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G = C23.629C24  order 128 = 27

346th central stem extension by C23 of C24

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

Aliases: C23.629C24, C24.421C23, C22.4022+ (1+4), C22.3042- (1+4), C22⋊C4⋊5Q8, C2.33(D4×Q8), (C2×Q8).120D4, C23.36(C2×Q8), C2.55(Q8⋊5D4), C2.52(D4⋊3Q8), C23⋊Q8.21C2, C2.53(C23⋊3D4), (C2×C42).680C22, (C23×C4).476C22, (C22×C4).198C23, C22.438(C22×D4), C23.8Q8.51C2, C23.Q8.32C2, C22.149(C22×Q8), (C22×Q8).198C22, C23.83C23⋊90C2, C23.78C23⋊52C2, C23.67C23⋊90C2, C24.C22.56C2, C23.81C23⋊102C2, C23.65C23⋊137C2, C23.63C23⋊150C2, C2.C42.335C22, C2.30(C22.57C24), C2.34(C23.41C23), C2.85(C22.36C24), (C2×C4⋊Q8)⋊23C2, (C2×C4).73(C2×Q8), (C2×C4).123(C2×D4), (C2×C22⋊Q8).47C2, (C2×C4).209(C4○D4), (C2×C4⋊C4).442C22, C22.491(C2×C4○D4), (C2×C22⋊C4).292C22, SmallGroup(128,1461)

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

Derived series
C1 — C23 — C23.629C24
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C23×C4 — C23.8Q8 — C23.629C24
Lower central
C1 — C23 — C23.629C24
Upper central
C1 — C23 — C23.629C24
Jennings
C1 — C23 — C23.629C24

Subgroups: 452 in 240 conjugacy classes, 104 normal (82 characteristic)
C1, C2 [×7], C2 [×2], C4 [×20], C22 [×7], C22 [×10], C2×C4 [×12], C2×C4 [×40], Q8 [×8], C23, C23 [×2], C23 [×6], C42 [×3], C22⋊C4 [×4], C22⋊C4 [×8], C4⋊C4 [×21], C22×C4 [×14], C22×C4 [×4], C2×Q8 [×4], C2×Q8 [×6], C24, C2.C42 [×12], C2×C42 [×2], C2×C22⋊C4 [×6], C2×C4⋊C4 [×12], C22⋊Q8 [×4], C4⋊Q8 [×4], C23×C4, C22×Q8 [×2], C23.8Q8 [×2], C23.63C23, C24.C22 [×2], C23.65C23, C23.67C23, C23⋊Q8, C23.78C23, C23.Q8, C23.81C23 [×2], C23.83C23, C2×C22⋊Q8, C2×C4⋊Q8, C23.629C24

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], C23⋊3D4, C22.36C24, C23.41C23, Q8⋊5D4, D4×Q8, D4⋊3Q8, C22.57C24, C23.629C24

Generators and relations
 G = < a,b,c,d,e,f,g | a2=b2=c2=f2=1, d2=g2=ba=ab, e2=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 9)(2 10)(3 11)(4 12)(5 37)(6 38)(7 39)(8 40)(13 45)(14 46)(15 47)(16 48)(17 49)(18 50)(19 51)(20 52)(21 43)(22 44)(23 41)(24 42)(25 55)(26 56)(27 53)(28 54)(29 59)(30 60)(31 57)(32 58)(33 64)(34 61)(35 62)(36 63)

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

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

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

(2 12)(4 10)(5 42)(6 23)(7 44)(8 21)(14 48)(16 46)(17 63)(18 35)(19 61)(20 33)(22 39)(24 37)(26 54)(28 56)(30 58)(32 60)(34 51)(36 49)(38 41)(40 43)(50 62)(52 64)

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

Matrix representation ►G ⊆ GL6(𝔽5)

100000
010000
004000
000400
000010
000001
,
400000
040000
001000
000100
000010
000001
,
100000
010000
001000
000100
000040
000004
,
010000
400000
000200
002000
000010
000001
,
400000
040000
000100
004000
000001
000010
,
100000
040000
001000
000100
000010
000004
,
300000
020000
003000
000200
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],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[0,4,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,0,0,2,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,4,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[1,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4],[3,0,0,0,0,0,0,2,0,0,0,0,0,0,3,0,0,0,0,0,0,2,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;
 

32 conjugacy classes

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

32 irreducible representations

dim111111111111122244
type+++++++++++++-++-
imageC1C2C2C2C2C2C2C2C2C2C2C2C2Q8D4C4○D42+ (1+4)2- (1+4)
kernelC23.629C24C23.8Q8C23.63C23C24.C22C23.65C23C23.67C23C23⋊Q8C23.78C23C23.Q8C23.81C23C23.83C23C2×C22⋊Q8C2×C4⋊Q8C22⋊C4C2×Q8C2×C4C22C22
# reps121211111211144422

In GAP, Magma, Sage, TeX 

C_2^3._{629}C_2^4
 
% in TeX
 
G:=Group("C2^3.629C2^4");
 
// GroupNames label
 
G:=SmallGroup(128,1461);
 
// by ID
 
G=gap.SmallGroup(128,1461);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,336,253,120,758,723,1571,346,192]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=f^2=1,d^2=g^2=b*a=a*b,e^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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