C2^4.408C2^3 - GroupNames

G = C₂⁴.₄₀₈C₂³ order 128 = 2⁷

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

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

Aliases: C₂⁴.₄₀₈C₂³, C₂³.₆₀₄C₂⁴, C₂².₂₈₁2_-¹⁺⁴, C₂².₃₇₈2₊¹⁺⁴, C₂²⋊C₄⋊₃Q₈, C₂.₂₉(D₄×Q₈), C₄⋊C₄.₁₁₉D₄, C₂³.₃₄(C₂×Q₈), C₂.₅₀(D₄⋊₃Q₈), C₂.₁₀₉(D₄⋊₅D₄), (C₂²×C₄).₈₇₉C₂³, (C₂³×C₄).₄₆₄C₂², (C₂×C₄²).₆₅₆C₂², C₂².₄₁₃(C₂²×D₄), C₂³.₈Q₈.₄₈C₂, C₂³.₇Q₈.₆₅C₂, C₂³.Q₈.₂₇C₂, C₂².₁₄₅(C₂²×Q₈), (C₂²×Q₈).₁₈₈C₂², C₂³.₈₁C₂³⋊₉₀C₂, C₂³.₇₈C₂³⋊₄₇C₂, C₂³.₆₇C₂³⋊₈₄C₂, C₂⁴.C₂².₅₁C₂, C₂³.₆₅C₂³⋊₁₂₄C₂, C₂.C₄².₃₁₀C₂², C₂.₄₇(C₂².₃₁C₂⁴), C₂.₄₁(C₂².₃₅C₂⁴), C₂.₃₀(C₂³.₄₁C₂³), C₂.₁₈(C₂².₅₆C₂⁴), (C₂×C₄).₆₉(C₂×Q₈), (C₂×C₄).₁₀₆(C₂×D₄), (C₂×C₄².C₂)⋊₂₂C₂, (C₂×C₂²⋊Q₈).₄₅C₂, (C₂×C₄).₄₃₀(C₄○D₄), (C₂×C₄⋊C₄).₄₁₇C₂², C₂².₄₆₆(C₂×C₄○D₄), (C₂×C₂²⋊C₄).₂₇₀C₂², SmallGroup(128,1436)

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

Derived series

C₁ — C₂³ — C₂⁴.₄₀₈C₂³

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂³×C₄ — C₂³.₇Q₈ — C₂⁴.₄₀₈C₂³

Lower central

C₁ — C₂³ — C₂⁴.₄₀₈C₂³

Upper central

C₁ — C₂³ — C₂⁴.₄₀₈C₂³

Jennings

C₁ — C₂³ — C₂⁴.₄₀₈C₂³

Generators and relations for C₂⁴.₄₀₈C₂³
G = < a,b,c,d,e,f,g | a²=b²=c²=f²=1, d²=e²=ba=ab, g²=a, 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: 436 in 236 conjugacy classes, 104 normal (82 characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₂×C₄, C₂×C₄, Q₈, C₂³, C₂³, C₂³, C₄², C₂²⋊C₄, C₂²⋊C₄, C₄⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂×Q₈, C₂⁴, C₂.C₄², C₂×C₄², C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₂²⋊Q₈, C₄².C₂, C₂³×C₄, C₂²×Q₈, C₂³.₇Q₈, C₂³.₈Q₈, C₂⁴.C₂², C₂³.₆₅C₂³, C₂³.₆₇C₂³, C₂³.₇₈C₂³, C₂³.Q₈, C₂³.₈₁C₂³, C₂×C₂²⋊Q₈, C₂×C₄².C₂, C₂⁴.₄₀₈C₂³
Quotients: C₁, C₂, C₂², D₄, Q₈, C₂³, C₂×D₄, C₂×Q₈, C₄○D₄, C₂⁴, C₂²×D₄, C₂²×Q₈, C₂×C₄○D₄, 2₊¹⁺⁴, 2_-¹⁺⁴, C₂².₃₁C₂⁴, C₂².₃₅C₂⁴, C₂³.₄₁C₂³, D₄⋊₅D₄, D₄×Q₈, D₄⋊₃Q₈, C₂².₅₆C₂⁴, C₂⁴.₄₀₈C₂³

Smallest permutation representation of C₂⁴.₄₀₈C₂³
►On 64 points

Generators in S₆₄

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

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

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

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

(2 12)(4 10)(5 42)(6 23)(7 44)(8 21)(14 48)(16 46)(17 63)(18 35)(19 61)(20 33)(22 39)(24 37)(26 54)(28 56)(30 58)(32 60)(34 51)(36 49)(38 41)(40 43)(50 62)(52 64)

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

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

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

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

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	1	2	2	2	4	4
type	+	+	+	+	+	+	+	+	+	+	+	-	+		+	-
image	C₁	C₂	C₂	C₂	C₂	C₂	C₂	C₂	C₂	C₂	C₂	Q₈	D₄	C₄○D₄	2₊¹⁺⁴	2_-¹⁺⁴
kernel	C₂⁴.₄₀₈C₂³	C₂³.₇Q₈	C₂³.₈Q₈	C₂⁴.C₂²	C₂³.₆₅C₂³	C₂³.₆₇C₂³	C₂³.₇₈C₂³	C₂³.Q₈	C₂³.₈₁C₂³	C₂×C₂²⋊Q₈	C₂×C₄².C₂	C₂²⋊C₄	C₄⋊C₄	C₂×C₄	C₂²	C₂²
# reps	1	1	1	2	2	1	1	2	3	1	1	4	4	4	2	2

Matrix representation of C₂⁴.₄₀₈C₂³ ►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

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

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

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	0	4

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

C₂⁴.₄₀₈C₂³ in GAP, Magma, Sage, TeX

C_2^4._{408}C_2^3

% in TeX

G:=Group("C2^4.408C2^3");

// GroupNames label

G:=SmallGroup(128,1436);

// by ID

G=gap.SmallGroup(128,1436);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,336,253,344,758,723,1571,346,80]);

// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=f^2=1,d^2=e^2=b*a=a*b,g^2=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 = C24.408C23 order 128 = 27

248th non-split extension by C24 of C23 acting via C23/C2=C22

G = C₂⁴.₄₀₈C₂³ order 128 = 2⁷

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