C2^2.98C2^5 - GroupNames

G = C₂².₉₈C₂⁵ order 128 = 2⁷

79^th central stem extension by C₂² of C₂⁵

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

Aliases: C₂².₉₈C₂⁵, C₄².₉₀C₂³, C₂³.₄₈C₂⁴, C₄.₁₀₇2_-¹⁺⁴, Q₈⋊₃Q₈⋊₁₉C₂, (C₂×C₄).₈₈C₂⁴, C₄⋊C₄.₄₉₃C₂³, C₄⋊Q₈.₃₅₈C₂², (C₂×D₄).₄₇₄C₂³, (C₄×D₄).₃₅₈C₂², C₂²⋊C₄.₂₅C₂³, (C₄×Q₈).₂₂₃C₂², (C₂×Q₈).₄₅₃C₂³, (C₂×C₄²).₉₅₀C₂², C₄²⋊₂C₂.₂C₂², C₄.₄D₄.₉₆C₂², C₂²⋊Q₈.₂₂₉C₂², C₂.₂₇(C₂×2_-¹⁺⁴), C₄².C₂.₈₂C₂², (C₂²×C₄).₁₂₁₀C₂³, (C₂²×Q₈).₃₆₁C₂², C₂².D₄.₉C₂², C₄²⋊C₂.₂₃₀C₂², C₂³.₃₇C₂³⋊₄₁C₂, C₂².₄₆C₂⁴⋊₁₈C₂, C₂².₃₅C₂⁴⋊₁₀C₂, C₂².₅₀C₂⁴⋊₂₂C₂, C₂³.₃₂C₂³⋊₁₄C₂, C₂³.₃₈C₂³.₁₆C₂, (C₂×C₄⋊Q₈)⋊₅₇C₂, C₂²⋊C₄ ○(C₄⋊Q₈), (C₄×C₄○D₄).₃₁C₂, C₄.₁₈₁(C₂×C₄○D₄), C₂².₃₁(C₂×C₄○D₄), C₂.₅₄(C₂²×C₄○D₄), (C₂×C₄).₃₀₇(C₄○D₄), (C₂×C₄⋊C₄).₇₀₇C₂², (C₂×C₄○D₄).₃₃₀C₂², SmallGroup(128,2241)

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

Derived series

C₁ — C₂² — C₂².₉₈C₂⁵

Chief series

C₁ — C₂ — C₂² — C₂×C₄ — C₂²×C₄ — C₂×C₄² — C₄×C₄○D₄ — C₂².₉₈C₂⁵

Lower central

C₁ — C₂² — C₂².₉₈C₂⁵

Upper central

C₁ — C₂² — C₂².₉₈C₂⁵

Jennings

C₁ — C₂² — C₂².₉₈C₂⁵

Generators and relations for C₂².₉₈C₂⁵
G = < a,b,c,d,e,f,g | a²=b²=f²=1, c²=g²=a, d²=e²=b, ab=ba, dcd^-1=gcg^-1=ac=ca, fdf=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: 604 in 496 conjugacy classes, 392 normal (18 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, C₂×Q₈, C₄○D₄, C₂×C₄², C₂×C₄², C₂×C₄⋊C₄, C₄²⋊C₂, C₄²⋊C₂, C₄×D₄, C₄×Q₈, C₂²⋊Q₈, C₂².D₄, C₄.₄D₄, C₄².C₂, C₄²⋊₂C₂, C₄⋊Q₈, C₂²×Q₈, C₂×C₄○D₄, C₄×C₄○D₄, C₂³.₃₂C₂³, C₂×C₄⋊Q₈, C₂³.₃₇C₂³, C₂³.₃₇C₂³, C₂³.₃₈C₂³, C₂².₃₅C₂⁴, C₂².₄₆C₂⁴, C₂².₅₀C₂⁴, Q₈⋊₃Q₈, C₂².₉₈C₂⁵
Quotients: C₁, C₂, C₂², C₂³, C₄○D₄, C₂⁴, C₂×C₄○D₄, 2_-¹⁺⁴, C₂⁵, C₂²×C₄○D₄, C₂×2_-¹⁺⁴, C₂².₉₈C₂⁵

Smallest permutation representation of C₂².₉₈C₂⁵
►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 15 51 59)(2 14 52 58)(3 13 49 57)(4 16 50 60)(5 27 36 43)(6 26 33 42)(7 25 34 41)(8 28 35 44)(9 19 53 63)(10 18 54 62)(11 17 55 61)(12 20 56 64)(21 29 37 45)(22 32 38 48)(23 31 39 47)(24 30 40 46)

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

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

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

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

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

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

44 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	4A	···	4P	4Q	···	4AJ
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

2_-¹⁺⁴

kernel

C₂².₉₈C₂⁵

C₄×C₄○D₄

C₂³.₃₂C₂³

C₂×C₄⋊Q₈

C₂³.₃₇C₂³

C₂³.₃₈C₂³

C₂².₃₅C₂⁴

C₂².₄₆C₂⁴

C₂².₅₀C₂⁴

Q₈⋊₃Q₈

C₂×C₄

C₄

# reps

Matrix representation of C₂².₉₈C₂⁵ ►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	1	0	0	0	0
1	0	0	0	0	0
0	0	3	0	2	0
0	0	3	4	2	3
0	0	0	0	2	0
0	0	3	1	4	1

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

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

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

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

C₂².₉₈C₂⁵ in GAP, Magma, Sage, TeX

C_2^2._{98}C_2^5

% in TeX

G:=Group("C2^2.98C2^5");

// GroupNames label

G:=SmallGroup(128,2241);

// by ID

G=gap.SmallGroup(128,2241);

# by ID

G:=PCGroup([7,-2,2,2,2,2,-2,2,224,477,456,1430,352,570,1684,102]);

// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^2=b^2=f^2=1,c^2=g^2=a,d^2=e^2=b,a*b=b*a,d*c*d^-1=g*c*g^-1=a*c=c*a,f*d*f=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 = C22.98C25 order 128 = 27

79th central stem extension by C22 of C25

G = C₂².₉₈C₂⁵ order 128 = 2⁷

79^th central stem extension by C₂² of C₂⁵