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G = C23.260C24  order 128 = 27

113rd central extension by C23 of C24

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

Aliases: C23.260C24, C24.227C23, C22.912+ (1+4), C22.662- (1+4), C22⋊Q8⋊22C4, C23.95(C22×C4), (C23×C4).315C22, (C2×C42).448C22, C22.151(C23×C4), (C22×C4).488C23, C23.7Q8.33C2, (C22×Q8).94C22, C23.34D4.12C2, C24.C22.9C2, C23.65C23⋊31C2, C23.63C23⋊26C2, C23.67C23⋊25C2, C2.40(C22.11C24), C2.C42.68C22, C2.3(C22.56C24), C2.2(C22.57C24), C2.20(C23.32C23), C2.40(C23.33C23), C4⋊C4.111(C2×C4), C22⋊C4.13(C2×C4), (C2×C4).57(C22×C4), (C2×Q8).113(C2×C4), (C2×C22⋊Q8).21C2, (C2×C4⋊C4).196C22, (C22×C4).317(C2×C4), (C2×C22⋊C4).42C22, SmallGroup(128,1110)

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

Derived series
C1 — C22 — C23.260C24
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C23×C4 — C2×C22⋊Q8 — C23.260C24
Lower central
C1 — C22 — C23.260C24
Upper central
C1 — C23 — C23.260C24
Jennings
C1 — C23 — C23.260C24

Subgroups: 412 in 232 conjugacy classes, 132 normal (22 characteristic)
C1, C2 [×3], C2 [×4], C2 [×2], C4 [×20], C22 [×3], C22 [×4], C22 [×10], C2×C4 [×12], C2×C4 [×40], Q8 [×4], C23, C23 [×2], C23 [×6], C42 [×4], C22⋊C4 [×8], C22⋊C4 [×4], C4⋊C4 [×12], C4⋊C4 [×6], C22×C4 [×2], C22×C4 [×16], C22×C4 [×2], C2×Q8 [×4], C2×Q8 [×2], C24, C2.C42 [×2], C2.C42 [×12], C2×C42 [×4], C2×C22⋊C4 [×2], C2×C22⋊C4 [×4], C2×C4⋊C4, C2×C4⋊C4 [×8], C22⋊Q8 [×8], C23×C4, C22×Q8, C23.7Q8, C23.34D4, C23.63C23 [×4], C24.C22 [×4], C23.65C23 [×2], C23.67C23 [×2], C2×C22⋊Q8, C23.260C24

Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], C22×C4 [×14], C24, C23×C4, 2+ (1+4) [×3], 2- (1+4) [×3], C22.11C24, C23.32C23, C23.33C23, C22.56C24 [×2], C22.57C24 [×2], C23.260C24

Generators and relations
 G = < a,b,c,d,e,f,g | a2=b2=c2=f2=1, d2=c, e2=g2=a, ab=ba, ac=ca, ede-1=ad=da, geg-1=ae=ea, af=fa, ag=ga, bc=cb, fdf=bd=db, be=eb, bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, cg=gc, gdg-1=abd, fef=abe, fg=gf >

Smallest permutation representation
►On 64 points
Generators in S64
(1 9)(2 10)(3 11)(4 12)(5 38)(6 39)(7 40)(8 37)(13 41)(14 42)(15 43)(16 44)(17 45)(18 46)(19 47)(20 48)(21 49)(22 50)(23 51)(24 52)(25 53)(26 54)(27 55)(28 56)(29 57)(30 58)(31 59)(32 60)(33 63)(34 64)(35 61)(36 62)

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

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

(2 52)(4 50)(5 38)(6 33)(7 40)(8 35)(10 24)(12 22)(14 28)(16 26)(17 59)(18 46)(19 57)(20 48)(29 47)(30 58)(31 45)(32 60)(34 64)(36 62)(37 61)(39 63)(42 56)(44 54)

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

Matrix representation ►G ⊆ GL9(𝔽5)

100000000
040000000
004000000
000400000
000040000
000004000
000000400
000000040
000000004
,
100000000
010000000
001000000
000100000
000010000
000004000
000000400
000000040
000000004
,
400000000
010000000
001000000
000100000
000010000
000001000
000000100
000000010
000000001
,
200000000
004000000
040000000
000010000
000100000
000002030
000000303
000004030
000000402
,
100000000
000100000
000010000
040000000
004000000
000000100
000004000
000000004
000000010
,
400000000
010000000
001000000
000400000
000040000
000001000
000000100
000002040
000000304
,
100000000
001000000
040000000
000040000
000100000
000003000
000000200
000000030
000000002

G:=sub<GL(9,GF(5))| [1,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1],[2,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,2,0,4,0,0,0,0,0,0,0,3,0,4,0,0,0,0,0,3,0,3,0,0,0,0,0,0,0,3,0,2],[1,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,4,0],[4,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,1,0,2,0,0,0,0,0,0,0,1,0,3,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,2] >;
 

38 conjugacy classes

class 1 2A···2G2H2I4A···4AB
order12···2224···4
size11···1444···4

38 irreducible representations

dim11111111144
type+++++++++-
imageC1C2C2C2C2C2C2C2C42+ (1+4)2- (1+4)
kernelC23.260C24C23.7Q8C23.34D4C23.63C23C24.C22C23.65C23C23.67C23C2×C22⋊Q8C22⋊Q8C22C22
# reps111442211633

In GAP, Magma, Sage, TeX 

C_2^3._{260}C_2^4
 
% in TeX
 
G:=Group("C2^3.260C2^4");
 
// GroupNames label
 
G:=SmallGroup(128,1110);
 
// by ID
 
G=gap.SmallGroup(128,1110);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,120,758,555,1571,346,80]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=f^2=1,d^2=c,e^2=g^2=a,a*b=b*a,a*c=c*a,e*d*e^-1=a*d=d*a,g*e*g^-1=a*e=e*a,a*f=f*a,a*g=g*a,b*c=c*b,f*d*f=b*d=d*b,b*e=e*b,b*f=f*b,b*g=g*b,c*d=d*c,c*e=e*c,c*f=f*c,c*g=g*c,g*d*g^-1=a*b*d,f*e*f=a*b*e,f*g=g*f>;
 
// generators/relations
 

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