C2^2.101C2^5 - GroupNames

G = C₂².₁₀₁C₂⁵ order 128 = 2⁷

82^nd central stem extension by C₂² of C₂⁵

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

Aliases: C₂³.₅₀C₂⁴, C₂².₁₀₁C₂⁵, C₄².₅₇₉C₂³, (C₂×C₄).₉₁C₂⁴, C₄⋊C₄.₄₉₆C₂³, C₄⋊Q₈.₃₄₆C₂², (C₄×D₄).₂₃₇C₂², (C₂×D₄).₃₀₈C₂³, (C₄×Q₈).₃₂₈C₂², (C₂×Q₈).₄₅₄C₂³, C₄⋊₁D₄.₁₈₇C₂², C₄⋊D₄.₁₁₅C₂², C₂²⋊C₄.₁₁₀C₂³, (C₂²×C₄).₃₇₁C₂³, (C₂×C₄²).₉₅₂C₂², C₂²⋊Q₈.₁₁₉C₂², C₂.₃₂(C₂.C₂⁵), C₂².₂₆C₂⁴⋊₄₂C₂, C₄²⋊₂C₂.₁₆C₂², C₄.₄D₄.₁₈₆C₂², C₄².C₂.₁₅₅C₂², C₂².₄₆C₂⁴⋊₁₉C₂, C₂².₃₅C₂⁴⋊₁₁C₂, C₂².₃₁C₂⁴⋊₁₈C₂, C₂³.₃₃C₂³⋊₂₄C₂, C₂².₃₄C₂⁴⋊₁₁C₂, C₂².₄₇C₂⁴⋊₁₈C₂, C₂³.₃₆C₂³⋊₃₂C₂, C₄²⋊C₂.₂₃₂C₂², C₂².D₄.₁₁C₂², (C₄×C₄○D₄)⋊₃₂C₂, C₄.₁₄₂(C₂×C₄○D₄), (C₂×C₄².C₂)⋊₄₆C₂, C₂².₃₇(C₂×C₄○D₄), C₂.₅₇(C₂²×C₄○D₄), (C₂×C₄).₃₀₉(C₄○D₄), (C₂×C₄⋊C₄).₇₀₉C₂², (C₂×C₄○D₄).₂₃₁C₂², SmallGroup(128,2244)

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

Derived series

C₁ — C₂² — C₂².₁₀₁C₂⁵

Chief series

C₁ — C₂ — C₂² — C₂×C₄ — C₂²×C₄ — C₂×C₄² — C₄×C₄○D₄ — C₂².₁₀₁C₂⁵

Lower central

C₁ — C₂² — C₂².₁₀₁C₂⁵

Upper central

C₁ — C₂² — C₂².₁₀₁C₂⁵

Jennings

C₁ — C₂² — C₂².₁₀₁C₂⁵

Generators and relations for C₂².₁₀₁C₂⁵
G = < a,b,c,d,e,f,g | a²=b²=c²=1, d²=b, e²=ba=ab, f²=g²=a, 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: 716 in 516 conjugacy classes, 388 normal (20 characteristic)
C₁, C₂, C₂, C₂, C₄, C₄, C₂², C₂², C₂², C₂×C₄, C₂×C₄, C₂×C₄, D₄, Q₈, C₂³, C₂³, C₄², C₄², C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂×Q₈, C₂×Q₈, C₄○D₄, C₂×C₄², C₂×C₄², C₂×C₄⋊C₄, C₄²⋊C₂, C₄²⋊C₂, C₄×D₄, C₄×Q₈, C₄⋊D₄, C₂²⋊Q₈, C₂².D₄, C₄.₄D₄, C₄².C₂, C₄²⋊₂C₂, C₄⋊₁D₄, C₄⋊Q₈, C₂×C₄○D₄, C₂×C₄○D₄, C₄×C₄○D₄, C₂³.₃₃C₂³, C₂×C₄².C₂, C₂³.₃₆C₂³, C₂².₂₆C₂⁴, C₂².₃₁C₂⁴, C₂².₃₄C₂⁴, C₂².₃₅C₂⁴, C₂².₄₆C₂⁴, C₂².₄₇C₂⁴, C₂².₁₀₁C₂⁵
Quotients: C₁, C₂, C₂², C₂³, C₄○D₄, C₂⁴, C₂×C₄○D₄, C₂⁵, C₂²×C₄○D₄, C₂.C₂⁵, C₂².₁₀₁C₂⁵

Smallest permutation representation of C₂².₁₀₁C₂⁵
►On 64 points

Generators in S₆₄

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

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

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

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

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

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

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

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

44 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	···	2K	4A	···	4P	4Q	···	4AF
order	1	2	2	2	2	2	2	···	2	4	···	4	4	···	4
size	1	1	1	1	2	2	4	···	4	2	···	2	4	···	4

44 irreducible representations

dim

type

image

C₁

C₂

C₄○D₄

C₂.C₂⁵

kernel

C₂².₁₀₁C₂⁵

C₄×C₄○D₄

C₂³.₃₃C₂³

C₂×C₄².C₂

C₂³.₃₆C₂³

C₂².₂₆C₂⁴

C₂².₃₁C₂⁴

C₂².₃₄C₂⁴

C₂².₃₅C₂⁴

C₂².₄₆C₂⁴

C₂².₄₇C₂⁴

C₂×C₄

C₂

# reps

Matrix representation of C₂².₁₀₁C₂⁵ ►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

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

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

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

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

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

C₂².₁₀₁C₂⁵ in GAP, Magma, Sage, TeX

C_2^2._{101}C_2^5

% in TeX

G:=Group("C2^2.101C2^5");

// GroupNames label

G:=SmallGroup(128,2244);

// by ID

G=gap.SmallGroup(128,2244);

# by ID

G:=PCGroup([7,-2,2,2,2,2,-2,2,477,456,1430,520,570,136,1684,102]);

// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=1,d^2=b,e^2=b*a=a*b,f^2=g^2=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 = C22.101C25 order 128 = 27

82nd central stem extension by C22 of C25

G = C₂².₁₀₁C₂⁵ order 128 = 2⁷

82^nd central stem extension by C₂² of C₂⁵