C2^3.84C2^3 - GroupNames

G = C₂³.₈₄C₂³ order 64 = 2⁶

10^th central stem extension by C₂³ of C₂³

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

Aliases: C₂³.₈₄C₂³, C₂.₇(C₄²:₂C₂), C₂².₄₄(C₄oD₄), C₂.C₄².₃C₂, (C₂²xC₄).₁₂C₂², 2-Sylow(Sz₈), SmallGroup(64,82)

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

Derived series

C₁ — C₂³ — C₂³.₈₄C₂³

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²xC₄ — C₂.C₄² — C₂³.₈₄C₂³

Lower central

C₁ — C₂³ — C₂³.₈₄C₂³

Upper central

C₁ — C₂³ — C₂³.₈₄C₂³

Jennings

C₁ — C₂³ — C₂³.₈₄C₂³

Generators and relations for C₂³.₈₄C₂³
G = < a,b,c,d,e,f | a²=b²=c²=1, d²=abc, e²=ba=ab, f²=a, ac=ca, ede^-1=ad=da, ae=ea, af=fa, bc=cb, fdf^-1=bd=db, be=eb, bf=fb, cd=dc, fef^-1=ce=ec, cf=fc >

Subgroups: 101 in 59 conjugacy classes, 31 normal (3 characteristic)
Quotients: C₁, C₂, C₂², C₂³, C₄oD₄, C₄²:₂C₂, C₂³.₈₄C₂³

C₁

C₂

C₂²

₄C₄

C₂³

₂C₂xC₄

C₂²xC₄

C₂.C₄²

C₂³.₈₄C₂³

Character table of C₂³.₈₄C₂³

class	1	2A	2B	2C	2D	2E	2F	2G	4A	4B	4C	4D	4E	4F	4G	4H	4I	4J	4K	4L	4M	4N
size	1	1	1	1	1	1	1	1	4	4	4	4	4	4	4	4	4	4	4	4	4	4
ρ₁	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	trivial
ρ₂	1	1	1	1	1	1	1	1	-1	-1	1	-1	1	1	-1	-1	-1	1	-1	1	1	-1	linear of order 2
ρ₃	1	1	1	1	1	1	1	1	1	1	1	-1	-1	-1	-1	1	1	1	-1	-1	-1	-1	linear of order 2
ρ₄	1	1	1	1	1	1	1	1	-1	-1	1	1	-1	-1	1	-1	-1	1	1	-1	-1	1	linear of order 2
ρ₅	1	1	1	1	1	1	1	1	-1	1	-1	-1	1	-1	1	-1	1	-1	-1	1	-1	1	linear of order 2
ρ₆	1	1	1	1	1	1	1	1	1	-1	-1	1	1	-1	-1	1	-1	-1	1	1	-1	-1	linear of order 2
ρ₇	1	1	1	1	1	1	1	1	-1	1	-1	1	-1	1	-1	-1	1	-1	1	-1	1	-1	linear of order 2
ρ₈	1	1	1	1	1	1	1	1	1	-1	-1	-1	-1	1	1	1	-1	-1	-1	-1	1	1	linear of order 2
ρ₉	2	-2	-2	2	-2	2	2	-2	0	0	0	0	0	0	2i	0	0	0	0	0	0	-2i	complex lifted from C₄oD₄
ρ₁₀	2	-2	2	2	2	-2	-2	-2	-2i	0	0	0	0	0	0	2i	0	0	0	0	0	0	complex lifted from C₄oD₄
ρ₁₁	2	2	-2	-2	2	2	-2	-2	0	2i	0	0	0	0	0	0	-2i	0	0	0	0	0	complex lifted from C₄oD₄
ρ₁₂	2	-2	-2	-2	2	-2	2	2	0	0	2i	0	0	0	0	0	0	-2i	0	0	0	0	complex lifted from C₄oD₄
ρ₁₃	2	-2	2	-2	-2	2	-2	2	0	0	0	0	-2i	0	0	0	0	0	0	2i	0	0	complex lifted from C₄oD₄
ρ₁₄	2	-2	2	2	2	-2	-2	-2	2i	0	0	0	0	0	0	-2i	0	0	0	0	0	0	complex lifted from C₄oD₄
ρ₁₅	2	2	-2	2	-2	-2	-2	2	0	0	0	0	0	2i	0	0	0	0	0	0	-2i	0	complex lifted from C₄oD₄
ρ₁₆	2	-2	-2	2	-2	2	2	-2	0	0	0	0	0	0	-2i	0	0	0	0	0	0	2i	complex lifted from C₄oD₄
ρ₁₇	2	2	-2	2	-2	-2	-2	2	0	0	0	0	0	-2i	0	0	0	0	0	0	2i	0	complex lifted from C₄oD₄
ρ₁₈	2	2	2	-2	-2	-2	2	-2	0	0	0	-2i	0	0	0	0	0	0	2i	0	0	0	complex lifted from C₄oD₄
ρ₁₉	2	2	-2	-2	2	2	-2	-2	0	-2i	0	0	0	0	0	0	2i	0	0	0	0	0	complex lifted from C₄oD₄
ρ₂₀	2	-2	2	-2	-2	2	-2	2	0	0	0	0	2i	0	0	0	0	0	0	-2i	0	0	complex lifted from C₄oD₄
ρ₂₁	2	2	2	-2	-2	-2	2	-2	0	0	0	2i	0	0	0	0	0	0	-2i	0	0	0	complex lifted from C₄oD₄
ρ₂₂	2	-2	-2	-2	2	-2	2	2	0	0	-2i	0	0	0	0	0	0	2i	0	0	0	0	complex lifted from C₄oD₄

Smallest permutation representation of C₂³.₈₄C₂³
►Regular action 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 50 42 36)(2 19 43 64)(3 52 44 34)(4 17 41 62)(5 58 23 15)(6 30 24 48)(7 60 21 13)(8 32 22 46)(9 33 56 51)(10 61 53 20)(11 35 54 49)(12 63 55 18)(14 40 57 27)(16 38 59 25)(26 45 39 31)(28 47 37 29)

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

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

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

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

C₂³.₈₄C₂³ is a maximal subgroup of
C₂⁴.₅₇₇C₂³ C₂³.₄₁₀C₂⁴ C₂⁴.₃₁₅C₂³ C₂³.₄₂₈C₂⁴ C₂⁴.₃₄₀C₂³ C₂³.₄₉₄C₂⁴ C₂³.₅₄₃C₂⁴ C₂³.₅₅₃C₂⁴ C₂³.₅₅₅C₂⁴ C₂³.₆₄₅C₂⁴ C₂⁴.₄₃₂C₂³ C₂³.₆₅₉C₂⁴ C₂³.₆₆₀C₂⁴ C₂³.₆₆₂C₂⁴ C₂⁴.₄₄₃C₂³ C₂³.₆₆₆C₂⁴ C₂³.₆₇₆C₂⁴ C₂³.₆₇₈C₂⁴ C₂³.₆₈₇C₂⁴ C₂³.₆₈₉C₂⁴ C₂³.₇₀₁C₂⁴ C₂³.₇₂₅C₂⁴ C₂³.₇₃₃C₂⁴ C₂³.₇₃₄C₂⁴ C₂³.₇₃₈C₂⁴ C₂³.₁₉(C₂xA₄) C₂³.F₈
C_2p.(C₄²:₂C₂): C₂³.₃₀₁C₂⁴ C₂⁴.₃₀₄C₂³ C₂³.₄₁₄C₂⁴ C₂³.₅₄₅C₂⁴ (C₂²xC₄).₃₀D₆ (C₂²xC₄).D₁₀ (C₂²xC₄).D₁₄ ...
C₂³.₈₄C₂³ is a maximal quotient of
C₂⁴.₄C₂³
(C₂²xC₄).D_2p: C₂⁴.₆₃₃C₂³ (C₂²xC₄).₃₀D₆ (C₂²xC₄).D₁₀ (C₂²xC₄).D₁₄ ...

Matrix representation of C₂³.₈₄C₂³ ►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	4	0
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	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	4	0
0	0	0	0	0	4

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

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	1	1
0	0	0	0	0	4

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

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

C₂³.₈₄C₂³ in GAP, Magma, Sage, TeX

C_2^3._{84}C_2^3

% in TeX

G:=Group("C2^3.84C2^3");

// GroupNames label

G:=SmallGroup(64,82);

// by ID

G=gap.SmallGroup(64,82);

# by ID

G:=PCGroup([6,-2,2,2,-2,2,2,336,121,151,362,332,50]);

// Polycyclic

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

// generators/relations

Export

Subgroup lattice of C₂³.₈₄C₂³ in TeX
Character table of C₂³.₈₄C₂³ in TeX

G = C23.84C23 order 64 = 26

10th central stem extension by C23 of C23

G = C₂³.₈₄C₂³ order 64 = 2⁶

10^th central stem extension by C₂³ of C₂³