C₂⁴