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G = C23.695C24order 128 = 27

412nd central stem extension by C23 of C24

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

Aliases: C24.93C23, C23.695C24, C22.4682+ 1+4, C22.3582- 1+4, C429C438C2, C23.Q888C2, (C2×C42).111C22, (C22×C4).605C23, C23.10D4.66C2, (C22×D4).283C22, C24.C22171C2, C23.83C23125C2, C23.63C23190C2, C2.36(C22.54C24), C2.C42.399C22, C2.47(C22.49C24), C2.117(C22.47C24), C2.113(C22.33C24), C2.118(C22.36C24), (C2×C4).236(C4○D4), (C2×C4⋊C4).505C22, C22.556(C2×C4○D4), (C2×C22⋊C4).326C22, SmallGroup(128,1527)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C23.695C24
C1C2C22C23C22×C4C2×C42C24.C22 — C23.695C24
C1C23 — C23.695C24
C1C23 — C23.695C24
C1C23 — C23.695C24

Generators and relations for C23.695C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=1, d2=abc, e2=f2=b, g2=ba=ab, 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: 436 in 210 conjugacy classes, 88 normal (22 characteristic)
C1, C2 [×3], C2 [×4], C2 [×2], C4 [×16], C22 [×3], C22 [×4], C22 [×14], C2×C4 [×6], C2×C4 [×36], D4 [×4], C23, C23 [×14], C42 [×4], C22⋊C4 [×18], C4⋊C4 [×11], C22×C4 [×3], C22×C4 [×10], C2×D4 [×3], C24 [×2], C2.C42 [×2], C2.C42 [×8], C2×C42, C2×C42 [×2], C2×C22⋊C4 [×12], C2×C4⋊C4 [×3], C2×C4⋊C4 [×6], C22×D4, C429C4, C23.63C23 [×2], C24.C22 [×6], C23.10D4, C23.10D4 [×2], C23.Q8 [×2], C23.83C23, C23.695C24
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], C4○D4 [×6], C24, C2×C4○D4 [×3], 2+ 1+4 [×3], 2- 1+4, C22.33C24, C22.36C24 [×2], C22.47C24 [×2], C22.49C24, C22.54C24, C23.695C24

Smallest permutation representation of C23.695C24
On 64 points
Generators in S64
(1 61)(2 62)(3 63)(4 64)(5 21)(6 22)(7 23)(8 24)(9 16)(10 13)(11 14)(12 15)(17 33)(18 34)(19 35)(20 36)(25 44)(26 41)(27 42)(28 43)(29 37)(30 38)(31 39)(32 40)(45 58)(46 59)(47 60)(48 57)(49 56)(50 53)(51 54)(52 55)
(1 35)(2 36)(3 33)(4 34)(5 50)(6 51)(7 52)(8 49)(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 53)(22 54)(23 55)(24 56)(25 47)(26 48)(27 45)(28 46)(41 57)(42 58)(43 59)(44 60)
(1 17)(2 18)(3 19)(4 20)(5 55)(6 56)(7 53)(8 54)(9 39)(10 40)(11 37)(12 38)(13 32)(14 29)(15 30)(16 31)(21 52)(22 49)(23 50)(24 51)(25 58)(26 59)(27 60)(28 57)(33 61)(34 62)(35 63)(36 64)(41 46)(42 47)(43 48)(44 45)
(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 43 35 59)(2 25 36 47)(3 41 33 57)(4 27 34 45)(5 29 50 9)(6 38 51 13)(7 31 52 11)(8 40 49 15)(10 22 30 54)(12 24 32 56)(14 23 39 55)(16 21 37 53)(17 48 63 26)(18 58 64 42)(19 46 61 28)(20 60 62 44)
(1 47 35 25)(2 26 36 48)(3 45 33 27)(4 28 34 46)(5 38 50 13)(6 14 51 39)(7 40 52 15)(8 16 49 37)(9 56 29 24)(10 21 30 53)(11 54 31 22)(12 23 32 55)(17 42 63 58)(18 59 64 43)(19 44 61 60)(20 57 62 41)
(1 53 19 5)(2 6 20 54)(3 55 17 7)(4 8 18 56)(9 46 37 43)(10 44 38 47)(11 48 39 41)(12 42 40 45)(13 25 30 60)(14 57 31 26)(15 27 32 58)(16 59 29 28)(21 61 50 35)(22 36 51 62)(23 63 52 33)(24 34 49 64)

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

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

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

32 conjugacy classes

class 1 2A···2G2H2I4A···4R4S4T4U4V
order12···2224···44444
size11···1884···48888

32 irreducible representations

dim1111111244
type++++++++-
imageC1C2C2C2C2C2C2C4○D42+ 1+42- 1+4
kernelC23.695C24C429C4C23.63C23C24.C22C23.10D4C23.Q8C23.83C23C2×C4C22C22
# reps11263211231

Matrix representation of C23.695C24 in GL6(𝔽5)

100000
010000
001000
000100
000040
000004
,
100000
010000
004000
000400
000010
000001
,
400000
040000
001000
000100
000010
000001
,
200000
020000
003300
000200
000033
000002
,
020000
300000
002000
000200
000022
000013
,
010000
100000
004400
002100
000010
000001
,
400000
040000
004400
002100
000020
000013

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

C23.695C24 in GAP, Magma, Sage, TeX

C_2^3._{695}C_2^4
% in TeX

G:=Group("C2^3.695C2^4");
// GroupNames label

G:=SmallGroup(128,1527);
// by ID

G=gap.SmallGroup(128,1527);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,784,253,232,758,723,184,1571,346,192]);
// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=1,d^2=a*b*c,e^2=f^2=b,g^2=b*a=a*b,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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