C2^2.88C2^5 - GroupNames

G = C₂².₈₈C₂⁵ order 128 = 2⁷

69^th central stem extension by C₂² of C₂⁵

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

Aliases: C₄⋊₂2_-¹⁺⁴, C₂².₈₈C₂⁵, C₂³.₄₄C₂⁴, C₄².₅₇₆C₂³, D₄ ○(C₄⋊Q₈), C₄○D₄⋊₁₁D₄, (D₄×Q₈)⋊₁₈C₂, Q₈ ○(C₄⋊₁D₄), D₄.₅₉(C₂×D₄), Q₈.₆₁(C₂×D₄), D₄⋊₆D₄⋊₂₂C₂, (C₂×C₄).₇₈C₂⁴, C₂.₃₄(D₄×C₂³), C₄⋊C₄.₂₉₄C₂³, C₄.₁₂₃(C₂²×D₄), C₄⋊Q₈.₃₄₀C₂², (C₄×D₄).₂₃₁C₂², (C₂×D₄).₄₇₁C₂³, C₂²⋊C₄.₂₃C₂³, (C₂×2_-¹⁺⁴)⋊₉C₂, (C₂×Q₈).₄₄₈C₂³, (C₄×Q₈).₃₂₇C₂², C₂².₁₅(C₂²×D₄), C₄⋊₁D₄.₁₉₅C₂², C₄⋊D₄.₂₂₅C₂², (C₂²×C₄).₃₆₀C₂³, (C₂×C₄²).₉₄₃C₂², C₂²⋊Q₈.₁₁₄C₂², C₂.₂₃(C₂×2_-¹⁺⁴), C₂².₂₆C₂⁴⋊₃₇C₂, C₄.₄D₄.₁₇₅C₂², (C₂²×Q₈).₃₅₉C₂², C₂².D₄.₇C₂², C₄²⋊C₂.₃₄₄C₂², C₂³.₃₈C₂³⋊₂₂C₂, (C₂×D₄)○(C₄⋊Q₈), (C₂×C₄⋊Q₈)⋊₅₅C₂, (C₄×C₄○D₄)⋊₂₇C₂, (C₂×Q₈)○(C₄⋊₁D₄), (C₂×C₄).₁₈₈(C₂×D₄), (C₂×C₄⋊C₄).₇₀₃C₂², (C₂×C₄○D₄).₂₂₈C₂², SmallGroup(128,2231)

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

Derived series

C₁ — C₂² — C₂².₈₈C₂⁵

Chief series

C₁ — C₂ — C₂² — C₂×C₄ — C₂²×C₄ — C₂×C₄² — C₄×C₄○D₄ — C₂².₈₈C₂⁵

Lower central

C₁ — C₂² — C₂².₈₈C₂⁵

Upper central

C₁ — C₂² — C₂².₈₈C₂⁵

Jennings

C₁ — C₂² — C₂².₈₈C₂⁵

Generators and relations for C₂².₈₈C₂⁵
G = < a,b,c,d,e,f,g | a²=b²=d²=f²=1, c²=g²=a, e²=b, ab=ba, dcd=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: 1052 in 738 conjugacy classes, 432 normal (10 characteristic)
C₁, C₂, C₂, C₂, C₄, C₄, C₂², C₂², C₂², C₂×C₄, C₂×C₄, C₂×C₄, D₄, D₄, Q₈, Q₈, C₂³, C₄², C₄², C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂×D₄, C₂×Q₈, C₂×Q₈, C₂×Q₈, C₄○D₄, C₄○D₄, C₂×C₄², C₂×C₄⋊C₄, C₄²⋊C₂, C₄×D₄, C₄×Q₈, C₄⋊D₄, C₂²⋊Q₈, C₂².D₄, C₄.₄D₄, C₄⋊₁D₄, C₄⋊Q₈, C₂²×Q₈, C₂×C₄○D₄, C₂×C₄○D₄, 2_-¹⁺⁴, C₄×C₄○D₄, C₂×C₄⋊Q₈, C₂².₂₆C₂⁴, C₂³.₃₈C₂³, D₄⋊₆D₄, D₄×Q₈, C₂×2_-¹⁺⁴, C₂².₈₈C₂⁵
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₂⁴, C₂²×D₄, 2_-¹⁺⁴, C₂⁵, D₄×C₂³, C₂×2_-¹⁺⁴, C₂².₈₈C₂⁵

Smallest permutation representation of C₂².₈₈C₂⁵
►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 49)(2 50)(3 51)(4 52)(5 27)(6 28)(7 25)(8 26)(9 22)(10 23)(11 24)(12 21)(13 29)(14 30)(15 31)(16 32)(17 36)(18 33)(19 34)(20 35)(37 56)(38 53)(39 54)(40 55)(41 60)(42 57)(43 58)(44 59)(45 62)(46 63)(47 64)(48 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 2)(3 4)(5 8)(6 7)(9 12)(10 11)(13 30)(14 29)(15 32)(16 31)(17 33)(18 36)(19 35)(20 34)(21 22)(23 24)(25 28)(26 27)(37 38)(39 40)(41 59)(42 58)(43 57)(44 60)(45 63)(46 62)(47 61)(48 64)(49 50)(51 52)(53 56)(54 55)

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

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

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

44 conjugacy classes

class	1	2A	2B	2C	2D	···	2I	2J	2K	2L	2M	4A	···	4L	4M	···	4AD
order	1	2	2	2	2	···	2	2	2	2	2	4	···	4	4	···	4
size	1	1	1	1	2	···	2	4	4	4	4	2	···	2	4	···	4

44 irreducible representations

dim	1	1	1	1	1	1	1	1	2	4
type	+	+	+	+	+	+	+	+	+	-
image	C₁	C₂	C₂	C₂	C₂	C₂	C₂	C₂	D₄	2_-¹⁺⁴
kernel	C₂².₈₈C₂⁵	C₄×C₄○D₄	C₂×C₄⋊Q₈	C₂².₂₆C₂⁴	C₂³.₃₈C₂³	D₄⋊₆D₄	D₄×Q₈	C₂×2_-¹⁺⁴	C₄○D₄	C₄
# reps	1	1	3	3	6	12	4	2	8	4

Matrix representation of C₂².₈₈C₂⁵ ►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

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

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

1	3	0	0	0	0
1	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

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	1	0	0	0
0	0	0	1	0	0

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

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

C₂².₈₈C₂⁵ in GAP, Magma, Sage, TeX

C_2^2._{88}C_2^5

% in TeX

G:=Group("C2^2.88C2^5");

// GroupNames label

G:=SmallGroup(128,2231);

// by ID

G=gap.SmallGroup(128,2231);

# by ID

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

// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^2=b^2=d^2=f^2=1,c^2=g^2=a,e^2=b,a*b=b*a,d*c*d=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 = C22.88C25 order 128 = 27

69th central stem extension by C22 of C25

G = C₂².₈₈C₂⁵ order 128 = 2⁷

69^th central stem extension by C₂² of C₂⁵