C2^3.78C2^3 - GroupNames

G = C₂³.₇₈C₂³ order 64 = 2⁶

4^th central stem extension by C₂³ of C₂³

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

Aliases: C₂³.₇₈C₂³, (C₂×C₄)⋊₁Q₈, (C₂×C₄).₁₅D₄, C₂.₄(C₄⋊Q₈), C₂.₇C₂²≀C₂, C₂².₇₁(C₂×D₄), C₂.₇(C₂²⋊Q₈), (C₂²×Q₈).₂C₂, C₂².₂₁(C₂×Q₈), C₂².₃₈(C₄○D₄), C₂.C₄².₈C₂, (C₂²×C₄).₂₇C₂², (C₂×C₄⋊C₄).₉C₂, SmallGroup(64,76)

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

Derived series

C₁ — C₂³ — C₂³.₇₈C₂³

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂×C₄⋊C₄ — C₂³.₇₈C₂³

Lower central

C₁ — C₂³ — C₂³.₇₈C₂³

Upper central

C₁ — C₂³ — C₂³.₇₈C₂³

Jennings

C₁ — C₂³ — C₂³.₇₈C₂³

Generators and relations for C₂³.₇₈C₂³
G = < a,b,c,d,e,f | a²=b²=c²=1, d²=e²=f²=a, ab=ba, 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: 141 in 91 conjugacy classes, 43 normal (7 characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₂×C₄, C₂×C₄, Q₈, C₂³, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂×Q₈, C₂.C₄², C₂×C₄⋊C₄, C₂²×Q₈, C₂³.₇₈C₂³
Quotients: C₁, C₂, C₂², D₄, Q₈, C₂³, C₂×D₄, C₂×Q₈, C₄○D₄, C₂²≀C₂, C₂²⋊Q₈, C₄⋊Q₈, C₂³.₇₈C₂³

Character table of C₂³.₇₈C₂³

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	-2	0	0	0	0	0	0	2	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₀	2	2	-2	2	-2	-2	-2	2	0	0	0	0	0	2	0	0	0	0	0	0	-2	0	orthogonal lifted from D₄
ρ₁₁	2	2	-2	2	-2	-2	-2	2	0	0	0	0	0	-2	0	0	0	0	0	0	2	0	orthogonal lifted from D₄
ρ₁₂	2	2	-2	-2	2	2	-2	-2	0	2	0	0	0	0	0	0	-2	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₃	2	2	2	-2	-2	-2	2	-2	0	0	0	2	0	0	0	0	0	0	-2	0	0	0	orthogonal lifted from D₄
ρ₁₄	2	2	2	-2	-2	-2	2	-2	0	0	0	-2	0	0	0	0	0	0	2	0	0	0	orthogonal lifted from D₄
ρ₁₅	2	-2	-2	2	-2	2	2	-2	0	0	0	0	0	0	2	0	0	0	0	0	0	-2	symplectic lifted from Q₈, Schur index 2
ρ₁₆	2	-2	-2	-2	2	-2	2	2	0	0	2	0	0	0	0	0	0	-2	0	0	0	0	symplectic lifted from Q₈, Schur index 2
ρ₁₇	2	-2	-2	2	-2	2	2	-2	0	0	0	0	0	0	-2	0	0	0	0	0	0	2	symplectic lifted from Q₈, Schur index 2
ρ₁₈	2	-2	2	-2	-2	2	-2	2	0	0	0	0	2	0	0	0	0	0	0	-2	0	0	symplectic lifted from Q₈, Schur index 2
ρ₁₉	2	-2	2	-2	-2	2	-2	2	0	0	0	0	-2	0	0	0	0	0	0	2	0	0	symplectic lifted from Q₈, Schur index 2
ρ₂₀	2	-2	-2	-2	2	-2	2	2	0	0	-2	0	0	0	0	0	0	2	0	0	0	0	symplectic lifted from Q₈, Schur index 2
ρ₂₁	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 C₄○D₄
ρ₂₂	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 C₄○D₄

Smallest permutation representation of C₂³.₇₈C₂³
►Regular action on 64 points

Generators in S₆₄

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

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

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

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

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

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

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

C₂³.₇₈C₂³ is a maximal subgroup of
C₂.₇C₂≀C₄ C₄⋊C₄.₆D₄ Q₈⋊D₄⋊C₂ (C₂×C₄)⋊Q₁₆ C₄².₁₆₂D₄ C₄²⋊₅Q₈ C₂⁴.₂₄₃C₂³ C₂⁴.₂₆₂C₂³ C₂⁴.₅₆₇C₂³ C₂⁴.₅₆₈C₂³ C₂³.₃₄₉C₂⁴ C₂³.₃₅₁C₂⁴ C₂⁴.₂₈₂C₂³ C₂³.₃₆₂C₂⁴ C₂⁴.₂₈₅C₂³ C₂³.₃₆₉C₂⁴ C₂³.₃₇₄C₂⁴ C₂³.₄₂₀C₂⁴ C₂⁴.₃₁₁C₂³ C₂³.₄₄₉C₂⁴ C₄²⋊₆Q₈ C₄²⋊₇Q₈ C₂³.₄₅₅C₂⁴ C₂³.₄₅₆C₂⁴ C₂⁴.₅₈₃C₂³ C₄².₁₇₆D₄ C₂⁴.₃₃₈C₂³ C₄².₁₇₉D₄ C₂³.₄₈₃C₂⁴ C₄².₁₈₁D₄ C₂³.₄₈₆C₂⁴ C₂³.₄₈₈C₂⁴ C₄².₁₈₃D₄ C₄².₁₈₄D₄ C₄²⋊₈Q₈ C₄²⋊₂₆D₄ C₄²⋊₉Q₈ C₂³.₅₁₄C₂⁴ C₂³.₅₂₇C₂⁴ C₄².₁₈₇D₄ C₄².₁₈₉D₄ C₄².₁₉₁D₄ C₄².₁₉₂D₄ C₂⁴.₃₇₄C₂³ C₂³.₅₅₀C₂⁴ C₂³.₅₅₉C₂⁴ C₄²⋊₁₀Q₈ C₄²⋊₃₂D₄ C₂⁴.₃₇₈C₂³ C₂⁴.₃₇₉C₂³ C₄²⋊₁₁Q₈ C₂³.₅₇₂C₂⁴ C₂³.₅₇₄C₂⁴ C₂⁴.₃₈₅C₂³ C₂³.₅₈₀C₂⁴ C₂³.₅₈₃C₂⁴ C₂⁴.₃₉₃C₂³ C₂³.₅₈₉C₂⁴ C₂³.₅₉₀C₂⁴ C₂³.₅₉₂C₂⁴ C₂⁴.₄₀₅C₂³ C₂³.₆₀₀C₂⁴ C₂⁴.₄₀₈C₂³ C₂³.₆₁₃C₂⁴ C₂³.₆₁₅C₂⁴ C₂³.₆₁₇C₂⁴ C₂³.₆₂₀C₂⁴ C₂⁴.₄₁₈C₂³ C₂⁴.₄₂₁C₂³ C₂³.₆₃₄C₂⁴ C₂³.₆₃₇C₂⁴ C₂⁴.₄₂₈C₂³ C₂³.₆₄₅C₂⁴ C₂³.₆₅₅C₂⁴ C₂³.₆₅₈C₂⁴ C₂³.₆₅₉C₂⁴ C₂³.₆₆₂C₂⁴ C₂³.₆₆₃C₂⁴ C₂³.₆₇₄C₂⁴ C₂³.₆₇₅C₂⁴ C₂⁴.₄₅₀C₂³ C₂³.₆₈₅C₂⁴ C₂³.₆₈₈C₂⁴ C₂³.₆₈₉C₂⁴ C₂³.₆₉₂C₂⁴ C₂³.₆₉₉C₂⁴ C₂³.₇₀₅C₂⁴ C₂³.₇₀₆C₂⁴ C₂³.₇₁₁C₂⁴ C₂³.₇₁₄C₂⁴ C₂⁴.₄₆₂C₂³ C₄².₁₉₉D₄ C₄².₂₀₀D₄ C₂³.₇₃₀C₂⁴ C₂³.₇₃₁C₂⁴ C₂³.₇₃₂C₂⁴ C₂³.₇₃₃C₂⁴ C₂³.₇₃₅C₂⁴ C₂³.₇₃₈C₂⁴ C₂³.₇₄₁C₂⁴ C₄²⋊₁₂Q₈ C₄²⋊₁₃Q₈ (C₂²×C₄).A₄
C_2p.C₂²≀C₂: C₂³.₂₈₈C₂⁴ C₂³.₃₀₉C₂⁴ C₂³.₃₃₄C₂⁴ C₂⁴.₃₆₁C₂³ (C₂×C₄)⋊Dic₆ (C₂×Dic₃)⋊Q₈ C₂².₅₂(S₃×Q₈) (C₂×Dic₅)⋊Q₈ ...
C₂³.₇₈C₂³ is a maximal quotient of
C₂⁴.₆₃₁C₂³ C₂⁴.₆₃₄C₂³ C₂⁴.₆₃₆C₂³ C₂⁴.₁₈₂C₂³
(C₂×C_4p)⋊Q₈: C₄⋊C₄⋊Q₈ (C₂×C₈)⋊Q₈ (C₂×C₄)⋊Dic₆ (C₂×Dic₃)⋊Q₈ (C₂×Dic₅)⋊Q₈ (C₂×C₄)⋊Dic₁₀ (C₂×Dic₇)⋊Q₈ (C₂×C₄)⋊Dic₁₄ ...
C_2p.(C₄⋊Q₈): C₂.(C₈⋊Q₈) M₄(2)⋊Q₈ C₄²⋊₃Q₈ C₂².₅₂(S₃×Q₈) C₁₀.C₂²≀C₂ C₁₄.C₂²≀C₂ ...

Matrix representation of C₂³.₇₈C₂³ ►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

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

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

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

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

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

C₂³.₇₈C₂³ in GAP, Magma, Sage, TeX

C_2^3._{78}C_2^3

% in TeX

G:=Group("C2^3.78C2^3");

// GroupNames label

G:=SmallGroup(64,76);

// by ID

G=gap.SmallGroup(64,76);

# by ID

G:=PCGroup([6,-2,2,2,-2,2,2,48,121,55,362,332,50]);

// Polycyclic

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

Character table of C₂³.₇₈C₂³ in TeX

G = C23.78C23 order 64 = 26

4th central stem extension by C23 of C23

G = C₂³.₇₈C₂³ order 64 = 2⁶

4^th central stem extension by C₂³ of C₂³