C2^4.455C2^3

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

Aliases: C₂⁴.₄₅₅C₂³, C₂³.₆₉₆C₂⁴, C₂².₃₅₉2_-⁽¹⁺⁴⁾, C₂².₄₆₉2₊⁽¹⁺⁴⁾, C₂³.₄Q₈⋊₆₂C₂, C₂³.₁₀₂(C₄○D₄), (C₂×C₄²).₁₁₂C₂², (C₂²×C₄).₆₀₆C₂³, (C₂³×C₄).₁₇₆C₂², C₂³.₈Q₈⋊₁₃₉C₂, C₂³.₁₁D₄⋊₁₂₁C₂, C₂³.₁₀D₄.₆₇C₂, C₂³.₂₃D₄.₇₆C₂, (C₂²×D₄).₂₈₄C₂², C₂⁴.C₂²⋊₁₇₂C₂, C₂⁴.₃C₂².₇₆C₂, C₂³.₆₃C₂³⋊₁₉₁C₂, C₂³.₆₅C₂³⋊₁₅₇C₂, C₂³.₈₃C₂³⋊₁₂₆C₂, C₂.₃₇(C₂².₅₄C₂⁴), C₂.C₄².₄₀₀C₂², C₂.₁₂₁(C₂².₄₅C₂⁴), C₂.₄₃(C₂².₅₃C₂⁴), C₂.₁₁₄(C₂².₃₃C₂⁴), C₂.₁₁₈(C₂².₄₇C₂⁴), C₂.₁₁₉(C₂².₃₆C₂⁴), (C₂×C₄).₂₃₇(C₄○D₄), (C₂×C₄⋊C₄).₅₀₆C₂², C₂².₅₅₇(C₂×C₄○D₄), (C₂×C₂²⋊C₄).₇₈C₂², SmallGroup(128,1528)

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

Derived series

C₁ — C₂³ — C₂⁴.₄₅₅C₂³

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂⁴ — C₂³×C₄ — C₂³.₂₃D₄ — C₂⁴.₄₅₅C₂³

Lower central

C₁ — C₂³ — C₂⁴.₄₅₅C₂³

Upper central

C₁ — C₂³ — C₂⁴.₄₅₅C₂³

Jennings

C₁ — C₂³ — C₂⁴.₄₅₅C₂³

Subgroups: 452 in 217 conjugacy classes, 88 normal (82 characteristic)
C₁, C₂^[×7], C₂^[×3], C₄^[×15], C₂²^[×7], C₂²^[×17], C₂×C₄^[×4], C₂×C₄^[×41], D₄^[×4], C₂³, C₂³^[×2], C₂³^[×13], C₄²^[×2], C₂²⋊C₄^[×15], C₄⋊C₄^[×11], C₂²×C₄^[×13], C₂²×C₄^[×3], C₂×D₄^[×5], C₂⁴^[×2], C₂.C₄²^[×12], C₂×C₄²^[×2], C₂×C₂²⋊C₄^[×11], C₂×C₄⋊C₄^[×8], C₂³×C₄, C₂²×D₄, C₂³.₈Q₈^[×2], C₂³.₂₃D₄, C₂³.₆₃C₂³^[×2], C₂⁴.C₂²^[×2], C₂³.₆₅C₂³, C₂⁴.₃C₂², C₂³.₁₀D₄, C₂³.₁₁D₄^[×3], C₂³.₄Q₈, C₂³.₈₃C₂³, C₂⁴.₄₅₅C₂³

Quotients:
C₁, C₂^[×15], C₂²^[×35], C₂³^[×15], C₄○D₄^[×6], C₂⁴, C₂×C₄○D₄^[×3], 2₊⁽¹⁺⁴⁾^[×3], 2_-⁽¹⁺⁴⁾, C₂².₃₃C₂⁴^[×2], C₂².₃₆C₂⁴, C₂².₄₅C₂⁴, C₂².₄₇C₂⁴, C₂².₅₃C₂⁴, C₂².₅₄C₂⁴, C₂⁴.₄₅₅C₂³

Generators and relations
G = < a,b,c,d,e,f,g | a²=b²=c²=d²=1, e²=d, f²=b, g²=bcd, faf^-1=ab=ba, ac=ca, ad=da, eae^-1=abc, ag=ga, bc=cb, 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 >

Smallest permutation representation
►On 64 points

Generators in S₆₄

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

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

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

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

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

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

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

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

Matrix representation ►G ⊆ GL₆(𝔽₅)

4	0	0	0	0	0
1	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

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

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

1	2	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	3	0
0	0	0	0	0	3

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

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

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

32 conjugacy classes

class	1	2A	···	2G	2H	2I	2J	4A	···	4P	4Q	···	4U
order	1	2	···	2	2	2	2	4	···	4	4	···	4
size	1	1	···	1	4	4	8	4	···	4	8	···	8

32 irreducible representations

dim

type

image

C₁

C₂

C₄○D₄

2₊⁽¹⁺⁴⁾

2_-⁽¹⁺⁴⁾

kernel

C₂⁴.₄₅₅C₂³

C₂³.₈Q₈

C₂³.₂₃D₄

C₂³.₆₃C₂³

C₂⁴.C₂²

C₂³.₆₅C₂³

C₂⁴.₃C₂²

C₂³.₁₀D₄

C₂³.₁₁D₄

C₂³.₄Q₈

C₂³.₈₃C₂³

C₂×C₄

C₂³

C₂²

# reps

In GAP, Magma, Sage, TeX

C_2^4._{455}C_2^3

% in TeX

G:=Group("C2^4.455C2^3");

// GroupNames label

G:=SmallGroup(128,1528);

// by ID

G=gap.SmallGroup(128,1528);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,120,758,723,604,1571,346]);

// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=1,e^2=d,f^2=b,g^2=b*c*d,f*a*f^-1=a*b=b*a,a*c=c*a,a*d=d*a,e*a*e^-1=a*b*c,a*g=g*a,b*c=c*b,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 = C₂⁴.₄₅₅C₂³ order 128 = 2⁷

295^th non-split extension by C₂⁴ of C₂³ acting via C₂³/C₂=C₂²

G = C24.455C23 order 128 = 27

295th non-split extension by C24 of C23 acting via C23/C2=C22

G = C₂⁴.₄₅₅C₂³ order 128 = 2⁷

295^th non-split extension by C₂⁴ of C₂³ acting via C₂³/C₂=C₂²