C2^2.107C2^5 - GroupNames

G = C₂².₁₀₇C₂⁵ order 128 = 2⁷

88^th central stem extension by C₂² of C₂⁵

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

Aliases: C₂³.₅₄C₂⁴, C₄².₉₈C₂³, C₂².₁₀₇C₂⁵, C₄.₁₁₀2_-¹⁺⁴, Q₈²⋊₁₀C₂, (D₄×Q₈)⋊₂₃C₂, Q₈⋊₅D₄⋊₂₂C₂, (C₂×C₄).₉₇C₂⁴, Q₈ ○₂(C₄.₄D₄), C₄⋊C₄.₅₀₀C₂³, Q₈.₄₂(C₄○D₄), C₄⋊Q₈.₃₄₈C₂², (C₂×D₄).₄₇₉C₂³, (C₄×D₄).₂₄₁C₂², C₂²⋊C₄.₃₀C₂³, (C₄×Q₈).₂₂₈C₂², (C₂×Q₈).₄₅₇C₂³, C₄⋊D₄.₁₁₇C₂², (C₂×C₄²).₉₅₆C₂², C₄.₄D₄.₉₉C₂², C₂²⋊Q₈.₂₃₂C₂², C₂.₃₁(C₂×2_-¹⁺⁴), C₂.₃₈(C₂.C₂⁵), C₄²⋊₂C₂.₂₀C₂², (C₂²×C₄).₁₂₁₁C₂³, C₄².C₂.₁₇₀C₂², (C₂²×Q₈).₃₆₄C₂², C₂³.₃₇C₂³⋊₄₅C₂, C₄²⋊C₂.₂₃₆C₂², C₂².₅₀C₂⁴⋊₂₇C₂, C₂².₃₆C₂⁴⋊₁₉C₂, C₂³.₃₆C₂³⋊₃₈C₂, C₂³.₃₂C₂³⋊₁₇C₂, C₂².₄₉C₂⁴⋊₁₆C₂, C₂².D₄.₃₂C₂², (C₄×C₄○D₄)⋊₃₆C₂, C₄.₂₈₀(C₂×C₄○D₄), (C₂×Q₈)○(C₄.₄D₄), C₂.₆₃(C₂²×C₄○D₄), (C₂×C₄○D₄).₃₃₂C₂², SmallGroup(128,2250)

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

Derived series

C₁ — C₂² — C₂².₁₀₇C₂⁵

Chief series

C₁ — C₂ — C₂² — C₂×C₄ — C₄² — C₂×C₄² — C₄×C₄○D₄ — C₂².₁₀₇C₂⁵

Lower central

C₁ — C₂² — C₂².₁₀₇C₂⁵

Upper central

C₁ — C₂² — C₂².₁₀₇C₂⁵

Jennings

C₁ — C₂² — C₂².₁₀₇C₂⁵

Generators and relations for C₂².₁₀₇C₂⁵
G = < a,b,c,d,e,f,g | a²=b²=1, c²=d²=e²=f²=a, g²=b, ab=ba, dcd^-1=gcg^-1=ac=ca, fdf^-1=ad=da, ae=ea, af=fa, ag=ga, ece^-1=bc=cb, bd=db, be=eb, bf=fb, bg=gb, cf=fc, de=ed, dg=gd, ef=fe, eg=ge, fg=gf >

Subgroups: 660 in 505 conjugacy classes, 390 normal (20 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₂×C₄, C₂×C₄, C₂×C₄, D₄, Q₈, Q₈, C₂³, C₄², C₄², C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂×Q₈, C₂×Q₈, C₂×Q₈, C₄○D₄, C₂×C₄², C₄²⋊C₂, C₄×D₄, C₄×Q₈, C₄×Q₈, C₄⋊D₄, C₂²⋊Q₈, C₂².D₄, C₄.₄D₄, C₄.₄D₄, C₄².C₂, C₄²⋊₂C₂, C₄⋊Q₈, C₂²×Q₈, C₂×C₄○D₄, C₄×C₄○D₄, C₂³.₃₂C₂³, C₂³.₃₆C₂³, C₂³.₃₇C₂³, C₂².₃₆C₂⁴, Q₈⋊₅D₄, D₄×Q₈, C₂².₄₉C₂⁴, C₂².₅₀C₂⁴, Q₈², C₂².₁₀₇C₂⁵
Quotients: C₁, C₂, C₂², C₂³, C₄○D₄, C₂⁴, C₂×C₄○D₄, 2_-¹⁺⁴, C₂⁵, C₂²×C₄○D₄, C₂×2_-¹⁺⁴, C₂.C₂⁵, C₂².₁₀₇C₂⁵

Smallest permutation representation of C₂².₁₀₇C₂⁵
►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 51)(2 52)(3 49)(4 50)(5 36)(6 33)(7 34)(8 35)(9 53)(10 54)(11 55)(12 56)(13 57)(14 58)(15 59)(16 60)(17 61)(18 62)(19 63)(20 64)(21 37)(22 38)(23 39)(24 40)(25 41)(26 42)(27 43)(28 44)(29 45)(30 46)(31 47)(32 48)

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

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

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

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

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

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

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

44 conjugacy classes

class	1	2A	2B	2C	2D	···	2H	4A	···	4R	4S	···	4AI
order	1	2	2	2	2	···	2	4	···	4	4	···	4
size	1	1	1	1	4	···	4	2	···	2	4	···	4

44 irreducible representations

dim

type

image

C₁

C₂

C₄○D₄

2_-¹⁺⁴

C₂.C₂⁵

kernel

C₂².₁₀₇C₂⁵

C₄×C₄○D₄

C₂³.₃₂C₂³

C₂³.₃₆C₂³

C₂³.₃₇C₂³

C₂².₃₆C₂⁴

Q₈⋊₅D₄

D₄×Q₈

C₂².₄₉C₂⁴

C₂².₅₀C₂⁴

Q₈²

Q₈

C₄

C₂

# reps

Matrix representation of C₂².₁₀₇C₂⁵ ►in GL₆(𝔽₅)

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

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

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

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

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

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

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

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

C₂².₁₀₇C₂⁵ in GAP, Magma, Sage, TeX

C_2^2._{107}C_2^5

% in TeX

G:=Group("C2^2.107C2^5");

// GroupNames label

G:=SmallGroup(128,2250);

// by ID

G=gap.SmallGroup(128,2250);

# by ID

G:=PCGroup([7,-2,2,2,2,2,-2,2,224,477,232,1430,184,570,136,1684,172]);

// Polycyclic

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

// generators/relations

G = C22.107C25 order 128 = 27

88th central stem extension by C22 of C25

G = C₂².₁₀₇C₂⁵ order 128 = 2⁷

88^th central stem extension by C₂² of C₂⁵