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G = C23.313C24  order 128 = 27

30th central stem extension by C23 of C24

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

Aliases: C24.5C23, C23.313C24, C22.912- (1+4), C22⋊C4.124D4, (C22×C4).373D4, C23.422(C2×D4), C2.11(D4⋊6D4), C23.Q8⋊2C2, C23.8Q8⋊29C2, C23.300(C4○D4), C23.34D4⋊20C2, C22.74(C4⋊D4), (C2×C42).462C22, (C22×C4).504C23, (C23×C4).331C22, C22.193(C22×D4), C24.C22⋊30C2, C23.81C23⋊3C2, C23.23D4.15C2, (C22×D4).119C22, C23.65C23⋊33C2, C2.20(C22.19C24), C2.C42.77C22, C2.10(C23.38C23), C2.12(C22.46C24), (C22×C4⋊C4)⋊17C2, (C2×C4).307(C2×D4), C2.17(C2×C4⋊D4), (C2×C42⋊C2)⋊19C2, (C2×C4).359(C4○D4), (C2×C4⋊C4).204C22, C22.192(C2×C4○D4), (C2×C22⋊C4).108C22, (C2×C22.D4).10C2, SmallGroup(128,1145)

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

Derived series
C1 — C23 — C23.313C24
Chief series
C1 — C2 — C22 — C23 — C24 — C23×C4 — C22×C4⋊C4 — C23.313C24
Lower central
C1 — C23 — C23.313C24
Upper central
C1 — C23 — C23.313C24
Jennings
C1 — C23 — C23.313C24

Subgroups: 548 in 305 conjugacy classes, 112 normal (34 characteristic)
C1, C2 [×3], C2 [×4], C2 [×5], C4 [×19], C22 [×3], C22 [×8], C22 [×19], C2×C4 [×12], C2×C4 [×53], D4 [×4], C23, C23 [×6], C23 [×11], C42 [×4], C22⋊C4 [×4], C22⋊C4 [×16], C4⋊C4 [×24], C22×C4 [×5], C22×C4 [×12], C22×C4 [×16], C2×D4 [×6], C24 [×2], C2.C42 [×8], C2×C42 [×2], C2×C22⋊C4 [×5], C2×C22⋊C4 [×4], C2×C4⋊C4 [×2], C2×C4⋊C4 [×10], C2×C4⋊C4 [×4], C42⋊C2 [×4], C22.D4 [×8], C23×C4 [×3], C22×D4, C23.34D4, C23.8Q8, C23.23D4, C24.C22 [×2], C23.65C23 [×2], C23.Q8 [×2], C23.81C23 [×2], C22×C4⋊C4, C2×C42⋊C2, C2×C22.D4 [×2], C23.313C24

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], C2×C4⋊D4, C22.19C24, C23.38C23, D4⋊6D4 [×2], C22.46C24 [×2], C23.313C24

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

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

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

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

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

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

Matrix representation ►G ⊆ GL6(𝔽5)

400000
040000
004000
000400
000010
000001
,
100000
010000
004000
000400
000040
000004
,
100000
010000
001000
000100
000040
000004
,
030000
200000
002000
000300
000001
000040
,
030000
300000
000200
002000
000003
000020
,
300000
030000
000300
003000
000001
000010
,
010000
100000
000100
001000
000040
000004

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

38 conjugacy classes

class 1 2A···2G2H2I2J2K2L4A4B4C4D4E···4V4W4X4Y
order12···22222244444···4444
size11···12222822224···4888

38 irreducible representations

dim1111111111122224
type+++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2C2D4D4C4○D4C4○D42- (1+4)
kernelC23.313C24C23.34D4C23.8Q8C23.23D4C24.C22C23.65C23C23.Q8C23.81C23C22×C4⋊C4C2×C42⋊C2C2×C22.D4C22⋊C4C22×C4C2×C4C23C22
# reps1111222211244842

In GAP, Magma, Sage, TeX 

C_2^3._{313}C_2^4
 
% in TeX
 
G:=Group("C2^3.313C2^4");
 
// GroupNames label
 
G:=SmallGroup(128,1145);
 
// by ID
 
G=gap.SmallGroup(128,1145);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,224,253,120,758,723,100,675]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=g^2=1,d^2=b,e^2=f^2=a,a*b=b*a,a*c=c*a,e*d*e^-1=g*d*g=a*d=d*a,a*e=e*a,a*f=f*a,a*g=g*a,b*c=c*b,f*d*f^-1=b*d=d*b,b*e=e*b,b*f=f*b,b*g=g*b,c*d=d*c,f*e*f^-1=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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