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G = C23.500C24order 128 = 27

217th central stem extension by C23 of C24

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

Aliases: C23.500C24, C24.351C23, C22.2812+ 1+4, (C22×C4)⋊13D4, C23.185(C2×D4), C23.61(C4○D4), C232D4.13C2, C23.8Q877C2, C23.23D462C2, C23.34D439C2, C23.11D452C2, C23.10D448C2, C2.19(C233D4), (C22×C4).122C23, (C23×C4).131C22, (C2×C42).587C22, C22.330(C22×D4), C24.C2298C2, (C22×D4).537C22, C2.73(C22.19C24), C23.63C23104C2, C2.66(C22.45C24), C2.C42.230C22, C2.76(C22.47C24), C2.30(C22.53C24), (C2×C4×D4)⋊49C2, (C2×C4).1198(C2×D4), (C2×C4).160(C4○D4), (C2×C4⋊C4).340C22, C22.376(C2×C4○D4), (C2×C22⋊C4).201C22, SmallGroup(128,1332)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C23.500C24
C1C2C22C23C22×C4C23×C4C2×C4×D4 — C23.500C24
C1C23 — C23.500C24
C1C23 — C23.500C24
C1C23 — C23.500C24

Generators and relations for C23.500C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=e2=f2=1, g2=db=bd, eae=ab=ba, ac=ca, faf=ad=da, ag=ga, bc=cb, geg-1=be=eb, bf=fb, bg=gb, cd=dc, fef=ce=ec, cf=fc, cg=gc, de=ed, df=fd, dg=gd, fg=gf >

Subgroups: 612 in 302 conjugacy classes, 100 normal (22 characteristic)
C1, C2 [×3], C2 [×4], C2 [×6], C4 [×16], C22 [×3], C22 [×4], C22 [×30], C2×C4 [×8], C2×C4 [×44], D4 [×20], C23, C23 [×6], C23 [×18], C42 [×4], C22⋊C4 [×20], C4⋊C4 [×9], C22×C4 [×2], C22×C4 [×14], C22×C4 [×10], C2×D4 [×17], C24, C24 [×2], C2.C42 [×2], C2.C42 [×8], C2×C42 [×2], C2×C22⋊C4 [×2], C2×C22⋊C4 [×10], C2×C4⋊C4, C2×C4⋊C4 [×4], C4×D4 [×8], C23×C4, C23×C4 [×2], C22×D4, C22×D4 [×2], C23.34D4, C23.8Q8 [×2], C23.23D4 [×2], C23.63C23 [×2], C24.C22 [×2], C232D4, C23.10D4 [×2], C23.11D4, C2×C4×D4 [×2], C23.500C24
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×8], C24, C22×D4, C2×C4○D4 [×4], 2+ 1+4 [×2], C22.19C24 [×2], C233D4, C22.45C24, C22.47C24 [×2], C22.53C24, C23.500C24

Smallest permutation representation of C23.500C24
On 64 points
Generators in S64
(1 44)(2 41)(3 42)(4 43)(5 45)(6 46)(7 47)(8 48)(9 52)(10 49)(11 50)(12 51)(13 54)(14 55)(15 56)(16 53)(17 37)(18 38)(19 39)(20 40)(21 33)(22 34)(23 35)(24 36)(25 31)(26 32)(27 29)(28 30)(57 64)(58 61)(59 62)(60 63)
(1 28)(2 25)(3 26)(4 27)(5 61)(6 62)(7 63)(8 64)(9 14)(10 15)(11 16)(12 13)(17 22)(18 23)(19 24)(20 21)(29 43)(30 44)(31 41)(32 42)(33 40)(34 37)(35 38)(36 39)(45 58)(46 59)(47 60)(48 57)(49 56)(50 53)(51 54)(52 55)
(1 23)(2 24)(3 21)(4 22)(5 12)(6 9)(7 10)(8 11)(13 61)(14 62)(15 63)(16 64)(17 27)(18 28)(19 25)(20 26)(29 37)(30 38)(31 39)(32 40)(33 42)(34 43)(35 44)(36 41)(45 51)(46 52)(47 49)(48 50)(53 57)(54 58)(55 59)(56 60)
(1 26)(2 27)(3 28)(4 25)(5 63)(6 64)(7 61)(8 62)(9 16)(10 13)(11 14)(12 15)(17 24)(18 21)(19 22)(20 23)(29 41)(30 42)(31 43)(32 44)(33 38)(34 39)(35 40)(36 37)(45 60)(46 57)(47 58)(48 59)(49 54)(50 55)(51 56)(52 53)
(1 58)(2 46)(3 60)(4 48)(5 44)(6 31)(7 42)(8 29)(9 39)(10 33)(11 37)(12 35)(13 38)(14 36)(15 40)(16 34)(17 53)(18 51)(19 55)(20 49)(21 56)(22 50)(23 54)(24 52)(25 59)(26 47)(27 57)(28 45)(30 61)(32 63)(41 62)(43 64)
(5 15)(6 16)(7 13)(8 14)(9 64)(10 61)(11 62)(12 63)(29 41)(30 42)(31 43)(32 44)(33 38)(34 39)(35 40)(36 37)(45 51)(46 52)(47 49)(48 50)(53 57)(54 58)(55 59)(56 60)
(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)

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

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

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

38 conjugacy classes

class 1 2A···2G2H···2M4A···4H4I···4T4U4V4W4X
order12···22···24···44···44444
size11···14···42···24···48888

38 irreducible representations

dim11111111112224
type++++++++++++
imageC1C2C2C2C2C2C2C2C2C2D4C4○D4C4○D42+ 1+4
kernelC23.500C24C23.34D4C23.8Q8C23.23D4C23.63C23C24.C22C232D4C23.10D4C23.11D4C2×C4×D4C22×C4C2×C4C23C22
# reps11222212124882

Matrix representation of C23.500C24 in GL6(𝔽5)

020000
300000
000100
001000
000043
000001
,
400000
040000
001000
000100
000010
000001
,
100000
010000
004000
000400
000010
000001
,
100000
010000
004000
000400
000040
000004
,
010000
100000
000400
004000
000010
000001
,
100000
010000
001000
000400
000010
000044
,
010000
400000
002000
000200
000030
000003

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

C23.500C24 in GAP, Magma, Sage, TeX

C_2^3._{500}C_2^4
% in TeX

G:=Group("C2^3.500C2^4");
// GroupNames label

G:=SmallGroup(128,1332);
// by ID

G=gap.SmallGroup(128,1332);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,253,758,723,675,304]);
// Polycyclic

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

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