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G = C23.391C24order 128 = 27

108th central stem extension by C23 of C24

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

Aliases: C23.391C24, C24.302C23, C22.1912+ 1+4, C22.1432- 1+4, C2.27D42, C4⋊C4.229D4, C22⋊C425D4, C42(C4.4D4), C23.42(C2×D4), C23⋊Q816C2, C2.42(D46D4), C2.34(Q85D4), C23.7Q857C2, C23.10D436C2, (C22×C4).826C23, (C23×C4).376C22, (C2×C42).519C22, C22.271(C22×D4), C24.C2265C2, C24.3C2248C2, (C22×D4).147C22, (C22×Q8).116C22, C2.C42.540C22, C2.21(C22.26C24), C2.13(C22.49C24), C2.28(C22.36C24), (C4×C4⋊C4)⋊69C2, (C2×C4⋊Q8)⋊10C2, (C2×C4).62(C2×D4), (C4×C22⋊C4)⋊74C2, (C2×C4.4D4)⋊13C2, (C2×C4⋊D4).32C2, C2.17(C2×C4.4D4), (C2×C4).122(C4○D4), (C2×C4⋊C4).261C22, C22.268(C2×C4○D4), (C2×C22⋊C4).156C22, SmallGroup(128,1223)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C23.391C24
C1C2C22C23C22×C4C23×C4C4×C22⋊C4 — C23.391C24
C1C23 — C23.391C24
C1C23 — C23.391C24
C1C23 — C23.391C24

Generators and relations for C23.391C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=f2=1, d2=c, e2=ba=ab, g2=a, ac=ca, ede-1=gdg-1=ad=da, ae=ea, af=fa, ag=ga, bc=cb, fdf=bd=db, be=eb, bf=fb, bg=gb, cd=dc, fef=ce=ec, cf=fc, cg=gc, eg=ge, fg=gf >

Subgroups: 644 in 316 conjugacy classes, 112 normal (42 characteristic)
C1, C2 [×7], C2 [×4], C4 [×4], C4 [×16], C22 [×7], C22 [×24], C2×C4 [×16], C2×C4 [×32], D4 [×12], Q8 [×8], C23, C23 [×2], C23 [×20], C42 [×8], C22⋊C4 [×4], C22⋊C4 [×24], C4⋊C4 [×4], C4⋊C4 [×8], C22×C4 [×6], C22×C4 [×6], C22×C4 [×6], C2×D4 [×16], C2×Q8 [×10], C24, C24 [×2], C2.C42 [×2], C2.C42 [×4], C2×C42 [×2], C2×C42 [×2], C2×C22⋊C4 [×2], C2×C22⋊C4 [×12], C2×C4⋊C4 [×3], C2×C4⋊C4 [×2], C4⋊D4 [×4], C4.4D4 [×8], C4⋊Q8 [×4], C23×C4, C22×D4, C22×D4 [×2], C22×Q8 [×2], C4×C22⋊C4, C4×C4⋊C4, C23.7Q8, C24.C22 [×2], C24.3C22 [×2], C23⋊Q8 [×2], C23.10D4 [×2], C2×C4⋊D4, C2×C4.4D4 [×2], C2×C4⋊Q8, C23.391C24
Quotients: C1, C2 [×15], C22 [×35], D4 [×8], C23 [×15], C2×D4 [×12], C4○D4 [×6], C24, C4.4D4 [×4], C22×D4 [×2], C2×C4○D4 [×3], 2+ 1+4, 2- 1+4, C2×C4.4D4, C22.26C24, C22.36C24, D42, D46D4, Q85D4, C22.49C24, C23.391C24

Smallest permutation representation of C23.391C24
On 64 points
Generators in S64
(1 21)(2 22)(3 23)(4 24)(5 46)(6 47)(7 48)(8 45)(9 58)(10 59)(11 60)(12 57)(13 54)(14 55)(15 56)(16 53)(17 36)(18 33)(19 34)(20 35)(25 43)(26 44)(27 41)(28 42)(29 39)(30 40)(31 37)(32 38)(49 64)(50 61)(51 62)(52 63)
(1 57)(2 58)(3 59)(4 60)(5 31)(6 32)(7 29)(8 30)(9 22)(10 23)(11 24)(12 21)(13 41)(14 42)(15 43)(16 44)(17 51)(18 52)(19 49)(20 50)(25 56)(26 53)(27 54)(28 55)(33 63)(34 64)(35 61)(36 62)(37 46)(38 47)(39 48)(40 45)
(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 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 12 37)(2 47 9 32)(3 7 10 39)(4 45 11 30)(6 58 38 22)(8 60 40 24)(13 61 27 20)(14 51 28 36)(15 63 25 18)(16 49 26 34)(17 55 62 42)(19 53 64 44)(21 46 57 31)(23 48 59 29)(33 56 52 43)(35 54 50 41)
(1 47)(2 39)(3 45)(4 37)(5 11)(6 21)(7 9)(8 23)(10 30)(12 32)(13 49)(14 20)(15 51)(16 18)(17 43)(19 41)(22 29)(24 31)(25 36)(26 63)(27 34)(28 61)(33 53)(35 55)(38 57)(40 59)(42 50)(44 52)(46 60)(48 58)(54 64)(56 62)
(1 35 21 20)(2 17 22 36)(3 33 23 18)(4 19 24 34)(5 54 46 13)(6 14 47 55)(7 56 48 15)(8 16 45 53)(9 62 58 51)(10 52 59 63)(11 64 60 49)(12 50 57 61)(25 39 43 29)(26 30 44 40)(27 37 41 31)(28 32 42 38)

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

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

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

38 conjugacy classes

class 1 2A···2G2H2I2J2K4A···4H4I···4X4Y4Z
order12···222224···44···444
size11···144882···24···488

38 irreducible representations

dim1111111111122244
type++++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2C2D4D4C4○D42+ 1+42- 1+4
kernelC23.391C24C4×C22⋊C4C4×C4⋊C4C23.7Q8C24.C22C24.3C22C23⋊Q8C23.10D4C2×C4⋊D4C2×C4.4D4C2×C4⋊Q8C22⋊C4C4⋊C4C2×C4C22C22
# reps11112222121441211

Matrix representation of C23.391C24 in GL6(𝔽5)

100000
010000
004000
000400
000010
000001
,
400000
040000
004000
000400
000010
000001
,
400000
040000
001000
000100
000040
000004
,
330000
020000
004000
000100
000020
000002
,
330000
020000
000200
003000
000034
000032
,
400000
210000
000200
003000
000042
000001
,
400000
040000
000100
004000
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],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[3,0,0,0,0,0,3,2,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,2,0,0,0,0,0,0,2],[3,0,0,0,0,0,3,2,0,0,0,0,0,0,0,3,0,0,0,0,2,0,0,0,0,0,0,0,3,3,0,0,0,0,4,2],[4,2,0,0,0,0,0,1,0,0,0,0,0,0,0,3,0,0,0,0,2,0,0,0,0,0,0,0,4,0,0,0,0,0,2,1],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,4,0,0,0,0,1,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4] >;

C23.391C24 in GAP, Magma, Sage, TeX

C_2^3._{391}C_2^4
% in TeX

G:=Group("C2^3.391C2^4");
// GroupNames label

G:=SmallGroup(128,1223);
// by ID

G=gap.SmallGroup(128,1223);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,344,758,723,675,80]);
// Polycyclic

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

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