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G = C23.617C24order 128 = 27

334th central stem extension by C23 of C24

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

Aliases: C23.617C24, C24.415C23, C22.3912+ 1+4, C22.2922- 1+4, (C2×D4).144D4, C23.73(C2×D4), C2.68(D46D4), C23.7Q898C2, C23.11D497C2, C2.50(C233D4), (C23×C4).471C22, (C22×C4).560C23, (C2×C42).668C22, C23.8Q8117C2, C22.426(C22×D4), C23.23D4.56C2, (C22×D4).250C22, (C22×Q8).194C22, C23.78C2349C2, C23.63C23144C2, C2.82(C22.45C24), C2.C42.323C22, C2.82(C22.36C24), C2.20(C22.57C24), (C2×C4).427(C2×D4), (C2×C4).434(C4○D4), (C2×C4⋊C4).430C22, (C2×C4.4D4).33C2, C22.479(C2×C4○D4), (C2×C22⋊C4).282C22, (C2×C22.D4).28C2, SmallGroup(128,1449)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C23.617C24
C1C2C22C23C24C23×C4C23.8Q8 — C23.617C24
C1C23 — C23.617C24
C1C23 — C23.617C24
C1C23 — C23.617C24

Generators and relations for C23.617C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=g2=1, e2=b, f2=ba=ab, ac=ca, 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 >

Subgroups: 516 in 256 conjugacy classes, 96 normal (22 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×16], C22 [×3], C22 [×4], C22 [×20], C2×C4 [×6], C2×C4 [×44], D4 [×4], Q8 [×4], C23, C23 [×4], C23 [×12], C42 [×2], C22⋊C4 [×19], C4⋊C4 [×12], C22×C4, C22×C4 [×12], C22×C4 [×9], C2×D4 [×4], C2×D4 [×2], C2×Q8 [×4], C24 [×2], C2.C42 [×12], C2×C42, C2×C22⋊C4 [×2], C2×C22⋊C4 [×8], C2×C4⋊C4 [×8], C22.D4 [×4], C4.4D4 [×4], C23×C4 [×2], C22×D4, C22×Q8, C23.7Q8 [×2], C23.8Q8 [×2], C23.23D4, C23.63C23 [×2], C23.78C23 [×2], C23.11D4 [×4], C2×C22.D4, C2×C4.4D4, C23.617C24
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×4], C24, C22×D4, C2×C4○D4 [×2], 2+ 1+4 [×2], 2- 1+4 [×2], C233D4, C22.36C24 [×2], D46D4 [×2], C22.45C24, C22.57C24, C23.617C24

Smallest permutation representation of C23.617C24
On 64 points
Generators in S64
(1 23)(2 24)(3 21)(4 22)(5 9)(6 10)(7 11)(8 12)(13 25)(14 26)(15 27)(16 28)(17 62)(18 63)(19 64)(20 61)(29 37)(30 38)(31 39)(32 40)(33 46)(34 47)(35 48)(36 45)(41 57)(42 58)(43 59)(44 60)(49 56)(50 53)(51 54)(52 55)
(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 27)(2 28)(3 25)(4 26)(5 37)(6 38)(7 39)(8 40)(9 29)(10 30)(11 31)(12 32)(13 21)(14 22)(15 23)(16 24)(17 51)(18 52)(19 49)(20 50)(33 58)(34 59)(35 60)(36 57)(41 45)(42 46)(43 47)(44 48)(53 61)(54 62)(55 63)(56 64)
(1 47)(2 35)(3 45)(4 33)(5 53)(6 51)(7 55)(8 49)(9 50)(10 54)(11 52)(12 56)(13 57)(14 42)(15 59)(16 44)(17 38)(18 31)(19 40)(20 29)(21 36)(22 46)(23 34)(24 48)(25 41)(26 58)(27 43)(28 60)(30 62)(32 64)(37 61)(39 63)
(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 5 21 11)(2 38 22 32)(3 7 23 9)(4 40 24 30)(6 14 12 28)(8 16 10 26)(13 31 27 37)(15 29 25 39)(17 48 64 33)(18 41 61 59)(19 46 62 35)(20 43 63 57)(34 52 45 53)(36 50 47 55)(42 54 60 49)(44 56 58 51)
(2 24)(4 22)(6 10)(8 12)(14 26)(16 28)(17 19)(18 61)(20 63)(30 38)(32 40)(33 35)(34 45)(36 47)(41 59)(42 44)(43 57)(46 48)(49 51)(50 55)(52 53)(54 56)(58 60)(62 64)

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

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

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

32 conjugacy classes

class 1 2A···2G2H2I2J2K4A···4N4O···4T
order12···222224···44···4
size11···144444···48···8

32 irreducible representations

dim1111111112244
type+++++++++++-
imageC1C2C2C2C2C2C2C2C2D4C4○D42+ 1+42- 1+4
kernelC23.617C24C23.7Q8C23.8Q8C23.23D4C23.63C23C23.78C23C23.11D4C2×C22.D4C2×C4.4D4C2×D4C2×C4C22C22
# reps1221224114822

Matrix representation of C23.617C24 in GL6(𝔽5)

400000
040000
001000
000100
000010
000001
,
100000
010000
001000
000100
000040
000004
,
100000
010000
004000
000400
000010
000001
,
030000
200000
004000
000400
000011
000004
,
040000
400000
000300
002000
000020
000002
,
300000
030000
000400
004000
000030
000042
,
400000
010000
001000
000100
000010
000034

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

C23.617C24 in GAP, Magma, Sage, TeX

C_2^3._{617}C_2^4
% in TeX

G:=Group("C2^3.617C2^4");
// GroupNames label

G:=SmallGroup(128,1449);
// by ID

G=gap.SmallGroup(128,1449);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,253,232,758,723,268,1571,346]);
// Polycyclic

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