C2^2.106C2^5 - GroupNames

G = C₂².₁₀₆C₂⁵ order 128 = 2⁷

87^th central stem extension by C₂² of C₂⁵

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

Aliases: C₄².₉₇C₂³, C₂³.₅₃C₂⁴, C₂².₁₀₆C₂⁵, C₄.₁₅₉2₊¹⁺⁴, D₄⋊₆D₄⋊₂₇C₂, Q₈⋊₁₃(C₄○D₄), Q₈⋊₆D₄⋊₂₁C₂, Q₈⋊₃Q₈⋊₂₂C₂, (C₂×C₄).₉₆C₂⁴, Q₈ ○₂(C₄².C₂), C₄⋊C₄.₄₉₉C₂³, C₄⋊Q₈.₃₄₇C₂², (C₂×D₄).₄₇₈C₂³, (C₄×D₄).₂₄₀C₂², (C₄×Q₈).₂₂₇C₂², (C₂×Q₈).₄₉₀C₂³, C₄⋊₁D₄.₁₁₅C₂², C₄⋊D₄.₂₂₉C₂², C₂²⋊C₄.₁₁₁C₂³, (C₂²×C₄).₃₇₆C₂³, (C₂×C₄²).₉₅₅C₂², C₂²⋊Q₈.₂₃₁C₂², C₂.₄₁(C₂×2₊¹⁺⁴), C₄².C₂.₈₃C₂², C₂.₃₇(C₂.C₂⁵), C₄²⋊₂C₂.₁₉C₂², C₂².₂₆C₂⁴⋊₄₄C₂, C₄.₄D₄.₁₈₈C₂², C₂².₄₇C₂⁴⋊₂₂C₂, C₂³.₃₃C₂³⋊₂₈C₂, C₂².₃₄C₂⁴⋊₁₃C₂, C₄²⋊C₂.₂₃₅C₂², C₂³.₃₆C₂³⋊₃₇C₂, C₂².D₄.₁₂C₂², (C₄×C₄○D₄)⋊₃₅C₂, C₄.₂₇₉(C₂×C₄○D₄), (C₂×Q₈)○(C₄².C₂), C₂.₆₂(C₂²×C₄○D₄), (C₂×C₄⋊C₄).₇₁₂C₂², (C₂×C₄○D₄).₂₃₄C₂², SmallGroup(128,2249)

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

Derived series

C₁ — C₂² — C₂².₁₀₆C₂⁵

Chief series

C₁ — C₂ — C₂² — C₂×C₄ — C₄² — C₂×C₄² — C₄×C₄○D₄ — C₂².₁₀₆C₂⁵

Lower central

C₁ — C₂² — C₂².₁₀₆C₂⁵

Upper central

C₁ — C₂² — C₂².₁₀₆C₂⁵

Jennings

C₁ — C₂² — C₂².₁₀₆C₂⁵

Generators and relations for C₂².₁₀₆C₂⁵
G = < a,b,c,d,e,f,g | a²=b²=c²=1, 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: 852 in 561 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, C₄⋊C₄, C₂²×C₄, C₂×D₄, C₂×Q₈, C₂×Q₈, C₄○D₄, C₂×C₄², C₂×C₄⋊C₄, C₄²⋊C₂, C₄×D₄, C₄×Q₈, C₄×Q₈, C₄⋊D₄, C₂²⋊Q₈, C₂².D₄, C₄.₄D₄, C₄².C₂, C₄².C₂, C₄²⋊₂C₂, C₄⋊₁D₄, C₄⋊Q₈, C₂×C₄○D₄, C₂×C₄○D₄, C₄×C₄○D₄, C₂³.₃₃C₂³, C₂³.₃₆C₂³, C₂².₂₆C₂⁴, C₂².₃₄C₂⁴, D₄⋊₆D₄, Q₈⋊₆D₄, Q₈⋊₆D₄, C₂².₄₇C₂⁴, Q₈⋊₃Q₈, C₂².₁₀₆C₂⁵
Quotients: C₁, C₂, C₂², C₂³, C₄○D₄, C₂⁴, C₂×C₄○D₄, 2₊¹⁺⁴, C₂⁵, C₂²×C₄○D₄, C₂×2₊¹⁺⁴, C₂.C₂⁵, C₂².₁₀₆C₂⁵

Smallest permutation representation of C₂².₁₀₆C₂⁵
►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 46)(2 45)(3 48)(4 47)(5 10)(6 9)(7 12)(8 11)(13 22)(14 21)(15 24)(16 23)(17 28)(18 27)(19 26)(20 25)(29 52)(30 51)(31 50)(32 49)(33 53)(34 56)(35 55)(36 54)(37 58)(38 57)(39 60)(40 59)(41 64)(42 63)(43 62)(44 61)

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

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

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

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

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

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

44 conjugacy classes

class	1	2A	2B	2C	2D	···	2L	4A	···	4R	4S	···	4AE
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

C₂².₁₀₆C₂⁵

C₄×C₄○D₄

C₂³.₃₃C₂³

C₂³.₃₆C₂³

C₂².₂₆C₂⁴

C₂².₃₄C₂⁴

D₄⋊₆D₄

Q₈⋊₆D₄

C₂².₄₇C₂⁴

Q₈⋊₃Q₈

Q₈

C₄

C₂

# reps

Matrix representation of C₂².₁₀₆C₂⁵ ►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	1	0	0	0
0	0	0	1	0	0
0	0	0	0	1	0
0	0	0	0	0	1

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

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

0	1	0	0	0	0
1	0	0	0	0	0
0	0	3	0	0	0
0	0	0	3	0	0
0	0	0	0	3	0
0	0	0	0	0	3

1	0	0	0	0	0
0	1	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

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

C₂².₁₀₆C₂⁵ in GAP, Magma, Sage, TeX

C_2^2._{106}C_2^5

% in TeX

G:=Group("C2^2.106C2^5");

// GroupNames label

G:=SmallGroup(128,2249);

// by ID

G=gap.SmallGroup(128,2249);

# by ID

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

// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=1,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 = C22.106C25 order 128 = 27

87th central stem extension by C22 of C25

G = C₂².₁₀₆C₂⁵ order 128 = 2⁷

87^th central stem extension by C₂² of C₂⁵