C2^4.421C2^3 - GroupNames

G = C₂⁴.₄₂₁C₂³ order 128 = 2⁷

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

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

Aliases: C₂⁴.₄₂₁C₂³, C₂³.₆₂₉C₂⁴, C₂².₄₀₂2₊¹⁺⁴, C₂².₃₀₄2_-¹⁺⁴, C₂²⋊C₄⋊₅Q₈, C₂.₃₃(D₄×Q₈), (C₂×Q₈).₁₂₀D₄, C₂³.₃₆(C₂×Q₈), C₂.₅₂(D₄⋊₃Q₈), C₂.₅₅(Q₈⋊₅D₄), C₂³⋊Q₈.₂₁C₂, C₂.₅₃(C₂³⋊₃D₄), (C₂³×C₄).₄₇₆C₂², (C₂²×C₄).₁₉₈C₂³, (C₂×C₄²).₆₈₀C₂², C₂².₄₃₈(C₂²×D₄), C₂³.Q₈.₃₂C₂, C₂³.₈Q₈.₅₁C₂, C₂².₁₄₉(C₂²×Q₈), (C₂²×Q₈).₁₉₈C₂², C₂³.₆₇C₂³⋊₉₀C₂, C₂³.₈₃C₂³⋊₉₀C₂, C₂³.₇₈C₂³⋊₅₂C₂, C₂⁴.C₂².₅₆C₂, C₂³.₆₃C₂³⋊₁₅₀C₂, C₂³.₈₁C₂³⋊₁₀₂C₂, C₂³.₆₅C₂³⋊₁₃₇C₂, C₂.C₄².₃₃₅C₂², C₂.₈₅(C₂².₃₆C₂⁴), C₂.₃₀(C₂².₅₇C₂⁴), C₂.₃₄(C₂³.₄₁C₂³), (C₂×C₄⋊Q₈)⋊₂₃C₂, (C₂×C₄).₇₃(C₂×Q₈), (C₂×C₄).₁₂₃(C₂×D₄), (C₂×C₂²⋊Q₈).₄₇C₂, (C₂×C₄).₂₀₉(C₄○D₄), (C₂×C₄⋊C₄).₄₄₂C₂², C₂².₄₉₁(C₂×C₄○D₄), (C₂×C₂²⋊C₄).₂₉₂C₂², SmallGroup(128,1461)

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

Derived series

C₁ — C₂³ — C₂⁴.₄₂₁C₂³

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂³×C₄ — C₂³.₈Q₈ — C₂⁴.₄₂₁C₂³

Lower central

C₁ — C₂³ — C₂⁴.₄₂₁C₂³

Upper central

C₁ — C₂³ — C₂⁴.₄₂₁C₂³

Jennings

C₁ — C₂³ — C₂⁴.₄₂₁C₂³

Generators and relations for C₂⁴.₄₂₁C₂³
G = < a,b,c,d,e,f,g | a²=b²=c²=f²=1, d²=g²=ba=ab, e²=a, ac=ca, ede^-1=ad=da, geg^-1=ae=ea, af=fa, ag=ga, bc=cb, fdf=bd=db, be=eb, bf=fb, bg=gb, cd=dc, fef=ce=ec, cf=fc, cg=gc, gdg^-1=abd, fg=gf >

Subgroups: 452 in 240 conjugacy classes, 104 normal (82 characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₂×C₄, C₂×C₄, Q₈, C₂³, C₂³, C₂³, C₄², C₂²⋊C₄, C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂×Q₈, C₂×Q₈, C₂⁴, C₂.C₄², C₂×C₄², C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₂²⋊Q₈, C₄⋊Q₈, C₂³×C₄, C₂²×Q₈, C₂³.₈Q₈, C₂³.₆₃C₂³, C₂⁴.C₂², C₂³.₆₅C₂³, C₂³.₆₇C₂³, C₂³⋊Q₈, C₂³.₇₈C₂³, C₂³.Q₈, C₂³.₈₁C₂³, C₂³.₈₃C₂³, C₂×C₂²⋊Q₈, C₂×C₄⋊Q₈, C₂⁴.₄₂₁C₂³
Quotients: C₁, C₂, C₂², D₄, Q₈, C₂³, C₂×D₄, C₂×Q₈, C₄○D₄, C₂⁴, C₂²×D₄, C₂²×Q₈, C₂×C₄○D₄, 2₊¹⁺⁴, 2_-¹⁺⁴, C₂³⋊₃D₄, C₂².₃₆C₂⁴, C₂³.₄₁C₂³, Q₈⋊₅D₄, D₄×Q₈, D₄⋊₃Q₈, C₂².₅₇C₂⁴, C₂⁴.₄₂₁C₂³

Smallest permutation representation of C₂⁴.₄₂₁C₂³
►On 64 points

Generators in S₆₄

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

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

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

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

(2 44)(4 42)(5 50)(6 17)(7 52)(8 19)(9 38)(10 35)(11 40)(12 33)(14 48)(16 46)(18 62)(20 64)(21 34)(22 39)(23 36)(24 37)(26 30)(28 32)(49 63)(51 61)(54 58)(56 60)

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

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

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

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

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	1	1	1	1	2	2	2	4	4
type	+	+	+	+	+	+	+	+	+	+	+	+	+	-	+		+	-
image	C₁	C₂	C₂	C₂	C₂	C₂	C₂	C₂	C₂	C₂	C₂	C₂	C₂	Q₈	D₄	C₄○D₄	2₊¹⁺⁴	2_-¹⁺⁴
kernel	C₂⁴.₄₂₁C₂³	C₂³.₈Q₈	C₂³.₆₃C₂³	C₂⁴.C₂²	C₂³.₆₅C₂³	C₂³.₆₇C₂³	C₂³⋊Q₈	C₂³.₇₈C₂³	C₂³.Q₈	C₂³.₈₁C₂³	C₂³.₈₃C₂³	C₂×C₂²⋊Q₈	C₂×C₄⋊Q₈	C₂²⋊C₄	C₂×Q₈	C₂×C₄	C₂²	C₂²
# reps	1	2	1	2	1	1	1	1	1	2	1	1	1	4	4	4	2	2

Matrix representation of C₂⁴.₄₂₁C₂³ ►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

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

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

1	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	4

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

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],[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],[0,4,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,0,0,2,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,4,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[1,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4],[3,0,0,0,0,0,0,2,0,0,0,0,0,0,3,0,0,0,0,0,0,2,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

C₂⁴.₄₂₁C₂³ in GAP, Magma, Sage, TeX

C_2^4._{421}C_2^3

% in TeX

G:=Group("C2^4.421C2^3");

// GroupNames label

G:=SmallGroup(128,1461);

// by ID

G=gap.SmallGroup(128,1461);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,336,253,120,758,723,1571,346,192]);

// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=f^2=1,d^2=g^2=b*a=a*b,e^2=a,a*c=c*a,e*d*e^-1=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=b*d=d*b,b*e=e*b,b*f=f*b,b*g=g*b,c*d=d*c,f*e*f=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 = C24.421C23 order 128 = 27

261st non-split extension by C24 of C23 acting via C23/C2=C22

G = C₂⁴.₄₂₁C₂³ order 128 = 2⁷

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