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G = C23.491C24order 128 = 27

208th central stem extension by C23 of C24

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

Aliases: C24.32C23, C23.491C24, C22.2732+ 1+4, C23.622(C2×D4), (C22×C4).396D4, (C2×C42).75C22, C23.333(C4○D4), C23.23D460C2, C23.10D446C2, C2.18(C233D4), (C22×C4).550C23, (C23×C4).410C22, C22.326(C22×D4), C24.C2292C2, C22.40(C4.4D4), (C22×D4).178C22, C23.83C2349C2, C2.69(C22.19C24), C2.C42.225C22, C2.69(C22.47C24), (C4×C22⋊C4)⋊93C2, (C22×C4⋊C4)⋊28C2, (C2×C4).366(C2×D4), (C2×C4⋊D4).37C2, C2.27(C2×C4.4D4), (C2×C4).403(C4○D4), (C2×C4⋊C4).880C22, C22.367(C2×C4○D4), (C2×C22⋊C4).196C22, SmallGroup(128,1323)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C23.491C24
C1C2C22C23C24C23×C4C22×C4⋊C4 — C23.491C24
C1C23 — C23.491C24
C1C23 — C23.491C24
C1C23 — C23.491C24

Generators and relations for C23.491C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=g2=1, d2=cb=bc, e2=ca=ac, f2=c, ab=ba, ede-1=gdg=ad=da, ae=ea, af=fa, ag=ga, fdf-1=bd=db, be=eb, bf=fb, bg=gb, cd=dc, fef-1=ce=ec, cf=fc, cg=gc, eg=ge, fg=gf >

Subgroups: 612 in 298 conjugacy classes, 104 normal (18 characteristic)
C1, C2 [×3], C2 [×4], C2 [×6], C4 [×16], C22, C22 [×10], C22 [×26], C2×C4 [×8], C2×C4 [×48], D4 [×12], C23, C23 [×6], C23 [×18], C42 [×2], C22⋊C4 [×20], C4⋊C4 [×10], C22×C4 [×2], C22×C4 [×14], C22×C4 [×12], C2×D4 [×16], C24, C24 [×2], C2.C42 [×10], C2×C42 [×2], C2×C22⋊C4 [×12], C2×C4⋊C4, C2×C4⋊C4 [×4], C2×C4⋊C4 [×4], C4⋊D4 [×4], C23×C4 [×3], C22×D4, C22×D4 [×2], C4×C22⋊C4, C23.23D4 [×4], C24.C22 [×4], C23.10D4 [×2], C23.83C23 [×2], C22×C4⋊C4, C2×C4⋊D4, C23.491C24
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×8], C24, C4.4D4 [×4], C22×D4, C2×C4○D4 [×4], 2+ 1+4 [×2], C22.19C24, C2×C4.4D4, C233D4, C22.47C24 [×4], C23.491C24

Smallest permutation representation of C23.491C24
On 64 points
Generators in S64
(1 10)(2 11)(3 12)(4 9)(5 28)(6 25)(7 26)(8 27)(13 24)(14 21)(15 22)(16 23)(17 64)(18 61)(19 62)(20 63)(29 38)(30 39)(31 40)(32 37)(33 51)(34 52)(35 49)(36 50)(41 56)(42 53)(43 54)(44 55)(45 60)(46 57)(47 58)(48 59)
(1 14)(2 15)(3 16)(4 13)(5 32)(6 29)(7 30)(8 31)(9 24)(10 21)(11 22)(12 23)(17 49)(18 50)(19 51)(20 52)(25 38)(26 39)(27 40)(28 37)(33 62)(34 63)(35 64)(36 61)(41 45)(42 46)(43 47)(44 48)(53 57)(54 58)(55 59)(56 60)
(1 16)(2 13)(3 14)(4 15)(5 30)(6 31)(7 32)(8 29)(9 22)(10 23)(11 24)(12 21)(17 51)(18 52)(19 49)(20 50)(25 40)(26 37)(27 38)(28 39)(33 64)(34 61)(35 62)(36 63)(41 47)(42 48)(43 45)(44 46)(53 59)(54 60)(55 57)(56 58)
(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 55 23 46)(2 41 24 58)(3 53 21 48)(4 43 22 60)(5 19 39 35)(6 63 40 50)(7 17 37 33)(8 61 38 52)(9 54 15 45)(10 44 16 57)(11 56 13 47)(12 42 14 59)(18 29 34 27)(20 31 36 25)(26 64 32 51)(28 62 30 49)
(1 13 16 2)(3 15 14 4)(5 27 30 38)(6 37 31 26)(7 25 32 40)(8 39 29 28)(9 12 22 21)(10 24 23 11)(17 50 51 20)(18 19 52 49)(33 63 64 36)(34 35 61 62)(41 57 47 55)(42 54 48 60)(43 59 45 53)(44 56 46 58)
(1 38)(2 30)(3 40)(4 32)(5 13)(6 21)(7 15)(8 23)(9 37)(10 29)(11 39)(12 31)(14 25)(16 27)(17 45)(18 57)(19 47)(20 59)(22 26)(24 28)(33 54)(34 44)(35 56)(36 42)(41 49)(43 51)(46 61)(48 63)(50 53)(52 55)(58 62)(60 64)

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

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

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

38 conjugacy classes

class 1 2A···2G2H2I2J2K2L2M4A4B4C4D4E···4V4W4X
order12···222222244444···444
size11···122228822224···488

38 irreducible representations

dim111111112224
type++++++++++
imageC1C2C2C2C2C2C2C2D4C4○D4C4○D42+ 1+4
kernelC23.491C24C4×C22⋊C4C23.23D4C24.C22C23.10D4C23.83C23C22×C4⋊C4C2×C4⋊D4C22×C4C2×C4C23C22
# reps114422114882

Matrix representation of C23.491C24 in GL6(𝔽5)

100000
010000
004000
000400
000010
000001
,
100000
010000
004000
000400
000040
000004
,
400000
040000
001000
000100
000010
000001
,
200000
020000
000100
004000
000033
000002
,
430000
110000
000300
003000
000040
000004
,
300000
220000
000100
001000
000022
000013
,
400000
040000
000100
001000
000040
000004

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

C23.491C24 in GAP, Magma, Sage, TeX

C_2^3._{491}C_2^4
% in TeX

G:=Group("C2^3.491C2^4");
// GroupNames label

G:=SmallGroup(128,1323);
// by ID

G=gap.SmallGroup(128,1323);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,672,253,568,758,723,352,675]);
// Polycyclic

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

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