C2^2.92C2^5 - GroupNames

G = C₂².₉₂C₂⁵ order 128 = 2⁷

73^rd central stem extension by C₂² of C₂⁵

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

Aliases: C₂².₉₂C₂⁵, C₄².₈₄C₂³, C₂³.₁₃₆C₂⁴, C₄.₁₀₆2_-¹⁺⁴, C₄.₁₅₅2₊¹⁺⁴, Q₈²⋊₉C₂, D₄ ○₂(C₄⋊Q₈), Q₈ ○₂(C₄⋊Q₈), C₄○D₄⋊₅Q₈, Q₈⋊₈(C₂×Q₈), D₄⋊₉(C₂×Q₈), (D₄×Q₈)⋊₂₀C₂, D₄⋊₃Q₈⋊₂₁C₂, Q₈⋊₃Q₈⋊₁₆C₂, (C₂×C₄).₈₂C₂⁴, C₄.₅₅(C₂²×Q₈), C₂.₁₇(Q₈×C₂³), C₄⋊C₄.₂₉₈C₂³, C₄⋊Q₈.₃₄₂C₂², (C₂×D₄).₅₀₆C₂³, (C₄×D₄).₂₃₃C₂², (C₄×Q₈).₂₂₀C₂², (C₂×Q₈).₂₈₈C₂³, C₂².₁₂(C₂²×Q₈), C₂²⋊C₄.₁₀₂C₂³, (C₂²×C₄).₃₆₃C₂³, (C₂×C₄²).₉₄₆C₂², C₂²⋊Q₈.₁₁₅C₂², C₂.₃₄(C₂×2₊¹⁺⁴), C₂.₂₅(C₂×2_-¹⁺⁴), C₄².C₂.₁₅₁C₂², (C₂²×Q₈).₃₆₀C₂², C₂³.₄₁C₂³⋊₁₆C₂, C₂³.₃₇C₂³⋊₃₆C₂, C₄²⋊C₂.₂₂₇C₂², C₂³.₃₃C₂³.₁₁C₂, (C₂×Q₈)○(C₄⋊Q₈), (C₂×C₄)⋊₄(C₂×Q₈), (C₂×C₄⋊Q₈)⋊₅₆C₂, (C₄×C₄○D₄).₂₉C₂, (C₂×C₄⋊C₄).₇₀₄C₂², (C₂×C₄○D₄).₃₂₈C₂², SmallGroup(128,2235)

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

Derived series

C₁ — C₂² — C₂².₉₂C₂⁵

Chief series

C₁ — C₂ — C₂² — C₂×C₄ — C₂²×C₄ — C₂×C₄² — C₄×C₄○D₄ — C₂².₉₂C₂⁵

Lower central

C₁ — C₂² — C₂².₉₂C₂⁵

Upper central

C₁ — C₂² — C₂².₉₂C₂⁵

Jennings

C₁ — C₂² — C₂².₉₂C₂⁵

Generators and relations for C₂².₉₂C₂⁵
G = < a,b,c,d,e,f,g | a²=b²=d²=f²=1, c²=e²=b, g²=a, 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: 700 in 530 conjugacy classes, 432 normal (16 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₂², C₂×C₄, C₂×C₄, C₂×C₄, D₄, Q₈, Q₈, C₂³, C₄², C₄², C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂×D₄, C₂×Q₈, C₂×Q₈, C₂×Q₈, C₄○D₄, C₂×C₄², C₂×C₄⋊C₄, C₄²⋊C₂, C₄×D₄, C₄×Q₈, C₄×Q₈, C₂²⋊Q₈, C₄².C₂, C₄⋊Q₈, C₄⋊Q₈, C₂²×Q₈, C₂×C₄○D₄, C₄×C₄○D₄, C₂³.₃₃C₂³, C₂×C₄⋊Q₈, C₂³.₃₇C₂³, C₂³.₄₁C₂³, D₄×Q₈, D₄⋊₃Q₈, Q₈⋊₃Q₈, Q₈², C₂².₉₂C₂⁵
Quotients: C₁, C₂, C₂², Q₈, C₂³, C₂×Q₈, C₂⁴, C₂²×Q₈, 2₊¹⁺⁴, 2_-¹⁺⁴, C₂⁵, Q₈×C₂³, C₂×2₊¹⁺⁴, C₂×2_-¹⁺⁴, C₂².₉₂C₂⁵

Smallest permutation representation of C₂².₉₂C₂⁵
►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 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)(2 28)(3 13)(4 26)(5 58)(6 47)(7 60)(8 45)(9 53)(10 42)(11 55)(12 44)(14 50)(16 52)(17 61)(18 38)(19 63)(20 40)(21 41)(22 54)(23 43)(24 56)(25 49)(27 51)(29 37)(30 62)(31 39)(32 64)(33 59)(34 48)(35 57)(36 46)

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

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

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

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

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

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

44 conjugacy classes

class	1	2A	2B	2C	2D	···	2I	4A	···	4L	4M	···	4AH
order	1	2	2	2	2	···	2	4	···	4	4	···	4
size	1	1	1	1	2	···	2	2	···	2	4	···	4

44 irreducible representations

dim	1	1	1	1	1	1	1	1	1	1	2	4	4
type	+	+	+	+	+	+	+	+	+	+	-	+	-
image	C₁	C₂	C₂	C₂	C₂	C₂	C₂	C₂	C₂	C₂	Q₈	2₊¹⁺⁴	2_-¹⁺⁴
kernel	C₂².₉₂C₂⁵	C₄×C₄○D₄	C₂³.₃₃C₂³	C₂×C₄⋊Q₈	C₂³.₃₇C₂³	C₂³.₄₁C₂³	D₄×Q₈	D₄⋊₃Q₈	Q₈⋊₃Q₈	Q₈²	C₄○D₄	C₄	C₄
# reps	1	1	2	3	3	6	6	6	2	2	8	2	2

Matrix representation of C₂².₉₂C₂⁵ ►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

2	0	0	0	0	0
0	3	0	0	0	0
0	0	0	0	2	0
0	0	0	0	0	2
0	0	2	0	0	0
0	0	0	2	0	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

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

1	0	0	0	0	0
0	1	0	0	0	0
0	0	0	1	0	0
0	0	1	0	0	0
0	0	0	0	0	1
0	0	0	0	1	0

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

C₂².₉₂C₂⁵ in GAP, Magma, Sage, TeX

C_2^2._{92}C_2^5

% in TeX

G:=Group("C2^2.92C2^5");

// GroupNames label

G:=SmallGroup(128,2235);

// by ID

G=gap.SmallGroup(128,2235);

# by ID

G:=PCGroup([7,-2,2,2,2,2,-2,2,448,477,1430,352,570,1684,102]);

// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^2=b^2=d^2=f^2=1,c^2=e^2=b,g^2=a,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 = C22.92C25 order 128 = 27

73rd central stem extension by C22 of C25

G = C₂².₉₂C₂⁵ order 128 = 2⁷

73^rd central stem extension by C₂² of C₂⁵