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G = C23.652C24  order 128 = 27

369th central stem extension by C23 of C24

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

Aliases: C24.80C23, C23.652C24, C22.4252+ (1+4), C23⋊2D4.29C2, C23.4Q8⋊60C2, C23.189(C4○D4), C23.34D4⋊58C2, (C2×C42).691C22, (C22×C4).573C23, (C23×C4).163C22, C23.10D4⋊100C2, C23.11D4⋊111C2, C23.23D4⋊102C2, C24.3C22⋊92C2, (C22×D4).269C22, C24.C22⋊160C2, C2.85(C22.32C24), C23.63C23⋊167C2, C2.27(C22.54C24), C2.C42.356C22, C2.104(C22.45C24), C2.38(C22.53C24), C2.54(C22.34C24), (C2×C4).216(C4○D4), (C2×C4⋊C4).463C22, C22.513(C2×C4○D4), (C2×C22⋊C4).67C22, SmallGroup(128,1484)

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

Derived series
C1 — C23 — C23.652C24
Chief series
C1 — C2 — C22 — C23 — C24 — C23×C4 — C23.23D4 — C23.652C24
Lower central
C1 — C23 — C23.652C24
Upper central
C1 — C23 — C23.652C24
Jennings
C1 — C23 — C23.652C24

Subgroups: 548 in 242 conjugacy classes, 88 normal (22 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×14], C22 [×3], C22 [×4], C22 [×24], C2×C4 [×4], C2×C4 [×38], D4 [×12], C23, C23 [×2], C23 [×20], C42 [×2], C22⋊C4 [×16], C4⋊C4 [×7], C22×C4 [×2], C22×C4 [×10], C22×C4 [×4], C2×D4 [×13], C24, C24 [×2], C2.C42 [×2], C2.C42 [×8], C2×C42 [×2], C2×C22⋊C4 [×2], C2×C22⋊C4 [×12], C2×C4⋊C4, C2×C4⋊C4 [×4], C23×C4, C22×D4, C22×D4 [×2], C23.34D4, C23.23D4 [×2], C23.63C23 [×2], C24.C22 [×2], C24.3C22 [×2], C23⋊2D4, C23.10D4 [×2], C23.11D4 [×2], C23.4Q8, C23.652C24

Quotients:
C1, C2 [×15], C22 [×35], C23 [×15], C4○D4 [×6], C24, C2×C4○D4 [×3], 2+ (1+4) [×4], C22.32C24, C22.34C24 [×2], C22.45C24 [×2], C22.53C24, C22.54C24, C23.652C24

Generators and relations
 G = < a,b,c,d,e,f,g | a2=b2=c2=g2=1, d2=ca=ac, e2=a, f2=ba=ab, ede-1=ad=da, geg=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, gdg=abd, fg=gf >

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

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

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

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

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

Matrix representation ►G ⊆ GL6(𝔽5)

400000
040000
001000
000100
000010
000001
,
100000
010000
001000
000100
000040
000004
,
100000
010000
004000
000400
000010
000001
,
210000
030000
003000
000300
000002
000030
,
420000
410000
003400
003200
000010
000001
,
200000
020000
001300
000400
000002
000020
,
400000
410000
004000
000400
000001
000010

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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[2,0,0,0,0,0,1,3,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,0,3,0,0,0,0,2,0],[4,4,0,0,0,0,2,1,0,0,0,0,0,0,3,3,0,0,0,0,4,2,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[2,0,0,0,0,0,0,2,0,0,0,0,0,0,1,0,0,0,0,0,3,4,0,0,0,0,0,0,0,2,0,0,0,0,2,0],[4,4,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;
 

32 conjugacy classes

class 1 2A···2G2H2I2J2K4A···4P4Q4R4S4T
order12···222224···44444
size11···144884···48888

32 irreducible representations

dim1111111111224
type+++++++++++
imageC1C2C2C2C2C2C2C2C2C2C4○D4C4○D42+ (1+4)
kernelC23.652C24C23.34D4C23.23D4C23.63C23C24.C22C24.3C22C23⋊2D4C23.10D4C23.11D4C23.4Q8C2×C4C23C22
# reps1122221221844

In GAP, Magma, Sage, TeX 

C_2^3._{652}C_2^4
 
% in TeX
 
G:=Group("C2^3.652C2^4");
 
// GroupNames label
 
G:=SmallGroup(128,1484);
 
// by ID
 
G=gap.SmallGroup(128,1484);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,560,253,120,758,723,268,1571,346]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=g^2=1,d^2=c*a=a*c,e^2=a,f^2=b*a=a*b,e*d*e^-1=a*d=d*a,g*e*g=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,g*d*g=a*b*d,f*g=g*f>;
 
// generators/relations
 

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