C2^4.454C2^3 - GroupNames

G = C₂⁴.₄₅₄C₂³ order 128 = 2⁷

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

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

Aliases: C₂⁴.₄₅₄C₂³, C₂³.₆₉₀C₂⁴, C₂².₄₆₃2₊¹⁺⁴, C₂².₃₅₃2_-¹⁺⁴, C₂²⋊C₄.₃Q₈, C₄²⋊₉C₄⋊₃₆C₂, C₂³.₄₅(C₂×Q₈), C₂.₆₃(D₄⋊₃Q₈), (C₂²×C₄).₂₁₆C₂³, (C₂³×C₄).₁₇₅C₂², (C₂×C₄²).₇₁₇C₂², C₂³.Q₈.₄₂C₂, C₂³.₈Q₈.₆₆C₂, C₂².₁₆₁(C₂²×Q₈), C₂⁴.C₂².₇₈C₂, C₂³.₆₃C₂³⋊₁₈₈C₂, C₂³.₈₁C₂³⋊₁₂₉C₂, C₂³.₆₅C₂³⋊₁₅₆C₂, C₂³.₈₃C₂³⋊₁₂₃C₂, C₂.₃₄(C₂².₅₄C₂⁴), C₂.C₄².₃₉₄C₂², C₂.₄₂(C₂³.₄₁C₂³), C₂.₁₁₆(C₂².₃₆C₂⁴), C₂.₁₁₅(C₂².₄₇C₂⁴), C₂.₁₀₈(C₂².₃₃C₂⁴), (C₂×C₄).₈₅(C₂×Q₈), (C₂×C₄).₂₃₂(C₄○D₄), (C₂×C₄⋊C₄).₅₀₀C₂², C₂².₅₅₁(C₂×C₄○D₄), (C₂×C₂²⋊C₄).₃₂₅C₂², SmallGroup(128,1522)

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

Derived series

C₁ — C₂³ — C₂⁴.₄₅₄C₂³

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂×C₄² — C₂⁴.C₂² — C₂⁴.₄₅₄C₂³

Lower central

C₁ — C₂³ — C₂⁴.₄₅₄C₂³

Upper central

C₁ — C₂³ — C₂⁴.₄₅₄C₂³

Jennings

C₁ — C₂³ — C₂⁴.₄₅₄C₂³

Generators and relations for C₂⁴.₄₅₄C₂³
G = < a,b,c,d,e,f,g | a²=b²=c²=e²=1, d²=abc, f²=b, g²=ba=ab, ac=ca, ede=ad=da, geg^-1=ae=ea, af=fa, ag=ga, bc=cb, fdf^-1=bd=db, be=eb, bf=fb, bg=gb, cd=dc, fef^-1=ce=ec, cf=fc, cg=gc, gdg^-1=abd, fg=gf >

Subgroups: 388 in 204 conjugacy classes, 96 normal (82 characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₂×C₄, C₂×C₄, C₂³, C₂³, C₂³, C₄², C₂²⋊C₄, C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂⁴, C₂.C₄², C₂×C₄², C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₂³×C₄, C₄²⋊₉C₄, C₂³.₈Q₈, C₂³.₆₃C₂³, C₂⁴.C₂², C₂³.₆₅C₂³, C₂³.Q₈, C₂³.₈₁C₂³, C₂³.₈₃C₂³, C₂⁴.₄₅₄C₂³
Quotients: C₁, C₂, C₂², Q₈, C₂³, C₂×Q₈, C₄○D₄, C₂⁴, C₂²×Q₈, C₂×C₄○D₄, 2₊¹⁺⁴, 2_-¹⁺⁴, C₂².₃₃C₂⁴, C₂².₃₆C₂⁴, C₂³.₄₁C₂³, C₂².₄₇C₂⁴, D₄⋊₃Q₈, C₂².₅₄C₂⁴, C₂⁴.₄₅₄C₂³

Smallest permutation representation of C₂⁴.₄₅₄C₂³
►On 64 points

Generators in S₆₄

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

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

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

(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)

(2 56)(4 54)(5 26)(6 22)(7 28)(8 24)(9 43)(11 41)(13 45)(15 47)(17 33)(18 61)(19 35)(20 63)(21 37)(23 39)(25 40)(27 38)(29 58)(31 60)(34 50)(36 52)(49 64)(51 62)

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

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

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

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

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

32 conjugacy classes

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

32 irreducible representations

dim	1	1	1	1	1	1	1	1	1	2	2	4	4
type	+	+	+	+	+	+	+	+	+	-		+	-
image	C₁	C₂	C₂	C₂	C₂	C₂	C₂	C₂	C₂	Q₈	C₄○D₄	2₊¹⁺⁴	2_-¹⁺⁴
kernel	C₂⁴.₄₅₄C₂³	C₄²⋊₉C₄	C₂³.₈Q₈	C₂³.₆₃C₂³	C₂⁴.C₂²	C₂³.₆₅C₂³	C₂³.Q₈	C₂³.₈₁C₂³	C₂³.₈₃C₂³	C₂²⋊C₄	C₂×C₄	C₂²	C₂²
# reps	1	1	3	2	2	1	2	3	1	4	8	3	1

Matrix representation of C₂⁴.₄₅₄C₂³ ►in GL₆(𝔽₅)

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

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

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

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

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

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

C₂⁴.₄₅₄C₂³ in GAP, Magma, Sage, TeX

C_2^4._{454}C_2^3

% in TeX

G:=Group("C2^4.454C2^3");

// GroupNames label

G:=SmallGroup(128,1522);

// by ID

G=gap.SmallGroup(128,1522);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,784,253,758,723,184,1571,346,192]);

// Polycyclic

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

// generators/relations

G = C24.454C23 order 128 = 27

294th non-split extension by C24 of C23 acting via C23/C2=C22

G = C₂⁴.₄₅₄C₂³ order 128 = 2⁷

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