C2^4.456C2^3 - GroupNames

G = C₂⁴.₄₅₆C₂³ order 128 = 2⁷

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

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

Aliases: C₂⁴.₄₅₆C₂³, C₂³.₇₀₄C₂⁴, C₂².₄₇₇2₊¹⁺⁴, (C₂×D₄)⋊₈Q₈, C₂³.₄₆(C₂×Q₈), C₂³⋊Q₈⋊₅₈C₂, C₂.₆₇(D₄⋊₃Q₈), (C₂²×C₄).₂₂₀C₂³, (C₂³×C₄).₁₇₈C₂², (C₂×C₄²).₇₂₄C₂², C₂³.₈Q₈⋊₁₄₁C₂, C₂.₂₀(C₂³⋊₂Q₈), C₂².₁₆₆(C₂²×Q₈), C₂³.₂₃D₄.₇₇C₂, (C₂²×D₄).₂₈₉C₂², (C₂²×Q₈).₂₂₅C₂², C₂.₁₃(C₂⁴⋊C₂²), C₂⁴.₃C₂².₇₈C₂, C₂³.₆₇C₂³⋊₁₀₄C₂, C₂³.₈₃C₂³⋊₁₂₈C₂, C₂.₁₀₉(C₂².₃₂C₂⁴), C₂.C₄².₄₀₈C₂², C₂.₄₇(C₂².₅₃C₂⁴), (C₂×C₄).₉₀(C₂×Q₈), (C₂×C₄).₂₄₅(C₄○D₄), (C₂×C₄⋊C₄).₅₁₄C₂², C₂².₅₆₅(C₂×C₄○D₄), (C₂×C₂²⋊C₄).₃₂₉C₂², SmallGroup(128,1536)

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

Derived series

C₁ — C₂³ — C₂⁴.₄₅₆C₂³

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂×C₄² — C₂⁴.₃C₂² — C₂⁴.₄₅₆C₂³

Lower central

C₁ — C₂³ — C₂⁴.₄₅₆C₂³

Upper central

C₁ — C₂³ — C₂⁴.₄₅₆C₂³

Jennings

C₁ — C₂³ — C₂⁴.₄₅₆C₂³

Generators and relations for C₂⁴.₄₅₆C₂³
G = < a,b,c,d,e,f,g | a²=b²=c²=e²=1, d²=abc, f²=cb=bc, g²=ba=ab, ac=ca, ede=ad=da, geg^-1=ae=ea, af=fa, ag=ga, 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: 516 in 240 conjugacy classes, 96 normal (14 characteristic)
C₁, C₂, C₂, C₂, C₄, C₂², C₂², C₂², C₂×C₄, C₂×C₄, D₄, Q₈, C₂³, C₂³, C₂³, C₄², C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂×Q₈, C₂⁴, C₂.C₄², C₂×C₄², C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₂×C₄⋊C₄, C₂³×C₄, C₂²×D₄, C₂²×Q₈, C₂³.₈Q₈, C₂³.₂₃D₄, C₂⁴.₃C₂², C₂³.₆₇C₂³, C₂³⋊Q₈, C₂³.₈₃C₂³, C₂⁴.₄₅₆C₂³
Quotients: C₁, C₂, C₂², Q₈, C₂³, C₂×Q₈, C₄○D₄, C₂⁴, C₂²×Q₈, C₂×C₄○D₄, 2₊¹⁺⁴, C₂².₃₂C₂⁴, C₂³⋊₂Q₈, D₄⋊₃Q₈, C₂².₅₃C₂⁴, C₂⁴⋊C₂², C₂⁴.₄₅₆C₂³

Smallest permutation representation of C₂⁴.₄₅₆C₂³
►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)

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

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

(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), (1,39)(2,8)(3,37)(4,6)(5,53)(7,55)(9,27)(10,23)(11,25)(12,21)(13,36)(14,64)(15,34)(16,62)(17,59)(18,31)(19,57)(20,29)(22,43)(24,41)(26,42)(28,44)(30,49)(32,51)(33,46)(35,48)(38,54)(40,56)(45,63)(47,61)(50,60)(52,58), (1,61,53,36)(2,49,54,19)(3,63,55,34)(4,51,56,17)(5,58,39,31)(6,48,40,14)(7,60,37,29)(8,46,38,16)(9,62,41,33)(10,50,42,20)(11,64,43,35)(12,52,44,18)(13,28,47,21)(15,26,45,23)(22,32,25,59)(24,30,27,57), (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), (1,39)(2,8)(3,37)(4,6)(5,53)(7,55)(9,27)(10,23)(11,25)(12,21)(13,36)(14,64)(15,34)(16,62)(17,59)(18,31)(19,57)(20,29)(22,43)(24,41)(26,42)(28,44)(30,49)(32,51)(33,46)(35,48)(38,54)(40,56)(45,63)(47,61)(50,60)(52,58), (1,61,53,36)(2,49,54,19)(3,63,55,34)(4,51,56,17)(5,58,39,31)(6,48,40,14)(7,60,37,29)(8,46,38,16)(9,62,41,33)(10,50,42,20)(11,64,43,35)(12,52,44,18)(13,28,47,21)(15,26,45,23)(22,32,25,59)(24,30,27,57), (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)], [(1,39),(2,8),(3,37),(4,6),(5,53),(7,55),(9,27),(10,23),(11,25),(12,21),(13,36),(14,64),(15,34),(16,62),(17,59),(18,31),(19,57),(20,29),(22,43),(24,41),(26,42),(28,44),(30,49),(32,51),(33,46),(35,48),(38,54),(40,56),(45,63),(47,61),(50,60),(52,58)], [(1,61,53,36),(2,49,54,19),(3,63,55,34),(4,51,56,17),(5,58,39,31),(6,48,40,14),(7,60,37,29),(8,46,38,16),(9,62,41,33),(10,50,42,20),(11,64,43,35),(12,52,44,18),(13,28,47,21),(15,26,45,23),(22,32,25,59),(24,30,27,57)], [(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	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
type	+	+	+	+	+	+	+	-		+
image	C₁	C₂	C₂	C₂	C₂	C₂	C₂	Q₈	C₄○D₄	2₊¹⁺⁴
kernel	C₂⁴.₄₅₆C₂³	C₂³.₈Q₈	C₂³.₂₃D₄	C₂⁴.₃C₂²	C₂³.₆₇C₂³	C₂³⋊Q₈	C₂³.₈₃C₂³	C₂×D₄	C₂×C₄	C₂²
# reps	1	4	2	1	2	4	2	4	8	4

Matrix representation of C₂⁴.₄₅₆C₂³ ►in GL₆(𝔽₅)

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
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

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

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

1	0	0	0	0	0
0	1	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	4	0

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	0	1
0	0	0	0	4	0

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

C₂⁴.₄₅₆C₂³ in GAP, Magma, Sage, TeX

C_2^4._{456}C_2^3

% in TeX

G:=Group("C2^4.456C2^3");

// GroupNames label

G:=SmallGroup(128,1536);

// by ID

G=gap.SmallGroup(128,1536);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,784,253,758,723,520,1571,346,192]);

// 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=c*b=b*c,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,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 = C24.456C23 order 128 = 27

296th non-split extension by C24 of C23 acting via C23/C2=C22

G = C₂⁴.₄₅₆C₂³ order 128 = 2⁷

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