C2^3.392C2^4 - GroupNames

G = C₂³.₃₉₂C₂⁴ order 128 = 2⁷

109^th central stem extension by C₂³ of C₂⁴

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

Aliases: C₂³.₃₉₂C₂⁴, C₂⁴.₃₀₃C₂³, C₂².₁₉₂2₊¹⁺⁴, C₂².₁₄₄2_-¹⁺⁴, C₂.₁₉(D₄×Q₈), C₄⋊C₄.₂₃₀D₄, C₂²⋊C₄⋊₁₅Q₈, C₂³.₁₇(C₂×Q₈), C₂.₆₃(D₄⋊₅D₄), C₂³⋊Q₈.₆C₂, C₂.₁₉(D₄⋊₃Q₈), C₄.₄₄(C₄.₄D₄), C₂².₈₅(C₂²×Q₈), (C₂×C₄²).₅₂₀C₂², (C₂³×C₄).₃₇₇C₂², (C₂²×C₄).₈₂₇C₂³, C₂².₂₇₂(C₂²×D₄), C₂³.₇Q₈.₄₃C₂, (C₂²×Q₈).₁₁₇C₂², C₂³.₆₇C₂³⋊₅₀C₂, C₂³.₈₃C₂³⋊₂₀C₂, C₂⁴.C₂².₁₉C₂, C₂.C₄².₁₄₄C₂², C₂.₂₁(C₂².₅₀C₂⁴), C₂.₂₉(C₂².₃₆C₂⁴), C₂.₁₉(C₂³.₃₇C₂³), (C₄×C₄⋊C₄)⋊₇₀C₂, (C₂×C₄⋊Q₈)⋊₁₁C₂, (C₂×C₄).₆₃(C₂×D₄), (C₂×C₄).₃₈(C₂×Q₈), (C₄×C₂²⋊C₄).₄₆C₂, C₂.₁₈(C₂×C₄.₄D₄), (C₂×C₂²⋊Q₈).₃₀C₂, (C₂×C₄).₁₂₃(C₄○D₄), (C₂×C₄⋊C₄).₂₆₂C₂², C₂².₂₆₉(C₂×C₄○D₄), (C₂×C₂²⋊C₄).₁₅₇C₂², SmallGroup(128,1224)

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

Derived series

C₁ — C₂³ — C₂³.₃₉₂C₂⁴

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂³×C₄ — C₄×C₂²⋊C₄ — C₂³.₃₉₂C₂⁴

Lower central

C₁ — C₂³ — C₂³.₃₉₂C₂⁴

Upper central

C₁ — C₂³ — C₂³.₃₉₂C₂⁴

Jennings

C₁ — C₂³ — C₂³.₃₉₂C₂⁴

Generators and relations for C₂³.₃₉₂C₂⁴
G = < a,b,c,d,e,f,g | a²=b²=c²=d²=1, e²=db=bd, f²=cb=bc, g²=b, ab=ba, eae^-1=ac=ca, faf^-1=ad=da, ag=ga, 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: 468 in 254 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, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂²×C₄, C₂×Q₈, C₂⁴, C₂.C₄², C₂.C₄², C₂×C₄², C₂×C₄², C₂×C₂²⋊C₄, C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₂×C₄⋊C₄, C₂²⋊Q₈, C₄⋊Q₈, C₂³×C₄, C₂²×Q₈, C₂²×Q₈, C₄×C₂²⋊C₄, C₄×C₄⋊C₄, C₂³.₇Q₈, C₂⁴.C₂², C₂³.₆₇C₂³, C₂³.₆₇C₂³, C₂³⋊Q₈, C₂³.₈₃C₂³, C₂×C₂²⋊Q₈, C₂×C₄⋊Q₈, C₂³.₃₉₂C₂⁴
Quotients: C₁, C₂, C₂², D₄, Q₈, C₂³, C₂×D₄, C₂×Q₈, C₄○D₄, C₂⁴, C₄.₄D₄, C₂²×D₄, C₂²×Q₈, C₂×C₄○D₄, 2₊¹⁺⁴, 2_-¹⁺⁴, C₂×C₄.₄D₄, C₂³.₃₇C₂³, C₂².₃₆C₂⁴, D₄⋊₅D₄, D₄×Q₈, D₄⋊₃Q₈, C₂².₅₀C₂⁴, C₂³.₃₉₂C₂⁴

Smallest permutation representation of C₂³.₃₉₂C₂⁴
►On 64 points

Generators in S₆₄

(2 59)(4 57)(5 61)(6 43)(7 63)(8 41)(9 21)(10 38)(11 23)(12 40)(14 32)(16 30)(17 27)(19 25)(22 35)(24 33)(34 37)(36 39)(42 52)(44 50)(46 56)(48 54)(49 62)(51 64)

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

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

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

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

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

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

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

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

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	2	2	2	4	4
type	+	+	+	+	+	+	+	+	+	+	-	+		+	-
image	C₁	C₂	C₂	C₂	C₂	C₂	C₂	C₂	C₂	C₂	Q₈	D₄	C₄○D₄	2₊¹⁺⁴	2_-¹⁺⁴
kernel	C₂³.₃₉₂C₂⁴	C₄×C₂²⋊C₄	C₄×C₄⋊C₄	C₂³.₇Q₈	C₂⁴.C₂²	C₂³.₆₇C₂³	C₂³⋊Q₈	C₂³.₈₃C₂³	C₂×C₂²⋊Q₈	C₂×C₄⋊Q₈	C₂²⋊C₄	C₄⋊C₄	C₂×C₄	C₂²	C₂²
# reps	1	1	1	1	2	4	2	2	1	1	4	4	12	1	1

Matrix representation of C₂³.₃₉₂C₂⁴ ►in GL₆(𝔽₅)

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	1	4

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

1	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	3
0	0	0	0	0	4

2	0	0	0	0	0
0	3	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	1

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

C₂³.₃₉₂C₂⁴ in GAP, Magma, Sage, TeX

C_2^3._{392}C_2^4

% in TeX

G:=Group("C2^3.392C2^4");

// GroupNames label

G:=SmallGroup(128,1224);

// by ID

G=gap.SmallGroup(128,1224);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,560,253,344,758,723,675,80]);

// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=1,e^2=d*b=b*d,f^2=c*b=b*c,g^2=b,a*b=b*a,e*a*e^-1=a*c=c*a,f*a*f^-1=a*d=d*a,a*g=g*a,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 = C23.392C24 order 128 = 27

109th central stem extension by C23 of C24

G = C₂³.₃₉₂C₂⁴ order 128 = 2⁷

109^th central stem extension by C₂³ of C₂⁴