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G = C23.604C24  order 128 = 27

321st central stem extension by C23 of C24

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

Aliases: C23.604C24, C24.408C23, C22.3782+ (1+4), C22.2812- (1+4), C22⋊C4⋊3Q8, C2.29(D4×Q8), C4⋊C4.119D4, C23.34(C2×Q8), C2.50(D4⋊3Q8), C2.109(D4⋊5D4), (C23×C4).464C22, (C2×C42).656C22, (C22×C4).879C23, C22.413(C22×D4), C23.7Q8.65C2, C23.8Q8.48C2, C23.Q8.27C2, C22.145(C22×Q8), (C22×Q8).188C22, C23.78C23⋊47C2, C23.67C23⋊84C2, C23.81C23⋊90C2, C24.C22.51C2, C23.65C23⋊124C2, C2.C42.310C22, C2.30(C23.41C23), C2.18(C22.56C24), C2.47(C22.31C24), C2.41(C22.35C24), (C2×C4).69(C2×Q8), (C2×C4).106(C2×D4), (C2×C42.C2)⋊22C2, (C2×C22⋊Q8).45C2, (C2×C4).430(C4○D4), (C2×C4⋊C4).417C22, C22.466(C2×C4○D4), (C2×C22⋊C4).270C22, SmallGroup(128,1436)

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

Derived series
C1 — C23 — C23.604C24
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C23×C4 — C23.7Q8 — C23.604C24
Lower central
C1 — C23 — C23.604C24
Upper central
C1 — C23 — C23.604C24
Jennings
C1 — C23 — C23.604C24

Subgroups: 436 in 236 conjugacy classes, 104 normal (82 characteristic)
C1, C2 [×7], C2 [×2], C4 [×20], C22 [×7], C22 [×10], C2×C4 [×12], C2×C4 [×40], Q8 [×4], C23, C23 [×2], C23 [×6], C42 [×3], C22⋊C4 [×4], C22⋊C4 [×7], C4⋊C4 [×4], C4⋊C4 [×22], C22×C4 [×14], C22×C4 [×5], C2×Q8 [×5], C24, C2.C42 [×10], C2×C42 [×2], C2×C22⋊C4 [×6], C2×C4⋊C4 [×15], C22⋊Q8 [×4], C42.C2 [×4], C23×C4, C22×Q8, C23.7Q8, C23.8Q8, C24.C22 [×2], C23.65C23 [×2], C23.67C23, C23.78C23, C23.Q8 [×2], C23.81C23 [×3], C2×C22⋊Q8, C2×C42.C2, C23.604C24

Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], Q8 [×4], C23 [×15], C2×D4 [×6], C2×Q8 [×6], C4○D4 [×2], C24, C22×D4, C22×Q8, C2×C4○D4, 2+ (1+4) [×2], 2- (1+4) [×2], C22.31C24, C22.35C24, C23.41C23, D4⋊5D4, D4×Q8, D4⋊3Q8, C22.56C24, C23.604C24

Generators and relations
 G = < a,b,c,d,e,f,g | a2=b2=c2=f2=1, d2=e2=ba=ab, g2=a, 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, fef=ce=ec, cf=fc, cg=gc, gdg-1=abd, fg=gf >

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

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

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

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

(2 44)(4 42)(5 50)(6 17)(7 52)(8 19)(9 38)(10 35)(11 40)(12 33)(14 48)(16 46)(18 62)(20 64)(21 34)(22 39)(23 36)(24 37)(26 30)(28 32)(49 63)(51 61)(54 58)(56 60)

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

Matrix representation ►G ⊆ GL6(𝔽5)

400000
040000
001000
000100
000010
000001
,
100000
010000
004000
000400
000010
000001
,
100000
010000
001000
000100
000040
000004
,
300000
020000
004300
001100
000010
000001
,
020000
200000
003000
000300
000001
000010
,
100000
010000
001000
004400
000010
000004
,
010000
400000
001000
004400
000040
000004

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

32 conjugacy classes

class 1 2A···2G2H2I4A···4P4Q···4V
order12···2224···44···4
size11···1444···48···8

32 irreducible representations

dim1111111111122244
type+++++++++++-++-
imageC1C2C2C2C2C2C2C2C2C2C2Q8D4C4○D42+ (1+4)2- (1+4)
kernelC23.604C24C23.7Q8C23.8Q8C24.C22C23.65C23C23.67C23C23.78C23C23.Q8C23.81C23C2×C22⋊Q8C2×C42.C2C22⋊C4C4⋊C4C2×C4C22C22
# reps1112211231144422

In GAP, Magma, Sage, TeX 

C_2^3._{604}C_2^4
 
% in TeX
 
G:=Group("C2^3.604C2^4");
 
// GroupNames label
 
G:=SmallGroup(128,1436);
 
// by ID
 
G=gap.SmallGroup(128,1436);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,336,253,344,758,723,1571,346,80]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=f^2=1,d^2=e^2=b*a=a*b,g^2=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,f*e*f=c*e=e*c,c*f=f*c,c*g=g*c,g*d*g^-1=a*b*d,f*g=g*f>;
 
// generators/relations
 

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