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G = C23.607C24  order 128 = 27

324th central stem extension by C23 of C24

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

Aliases: C24.66C23, C23.607C24, C22.2842- 1+4, C22.3812+ 1+4, C4⋊C4.120D4, C2.48(Q8⋊5D4), C2.112(D4⋊5D4), C23.7Q8⋊95C2, C23.Q8⋊67C2, C23.178(C4○D4), C23.11D4⋊91C2, (C23×C4).466C22, (C22×C4).882C23, (C2×C42).659C22, C23.8Q8⋊110C2, C22.416(C22×D4), C23.10D4.45C2, (C22×D4).242C22, C24.C22⋊137C2, C23.81C23⋊92C2, C24.3C22.62C2, C23.65C23⋊126C2, C23.63C23⋊138C2, C2.C42.313C22, C2.49(C22.31C24), C2.69(C22.33C24), C2.16(C22.57C24), C2.86(C22.46C24), C2.45(C22.34C24), (C2×C4).109(C2×D4), (C2×C42.C2)⋊23C2, (C2×C4).195(C4○D4), (C2×C4⋊C4).420C22, C22.469(C2×C4○D4), (C2×C22⋊C4).273C22, (C2×C22.D4).27C2, SmallGroup(128,1439)

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

Derived series
C1 — C23 — C23.607C24
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C23×C4 — C23.7Q8 — C23.607C24
Lower central
C1 — C23 — C23.607C24
Upper central
C1 — C23 — C23.607C24
Jennings
C1 — C23 — C23.607C24

Generators and relations for C23.607C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=1, d2=e2=ba=ab, f2=g2=a, ac=ca, ede-1=ad=da, geg-1=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-1=abd, fg=gf >

Subgroups: 484 in 245 conjugacy classes, 96 normal (82 characteristic)
C1, C2, C2, C4, C22, C22, C2×C4, C2×C4, D4, C23, C23, C23, C42, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C24, C2.C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C22.D4, C42.C2, C23×C4, C22×D4, C23.7Q8, C23.8Q8, C23.63C23, C24.C22, C23.65C23, C24.3C22, C23.10D4, C23.Q8, C23.11D4, C23.81C23, C2×C22.D4, C2×C42.C2, C23.607C24
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C22×D4, C2×C4○D4, 2+ 1+4, 2- 1+4, C22.31C24, C22.33C24, C22.34C24, D4⋊5D4, Q8⋊5D4, C22.46C24, C22.57C24, C23.607C24

Smallest permutation representation of C23.607C24
►On 64 points
Generators in S64
(1 25)(2 26)(3 27)(4 28)(5 19)(6 20)(7 17)(8 18)(9 15)(10 16)(11 13)(12 14)(21 62)(22 63)(23 64)(24 61)(29 33)(30 34)(31 35)(32 36)(37 49)(38 50)(39 51)(40 52)(41 55)(42 56)(43 53)(44 54)(45 60)(46 57)(47 58)(48 59)

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

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

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

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

32 conjugacy classes

class 1 2A···2G2H2I2J4A···4P4Q···4U
order12···22224···44···4
size11···14484···48···8

32 irreducible representations

dim111111111111122244
type+++++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2C2C2C2D4C4○D4C4○D42+ 1+42- 1+4
kernelC23.607C24C23.7Q8C23.8Q8C23.63C23C24.C22C23.65C23C24.3C22C23.10D4C23.Q8C23.11D4C23.81C23C2×C22.D4C2×C42.C2C4⋊C4C2×C4C23C22C22
# reps111121113111144422

Matrix representation of C23.607C24 ►in GL6(𝔽5)

100000
010000
001000
000100
000040
000004
,
400000
040000
001000
000100
000010
000001
,
100000
010000
004000
000400
000010
000001
,
040000
100000
001000
000100
000020
000003
,
200000
020000
004300
000100
000003
000030
,
010000
100000
001000
004400
000030
000003
,
010000
100000
004000
000400
000001
000040

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

C23.607C24 in GAP, Magma, Sage, TeX 

C_2^3._{607}C_2^4
 
% in TeX
 
G:=Group("C2^3.607C2^4");
 
// GroupNames label
 
G:=SmallGroup(128,1439);
 
// by ID
 
G=gap.SmallGroup(128,1439);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,336,253,344,758,723,100,1571,346,80]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=1,d^2=e^2=b*a=a*b,f^2=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^-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^-1=a*b*d,f*g=g*f>;
 
// generators/relations
 

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