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G = C23.326C24  order 128 = 27

43rd central stem extension by C23 of C24

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

Aliases: C23.326C24, C24.259C23, C22.1002- (1+4), C22.1392+ (1+4), (C2×Q8)⋊23D4, C4⋊C4.319D4, C4.30(C4⋊D4), C2.9(Q8⋊6D4), C2.26(D4⋊5D4), C23.Q8⋊7C2, C2.13(Q8⋊5D4), C23.24(C4○D4), C23.10D4⋊17C2, (C2×C42).473C22, (C23×C4).339C22, (C22×C4).507C23, C22.206(C22×D4), C24.C22⋊36C2, C24.3C22⋊33C2, (C22×D4).124C22, (C22×Q8).418C22, C23.65C23⋊43C2, C2.C42.87C22, C2.22(C23.36C23), C2.14(C22.36C24), (C2×C4×Q8)⋊15C2, (C2×C4).49(C2×D4), (C2×C22⋊Q8)⋊6C2, (C4×C22⋊C4)⋊52C2, (C2×C4.4D4)⋊4C2, C2.24(C2×C4⋊D4), (C2×C4⋊D4).25C2, (C2×C4).653(C4○D4), (C2×C4⋊C4).213C22, C22.205(C2×C4○D4), (C2×C22⋊C4).116C22, SmallGroup(128,1158)

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

Derived series
C1 — C23 — C23.326C24
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C23×C4 — C4×C22⋊C4 — C23.326C24
Lower central
C1 — C23 — C23.326C24
Upper central
C1 — C23 — C23.326C24
Jennings
C1 — C23 — C23.326C24

Subgroups: 628 in 316 conjugacy classes, 112 normal (42 characteristic)
C1, C2 [×7], C2 [×4], C4 [×4], C4 [×16], C22 [×7], C22 [×24], C2×C4 [×16], C2×C4 [×32], D4 [×12], Q8 [×8], C23, C23 [×2], C23 [×20], C42 [×8], C22⋊C4 [×26], C4⋊C4 [×4], C4⋊C4 [×12], C22×C4 [×6], C22×C4 [×6], C22×C4 [×6], C2×D4 [×16], C2×Q8 [×4], C2×Q8 [×4], C24, C24 [×2], C2.C42 [×2], C2.C42 [×2], C2×C42 [×2], C2×C42 [×2], C2×C22⋊C4 [×2], C2×C22⋊C4 [×12], C2×C4⋊C4 [×2], C2×C4⋊C4 [×6], C4×Q8 [×4], C4⋊D4 [×4], C22⋊Q8 [×4], C4.4D4 [×4], C23×C4, C22×D4, C22×D4 [×2], C22×Q8, C4×C22⋊C4, C24.C22 [×2], C23.65C23, C24.3C22, C24.3C22 [×2], C23.10D4 [×2], C23.Q8 [×2], C2×C4×Q8, C2×C4⋊D4, C2×C22⋊Q8, C2×C4.4D4, C23.326C24

Quotients:
C1, C2 [×15], C22 [×35], D4 [×8], C23 [×15], C2×D4 [×12], C4○D4 [×6], C24, C4⋊D4 [×4], C22×D4 [×2], C2×C4○D4 [×3], 2+ (1+4), 2- (1+4), C2×C4⋊D4, C23.36C23, C22.36C24, D4⋊5D4, Q8⋊5D4 [×2], Q8⋊6D4, C23.326C24

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

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

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

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

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

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

Matrix representation ►G ⊆ GL6(𝔽5)

100000
010000
004000
000400
000010
000001
,
400000
040000
004000
000400
000040
000004
,
100000
010000
001000
000100
000040
000004
,
330000
020000
003400
000200
000004
000010
,
300000
030000
004200
004100
000004
000010
,
220000
130000
003400
003200
000001
000010
,
100000
010000
001300
001400
000010
000001

G:=sub<GL(6,GF(5))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[3,0,0,0,0,0,3,2,0,0,0,0,0,0,3,0,0,0,0,0,4,2,0,0,0,0,0,0,0,1,0,0,0,0,4,0],[3,0,0,0,0,0,0,3,0,0,0,0,0,0,4,4,0,0,0,0,2,1,0,0,0,0,0,0,0,1,0,0,0,0,4,0],[2,1,0,0,0,0,2,3,0,0,0,0,0,0,3,3,0,0,0,0,4,2,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,3,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;
 

38 conjugacy classes

class 1 2A···2G2H2I2J2K4A···4H4I···4X4Y4Z
order12···222224···44···444
size11···144882···24···488

38 irreducible representations

dim11111111111222244
type++++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2C2D4D4C4○D4C4○D42+ (1+4)2- (1+4)
kernelC23.326C24C4×C22⋊C4C24.C22C23.65C23C24.3C22C23.10D4C23.Q8C2×C4×Q8C2×C4⋊D4C2×C22⋊Q8C2×C4.4D4C4⋊C4C2×Q8C2×C4C23C22C22
# reps11213221111448411

In GAP, Magma, Sage, TeX 

C_2^3._{326}C_2^4
 
% in TeX
 
G:=Group("C2^3.326C2^4");
 
// GroupNames label
 
G:=SmallGroup(128,1158);
 
// by ID
 
G=gap.SmallGroup(128,1158);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,224,253,232,758,723,675,80]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=f^2=1,d^2=e^2=b,g^2=a,a*b=b*a,a*c=c*a,e*d*e^-1=g*d*g^-1=a*d=d*a,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,f*e*f=c*e=e*c,c*f=f*c,c*g=g*c,e*g=g*e,f*g=g*f>;
 
// generators/relations
 

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