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G = C23.663C24order 128 = 27

380th central stem extension by C23 of C24

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

Aliases: C23.663C24, C24.441C23, C22.4362+ 1+4, C22.3292- 1+4, C22⋊C47Q8, C428C462C2, C23.40(C2×Q8), C2.57(D43Q8), C23⋊Q8.26C2, (C22×C4).208C23, (C2×C42).695C22, (C23×C4).168C22, C23.8Q8.58C2, C22.155(C22×Q8), (C22×Q8).214C22, C23.78C2357C2, C23.67C2399C2, C2.10(C24⋊C22), C24.C22.65C2, C23.83C23105C2, C2.C42.367C22, C2.115(C22.45C24), C2.38(C23.41C23), C2.104(C22.36C24), (C2×C4).79(C2×Q8), (C2×C4).457(C4○D4), (C2×C4⋊C4).473C22, C22.524(C2×C4○D4), (C2×C22⋊C4).310C22, SmallGroup(128,1495)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C23.663C24
C1C2C22C23C22×C4C2×C42C23.67C23 — C23.663C24
C1C23 — C23.663C24
C1C23 — C23.663C24
C1C23 — C23.663C24

Generators and relations for C23.663C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=1, e2=b, f2=c, g2=cb=bc, ab=ba, gag-1=ac=ca, eae-1=ad=da, faf-1=acd, bd=db, be=eb, bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, cg=gc, fef-1=de=ed, df=fd, dg=gd, geg-1=bde, gfg-1=cdf >

Subgroups: 420 in 212 conjugacy classes, 96 normal (22 characteristic)
C1, C2 [×3], C2 [×4], C2 [×2], C4 [×18], C22 [×3], C22 [×4], C22 [×10], C2×C4 [×8], C2×C4 [×42], Q8 [×8], C23, C23 [×2], C23 [×6], C42 [×4], C22⋊C4 [×4], C22⋊C4 [×7], C4⋊C4 [×8], C22×C4 [×2], C22×C4 [×12], C22×C4 [×3], C2×Q8 [×8], C24, C2.C42 [×2], C2.C42 [×14], C2×C42 [×2], C2×C22⋊C4 [×2], C2×C22⋊C4 [×4], C2×C4⋊C4 [×8], C23×C4, C22×Q8 [×2], C428C4 [×2], C23.8Q8, C23.8Q8 [×2], C24.C22 [×2], C23.67C23 [×2], C23⋊Q8 [×2], C23.78C23 [×2], C23.83C23 [×2], C23.663C24
Quotients: C1, C2 [×15], C22 [×35], Q8 [×4], C23 [×15], C2×Q8 [×6], C4○D4 [×4], C24, C22×Q8, C2×C4○D4 [×2], 2+ 1+4 [×3], 2- 1+4, C22.36C24 [×2], C23.41C23, C22.45C24, D43Q8 [×2], C24⋊C22, C23.663C24

Smallest permutation representation of C23.663C24
On 64 points
Generators in S64
(2 46)(4 48)(6 55)(8 53)(10 52)(12 50)(13 42)(14 31)(15 44)(16 29)(17 21)(18 63)(19 23)(20 61)(22 33)(24 35)(26 37)(28 39)(30 59)(32 57)(34 64)(36 62)(41 58)(43 60)
(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(13 15)(14 16)(17 19)(18 20)(21 23)(22 24)(25 27)(26 28)(29 31)(30 32)(33 35)(34 36)(37 39)(38 40)(41 43)(42 44)(45 47)(46 48)(49 51)(50 52)(53 55)(54 56)(57 59)(58 60)(61 63)(62 64)
(1 49)(2 50)(3 51)(4 52)(5 38)(6 39)(7 40)(8 37)(9 47)(10 48)(11 45)(12 46)(13 42)(14 43)(15 44)(16 41)(17 21)(18 22)(19 23)(20 24)(25 56)(26 53)(27 54)(28 55)(29 58)(30 59)(31 60)(32 57)(33 63)(34 64)(35 61)(36 62)
(1 45)(2 46)(3 47)(4 48)(5 54)(6 55)(7 56)(8 53)(9 51)(10 52)(11 49)(12 50)(13 59)(14 60)(15 57)(16 58)(17 36)(18 33)(19 34)(20 35)(21 62)(22 63)(23 64)(24 61)(25 40)(26 37)(27 38)(28 39)(29 41)(30 42)(31 43)(32 44)
(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 63 49 33)(2 23 50 19)(3 61 51 35)(4 21 52 17)(5 16 38 41)(6 59 39 30)(7 14 40 43)(8 57 37 32)(9 20 47 24)(10 36 48 62)(11 18 45 22)(12 34 46 64)(13 28 42 55)(15 26 44 53)(25 31 56 60)(27 29 54 58)
(1 13 51 44)(2 58 52 31)(3 15 49 42)(4 60 50 29)(5 17 40 23)(6 35 37 63)(7 19 38 21)(8 33 39 61)(9 32 45 59)(10 43 46 16)(11 30 47 57)(12 41 48 14)(18 28 24 53)(20 26 22 55)(25 64 54 36)(27 62 56 34)

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

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

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

32 conjugacy classes

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

32 irreducible representations

dim111111112244
type++++++++-+-
imageC1C2C2C2C2C2C2C2Q8C4○D42+ 1+42- 1+4
kernelC23.663C24C428C4C23.8Q8C24.C22C23.67C23C23⋊Q8C23.78C23C23.83C23C22⋊C4C2×C4C22C22
# reps123222224831

Matrix representation of C23.663C24 in GL6(𝔽5)

100000
040000
001000
000100
000010
000004
,
100000
010000
004000
000400
000010
000001
,
400000
040000
001000
000100
000010
000001
,
100000
010000
001000
000100
000040
000004
,
400000
040000
000300
003000
000003
000020
,
030000
300000
004000
000400
000001
000010
,
010000
400000
000100
004000
000010
000004

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

C23.663C24 in GAP, Magma, Sage, TeX

C_2^3._{663}C_2^4
% in TeX

G:=Group("C2^3.663C2^4");
// GroupNames label

G:=SmallGroup(128,1495);
// by ID

G=gap.SmallGroup(128,1495);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,784,253,120,758,723,184,1571,346,80]);
// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=1,e^2=b,f^2=c,g^2=c*b=b*c,a*b=b*a,g*a*g^-1=a*c=c*a,e*a*e^-1=a*d=d*a,f*a*f^-1=a*c*d,b*d=d*b,b*e=e*b,b*f=f*b,b*g=g*b,c*d=d*c,c*e=e*c,c*f=f*c,c*g=g*c,f*e*f^-1=d*e=e*d,d*f=f*d,d*g=g*d,g*e*g^-1=b*d*e,g*f*g^-1=c*d*f>;
// generators/relations

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