C2^4.385C2^3 - GroupNames

G = C₂⁴.₃₈₅C₂³ order 128 = 2⁷

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

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

Aliases: C₂⁴.₃₈₅C₂³, C₂³.₅₇₇C₂⁴, C₂².₃₅₁2₊¹⁺⁴, C₂².₂₆₂2_-¹⁺⁴, C₂.₄₁D₄², C₂²⋊C₄⋊₂Q₈, C₂.₂₆(D₄×Q₈), C₄⋊C₄.₁₁₆D₄, C₂³.₃₁(C₂×Q₈), C₂³⋊Q₈.₁₉C₂, (C₂³×C₄).₄₄₇C₂², (C₂²×C₄).₈₆₆C₂³, (C₂×C₄²).₆₃₆C₂², C₂².₃₈₆(C₂²×D₄), C₂³.₈Q₈.₄₄C₂, C₂³.₄Q₈.₁₉C₂, C₂².₁₄₂(C₂²×Q₈), (C₂²×Q₈).₁₇₅C₂², C₂³.₇₈C₂³⋊₃₈C₂, C₂³.₈₁C₂³⋊₇₆C₂, C₂⁴.C₂².₄₇C₂, C₂³.₆₃C₂³⋊₁₂₇C₂, C₂.C₄².₂₈₈C₂², C₂.₈(C₂².₅₇C₂⁴), C₂.₂₉(C₂³.₄₁C₂³), C₂.₅₃(C₂³.₃₈C₂³), (C₂×C₄⋊Q₈)⋊₂₁C₂, (C₂×C₄).₈₅(C₂×D₄), (C₂×C₄).₆₆(C₂×Q₈), (C₂×C₂²⋊Q₈).₄₁C₂, (C₂×C₄⋊C₄).₃₉₅C₂², (C₂×C₂²⋊C₄).₂₄₈C₂², SmallGroup(128,1409)

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

Derived series

C₁ — C₂³ — C₂⁴.₃₈₅C₂³

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂³×C₄ — C₂×C₂²⋊Q₈ — C₂⁴.₃₈₅C₂³

Lower central

C₁ — C₂³ — C₂⁴.₃₈₅C₂³

Upper central

C₁ — C₂³ — C₂⁴.₃₈₅C₂³

Jennings

C₁ — C₂³ — C₂⁴.₃₈₅C₂³

Generators and relations for C₂⁴.₃₈₅C₂³
G = < a,b,c,d,e,f,g | a²=b²=c²=f²=1, d²=e²=g²=a, ab=ba, ac=ca, ede^-1=ad=da, geg^-1=ae=ea, af=fa, ag=ga, bc=cb, fdf=bd=db, be=eb, bf=fb, bg=gb, cd=dc, fef=ce=ec, cf=fc, cg=gc, gdg^-1=abd, fg=gf >

Subgroups: 500 in 272 conjugacy classes, 112 normal (22 characteristic)
C₁, C₂, C₂, C₂, C₄, C₂², 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×C₂²⋊C₄, C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₂×C₄⋊C₄, C₂²⋊Q₈, C₄⋊Q₈, C₂³×C₄, C₂²×Q₈, C₂²×Q₈, C₂³.₈Q₈, C₂³.₆₃C₂³, C₂⁴.C₂², C₂³⋊Q₈, C₂³.₇₈C₂³, C₂³.₈₁C₂³, C₂³.₄Q₈, C₂×C₂²⋊Q₈, C₂×C₄⋊Q₈, C₂⁴.₃₈₅C₂³
Quotients: C₁, C₂, C₂², D₄, Q₈, C₂³, C₂×D₄, C₂×Q₈, C₂⁴, C₂²×D₄, C₂²×Q₈, 2₊¹⁺⁴, 2_-¹⁺⁴, C₂³.₃₈C₂³, C₂³.₄₁C₂³, D₄², D₄×Q₈, C₂².₅₇C₂⁴, C₂⁴.₃₈₅C₂³

Smallest permutation representation of C₂⁴.₃₈₅C₂³
►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 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 47 3 45)(2 46 4 48)(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)(53 63 55 61)(54 62 56 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 43 3 41)(2 14 4 16)(5 35 7 33)(6 63 8 61)(9 15 11 13)(10 42 12 44)(17 21 19 23)(18 52 20 50)(22 46 24 48)(25 31 27 29)(26 60 28 58)(30 54 32 56)(34 38 36 40)(37 62 39 64)(45 49 47 51)(53 59 55 57)

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

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

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

32 conjugacy classes

class	1	2A	···	2G	2H	2I	4A	···	4P	4Q	···	4V
order	1	2	···	2	2	2	4	···	4	4	···	4
size	1	1	···	1	4	4	4	···	4	8	···	8

32 irreducible representations

dim	1	1	1	1	1	1	1	1	1	1	2	2	4	4
type	+	+	+	+	+	+	+	+	+	+	-	+	+	-
image	C₁	C₂	C₂	C₂	C₂	C₂	C₂	C₂	C₂	C₂	Q₈	D₄	2₊¹⁺⁴	2_-¹⁺⁴
kernel	C₂⁴.₃₈₅C₂³	C₂³.₈Q₈	C₂³.₆₃C₂³	C₂⁴.C₂²	C₂³⋊Q₈	C₂³.₇₈C₂³	C₂³.₈₁C₂³	C₂³.₄Q₈	C₂×C₂²⋊Q₈	C₂×C₄⋊Q₈	C₂²⋊C₄	C₄⋊C₄	C₂²	C₂²
# reps	1	1	2	2	1	2	2	1	2	2	4	8	1	3

Matrix representation of C₂⁴.₃₈₅C₂³ ►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	4	0	0	0
0	0	0	4	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	3	0	0	0	0
3	0	0	0	0	0
0	0	1	2	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
4	0	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	3	3
0	0	0	0	4	2

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

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

C₂⁴.₃₈₅C₂³ in GAP, Magma, Sage, TeX

C_2^4._{385}C_2^3

% in TeX

G:=Group("C2^4.385C2^3");

// GroupNames label

G:=SmallGroup(128,1409);

// by ID

G=gap.SmallGroup(128,1409);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,112,253,120,758,723,1571,346,80]);

// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=f^2=1,d^2=e^2=g^2=a,a*b=b*a,a*c=c*a,e*d*e^-1=a*d=d*a,g*e*g^-1=a*e=e*a,a*f=f*a,a*g=g*a,b*c=c*b,f*d*f=b*d=d*b,b*e=e*b,b*f=f*b,b*g=g*b,c*d=d*c,f*e*f=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 = C24.385C23 order 128 = 27

225th non-split extension by C24 of C23 acting via C23/C2=C22

G = C₂⁴.₃₈₅C₂³ order 128 = 2⁷

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