C2^3.323C2^4 - GroupNames

G = C₂³.₃₂₃C₂⁴ order 128 = 2⁷

40^th central stem extension by C₂³ of C₂⁴

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

Aliases: C₂³.₃₂₃C₂⁴, C₂⁴.₂₅₇C₂³, C₂².₉₈2_-¹⁺⁴, C₂².₁₃₆2₊¹⁺⁴, C₂.₈(D₄×Q₈), C₂²⋊C₄⋊₁₀Q₈, (C₂×Q₈).₂₂₀D₄, C₄²⋊₉C₄⋊₁₆C₂, C₂³.₁₀(C₂×Q₈), C₂.₈(Q₈⋊₆D₄), C₂.₁₀(D₄⋊₃Q₈), C₂³.₄Q₈.₃C₂, C₂².₆₅(C₂²×Q₈), (C₂×C₄²).₄₇₁C₂², (C₂³×C₄).₃₃₇C₂², (C₂²×C₄).₇₉₃C₂³, C₂².₂₀₃(C₂²×D₄), C₂³.₇Q₈.₃₅C₂, C₂³.₈₁C₂³⋊₆C₂, C₄.₄₉(C₂².D₄), (C₂²×Q₈).₄₁₇C₂², C₂³.₆₇C₂³⋊₃₂C₂, C₂³.₆₅C₂³⋊₄₁C₂, C₂³.₆₃C₂³⋊₃₀C₂, C₂⁴.C₂².₁₂C₂, C₂.C₄².₈₆C₂², C₂.₁₇(C₂².₄₆C₂⁴), C₂.₁₃(C₂².₃₆C₂⁴), C₂.₁₀(C₂³.₃₇C₂³), (C₂×C₄×Q₈)⋊₁₄C₂, (C₂×C₄).₂₆(C₂×Q₈), (C₂×C₄).₃₁₂(C₂×D₄), (C₂×C₄).₉₇(C₄○D₄), (C₄×C₂²⋊C₄).₃₆C₂, (C₂×C₂²⋊Q₈).₂₃C₂, (C₂×C₄⋊C₄).₂₁₁C₂², C₂².₂₀₂(C₂×C₄○D₄), C₂.₂₀(C₂×C₂².D₄), (C₂×C₂²⋊C₄).₄₉₁C₂², SmallGroup(128,1155)

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

Derived series

C₁ — C₂³ — C₂³.₃₂₃C₂⁴

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂³×C₄ — C₄×C₂²⋊C₄ — C₂³.₃₂₃C₂⁴

Lower central

C₁ — C₂³ — C₂³.₃₂₃C₂⁴

Upper central

C₁ — C₂³ — C₂³.₃₂₃C₂⁴

Jennings

C₁ — C₂³ — C₂³.₃₂₃C₂⁴

Generators and relations for C₂³.₃₂₃C₂⁴
G = < a,b,c,d,e,f,g | a²=b²=c²=d²=1, e²=g²=b, f²=cb=bc, ab=ba, eae^-1=ac=ca, faf^-1=ad=da, ag=ga, bd=db, fef^-1=geg^-1=be=eb, bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, cg=gc, de=ed, df=fd, dg=gd, fg=gf >

Subgroups: 436 in 250 conjugacy classes, 112 normal (42 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₂×C₄, C₂×C₄, Q₈, C₂³, C₂³, C₂³, C₄², C₂²⋊C₄, C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂²×C₄, C₂×Q₈, C₂×Q₈, C₂⁴, C₂.C₄², C₂.C₄², C₂×C₄², C₂×C₄², C₂×C₂²⋊C₄, C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₂×C₄⋊C₄, C₄×Q₈, C₂²⋊Q₈, C₂³×C₄, C₂²×Q₈, C₄×C₂²⋊C₄, C₂³.₇Q₈, C₄²⋊₉C₄, C₂³.₆₃C₂³, C₂⁴.C₂², C₂³.₆₅C₂³, C₂³.₆₇C₂³, C₂³.₈₁C₂³, C₂³.₄Q₈, C₂×C₄×Q₈, C₂×C₂²⋊Q₈, C₂³.₃₂₃C₂⁴
Quotients: C₁, C₂, C₂², D₄, Q₈, C₂³, C₂×D₄, C₂×Q₈, C₄○D₄, C₂⁴, C₂².D₄, C₂²×D₄, C₂²×Q₈, C₂×C₄○D₄, 2₊¹⁺⁴, 2_-¹⁺⁴, C₂×C₂².D₄, C₂³.₃₇C₂³, C₂².₃₆C₂⁴, D₄×Q₈, Q₈⋊₆D₄, C₂².₄₆C₂⁴, D₄⋊₃Q₈, C₂³.₃₂₃C₂⁴

Smallest permutation representation of C₂³.₃₂₃C₂⁴
►On 64 points

Generators in S₆₄

(2 10)(4 12)(5 24)(6 49)(7 22)(8 51)(14 42)(16 44)(17 36)(18 62)(19 34)(20 64)(21 40)(23 38)(26 56)(28 54)(30 60)(32 58)(33 46)(35 48)(37 50)(39 52)(45 61)(47 63)

(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 9)(2 10)(3 11)(4 12)(5 39)(6 40)(7 37)(8 38)(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 55)(26 56)(27 53)(28 54)(29 59)(30 60)(31 57)(32 58)(33 62)(34 63)(35 64)(36 61)

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

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

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

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

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

38 conjugacy classes

class	1	2A	···	2G	2H	2I	4A	···	4H	4I	···	4X	4Y	4Z	4AA	4AB
order	1	2	···	2	2	2	4	···	4	4	···	4	4	4	4	4
size	1	1	···	1	4	4	2	···	2	4	···	4	8	8	8	8

38 irreducible representations

dim	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	Q₈	D₄	C₄○D₄	2₊¹⁺⁴	2_-¹⁺⁴
kernel	C₂³.₃₂₃C₂⁴	C₄×C₂²⋊C₄	C₂³.₇Q₈	C₄²⋊₉C₄	C₂³.₆₃C₂³	C₂⁴.C₂²	C₂³.₆₅C₂³	C₂³.₆₇C₂³	C₂³.₈₁C₂³	C₂³.₄Q₈	C₂×C₄×Q₈	C₂×C₂²⋊Q₈	C₂²⋊C₄	C₂×Q₈	C₂×C₄	C₂²	C₂²
# reps	1	1	1	1	2	2	1	1	2	2	1	1	4	4	12	1	1

Matrix representation of C₂³.₃₂₃C₂⁴ ►in GL₆(𝔽₅)

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

4	0	0	0	0	0
0	4	0	0	0	0
0	0	4	0	0	0
0	0	0	4	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

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

4	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	2	0
0	0	0	0	0	2

2	0	0	0	0	0
0	3	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

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

C₂³.₃₂₃C₂⁴ in GAP, Magma, Sage, TeX

C_2^3._{323}C_2^4

% in TeX

G:=Group("C2^3.323C2^4");

// GroupNames label

G:=SmallGroup(128,1155);

// by ID

G=gap.SmallGroup(128,1155);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,112,253,344,758,723,675,80]);

// Polycyclic

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

// generators/relations

G = C23.323C24 order 128 = 27

40th central stem extension by C23 of C24

G = C₂³.₃₂₃C₂⁴ order 128 = 2⁷

40^th central stem extension by C₂³ of C₂⁴