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G = C23.322C24order 128 = 27

39th central stem extension by C23 of C24

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

Aliases: C23.322C24, C24.256C23, C22.972- 1+4, C22.1352+ 1+4, C2.11D42, C22⋊C421D4, (C2×D4).283D4, C23.37(C2×D4), C429C415C2, C2.15(D46D4), C23.11D48C2, C41(C22.D4), (C23×C4).74C22, C23.8Q832C2, C23.7Q841C2, C23.10D414C2, (C2×C42).470C22, (C22×C4).505C23, C22.202(C22×D4), C24.3C2231C2, (C22×D4).123C22, C23.65C2340C2, C2.C42.85C22, C2.12(C22.26C24), C2.15(C22.47C24), C2.12(C22.36C24), (C2×C4×D4)⋊25C2, (C2×C4).47(C2×D4), (C4×C22⋊C4)⋊51C2, (C2×C4).96(C4○D4), (C2×C4⋊D4).23C2, (C2×C4⋊C4).210C22, C22.201(C2×C4○D4), (C2×C22.D4)⋊7C2, C2.19(C2×C22.D4), (C2×C22⋊C4).114C22, SmallGroup(128,1154)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C23.322C24
C1C2C22C23C22×C4C23×C4C4×C22⋊C4 — C23.322C24
C1C23 — C23.322C24
C1C23 — C23.322C24
C1C23 — C23.322C24

Generators and relations for C23.322C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=f2=1, e2=cb=bc, g2=b, eae-1=gag-1=ab=ba, faf=ac=ca, ad=da, bd=db, be=eb, bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, cg=gc, fef=de=ed, df=fd, dg=gd, eg=ge, fg=gf >

Subgroups: 644 in 326 conjugacy classes, 112 normal (42 characteristic)
C1, C2 [×7], C2 [×6], C4 [×4], C4 [×14], C22 [×7], C22 [×30], C2×C4 [×12], C2×C4 [×42], D4 [×16], C23, C23 [×6], C23 [×18], C42 [×4], C22⋊C4 [×4], C22⋊C4 [×22], C4⋊C4 [×16], C22×C4 [×6], C22×C4 [×6], C22×C4 [×16], C2×D4 [×4], C2×D4 [×14], C24, C24 [×2], C2.C42 [×2], C2.C42 [×4], C2×C42 [×2], C2×C22⋊C4 [×2], C2×C22⋊C4 [×10], C2×C4⋊C4 [×3], C2×C4⋊C4 [×6], C4×D4 [×4], C4⋊D4 [×4], C22.D4 [×8], C23×C4, C23×C4 [×2], C22×D4, C22×D4 [×2], C4×C22⋊C4, C23.7Q8, C429C4, C23.8Q8 [×2], C23.65C23, C24.3C22, C23.10D4 [×2], C23.11D4 [×2], C2×C4×D4, C2×C4⋊D4, C2×C22.D4 [×2], C23.322C24
Quotients: C1, C2 [×15], C22 [×35], D4 [×8], C23 [×15], C2×D4 [×12], C4○D4 [×6], C24, C22.D4 [×4], C22×D4 [×2], C2×C4○D4 [×3], 2+ 1+4, 2- 1+4, C2×C22.D4, C22.26C24, C22.36C24, D42, D46D4 [×2], C22.47C24, C23.322C24

Smallest permutation representation of C23.322C24
On 64 points
Generators in S64
(1 47)(2 16)(3 45)(4 14)(5 64)(6 34)(7 62)(8 36)(9 15)(10 48)(11 13)(12 46)(17 43)(18 22)(19 41)(20 24)(21 49)(23 51)(25 31)(26 58)(27 29)(28 60)(30 54)(32 56)(33 37)(35 39)(38 61)(40 63)(42 52)(44 50)(53 59)(55 57)
(1 9)(2 10)(3 11)(4 12)(5 37)(6 38)(7 39)(8 40)(13 45)(14 46)(15 47)(16 48)(17 49)(18 50)(19 51)(20 52)(21 43)(22 44)(23 41)(24 42)(25 55)(26 56)(27 53)(28 54)(29 59)(30 60)(31 57)(32 58)(33 64)(34 61)(35 62)(36 63)
(1 11)(2 12)(3 9)(4 10)(5 39)(6 40)(7 37)(8 38)(13 47)(14 48)(15 45)(16 46)(17 51)(18 52)(19 49)(20 50)(21 41)(22 42)(23 43)(24 44)(25 53)(26 54)(27 55)(28 56)(29 57)(30 58)(31 59)(32 60)(33 62)(34 63)(35 64)(36 61)
(1 25)(2 26)(3 27)(4 28)(5 22)(6 23)(7 24)(8 21)(9 55)(10 56)(11 53)(12 54)(13 59)(14 60)(15 57)(16 58)(17 63)(18 64)(19 61)(20 62)(29 45)(30 46)(31 47)(32 48)(33 50)(34 51)(35 52)(36 49)(37 44)(38 41)(39 42)(40 43)
(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 49)(2 33)(3 51)(4 35)(5 14)(6 57)(7 16)(8 59)(9 17)(10 64)(11 19)(12 62)(13 21)(15 23)(18 56)(20 54)(22 60)(24 58)(25 36)(26 50)(27 34)(28 52)(29 40)(30 44)(31 38)(32 42)(37 46)(39 48)(41 47)(43 45)(53 61)(55 63)
(1 15 9 47)(2 16 10 48)(3 13 11 45)(4 14 12 46)(5 62 37 35)(6 63 38 36)(7 64 39 33)(8 61 40 34)(17 41 49 23)(18 42 50 24)(19 43 51 21)(20 44 52 22)(25 57 55 31)(26 58 56 32)(27 59 53 29)(28 60 54 30)

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

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

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

38 conjugacy classes

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

38 irreducible representations

dim11111111111122244
type+++++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2C2C2D4D4C4○D42+ 1+42- 1+4
kernelC23.322C24C4×C22⋊C4C23.7Q8C429C4C23.8Q8C23.65C23C24.3C22C23.10D4C23.11D4C2×C4×D4C2×C4⋊D4C2×C22.D4C22⋊C4C2×D4C2×C4C22C22
# reps111121122112441211

Matrix representation of C23.322C24 in GL6(𝔽5)

100000
010000
001200
000400
000040
000031
,
100000
010000
004000
000400
000010
000001
,
100000
010000
001000
000100
000040
000004
,
400000
040000
001000
000100
000040
000004
,
100000
340000
001200
004400
000030
000012
,
110000
040000
004000
000400
000032
000012
,
100000
010000
001200
004400
000040
000004

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

C23.322C24 in GAP, Magma, Sage, TeX

C_2^3._{322}C_2^4
% in TeX

G:=Group("C2^3.322C2^4");
// GroupNames label

G:=SmallGroup(128,1154);
// by ID

G=gap.SmallGroup(128,1154);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,253,344,758,723,675,80]);
// Polycyclic

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

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