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G = C24.327C23order 128 = 27

167th non-split extension by C24 of C23 acting via C23/C2=C22

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

Aliases: C24.327C23, C23.454C24, C22.2392+ 1+4, C4⋊C428D4, C2.74(D45D4), C2.26(Q86D4), C232D4.10C2, C23.Q829C2, C23.153(C4○D4), C23.10D443C2, C23.23D456C2, (C2×C42).559C22, (C23×C4).114C22, (C22×C4).540C23, C22.305(C22×D4), C24.C2281C2, C24.3C2257C2, (C22×D4).169C22, C23.81C2339C2, C2.22(C22.32C24), C2.47(C22.45C24), C2.C42.191C22, C2.34(C22.26C24), C2.78(C23.36C23), C2.54(C22.47C24), (C4×C4⋊C4)⋊92C2, (C2×C4).79(C2×D4), (C4×C22⋊C4)⋊85C2, (C2×C422C2)⋊10C2, (C2×C4).387(C4○D4), (C2×C4⋊C4).872C22, C22.330(C2×C4○D4), (C2×C22.D4)⋊21C2, (C2×C22⋊C4).181C22, SmallGroup(128,1286)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C24.327C23
C1C2C22C23C22×C4C23×C4C4×C22⋊C4 — C24.327C23
C1C23 — C24.327C23
C1C23 — C24.327C23
C1C23 — C24.327C23

Generators and relations for C24.327C23
 G = < a,b,c,d,e,f,g | a2=b2=c2=f2=1, d2=a, e2=ca=ac, g2=b, ab=ba, 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: 580 in 280 conjugacy classes, 100 normal (82 characteristic)
C1, C2, C2, C4, C22, C22, C2×C4, C2×C4, D4, C23, C23, C23, C42, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C24, C2.C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C22.D4, C422C2, C23×C4, C22×D4, C4×C22⋊C4, C4×C4⋊C4, C23.23D4, C24.C22, C24.3C22, C232D4, C23.10D4, C23.Q8, C23.81C23, C2×C22.D4, C2×C422C2, C24.327C23
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C22×D4, C2×C4○D4, 2+ 1+4, C23.36C23, C22.26C24, C22.32C24, D45D4, Q86D4, C22.45C24, C22.47C24, C24.327C23

Smallest permutation representation of C24.327C23
On 64 points
Generators in S64
(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 17)(2 18)(3 19)(4 20)(5 37)(6 38)(7 39)(8 40)(9 24)(10 21)(11 22)(12 23)(13 52)(14 49)(15 50)(16 51)(25 54)(26 55)(27 56)(28 53)(29 60)(30 57)(31 58)(32 59)(33 64)(34 61)(35 62)(36 63)(41 45)(42 46)(43 47)(44 48)
(1 47)(2 48)(3 45)(4 46)(5 35)(6 36)(7 33)(8 34)(9 50)(10 51)(11 52)(12 49)(13 22)(14 23)(15 24)(16 21)(17 43)(18 44)(19 41)(20 42)(25 58)(26 59)(27 60)(28 57)(29 56)(30 53)(31 54)(32 55)(37 62)(38 63)(39 64)(40 61)
(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 45 30)(2 54 46 29)(3 53 47 32)(4 56 48 31)(5 52 33 9)(6 51 34 12)(7 50 35 11)(8 49 36 10)(13 64 24 37)(14 63 21 40)(15 62 22 39)(16 61 23 38)(17 26 41 57)(18 25 42 60)(19 28 43 59)(20 27 44 58)
(1 24)(2 10)(3 22)(4 12)(5 59)(6 29)(7 57)(8 31)(9 17)(11 19)(13 45)(14 42)(15 47)(16 44)(18 21)(20 23)(25 61)(26 35)(27 63)(28 33)(30 39)(32 37)(34 54)(36 56)(38 60)(40 58)(41 52)(43 50)(46 49)(48 51)(53 64)(55 62)
(1 21 17 10)(2 24 18 9)(3 23 19 12)(4 22 20 11)(5 54 37 25)(6 53 38 28)(7 56 39 27)(8 55 40 26)(13 42 52 46)(14 41 49 45)(15 44 50 48)(16 43 51 47)(29 64 60 33)(30 63 57 36)(31 62 58 35)(32 61 59 34)

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

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

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

38 conjugacy classes

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

38 irreducible representations

dim1111111111112224
type++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2C2D4C4○D4C4○D42+ 1+4
kernelC24.327C23C4×C22⋊C4C4×C4⋊C4C23.23D4C24.C22C24.3C22C232D4C23.10D4C23.Q8C23.81C23C2×C22.D4C2×C422C2C4⋊C4C2×C4C23C22
# reps11113311111141242

Matrix representation of C24.327C23 in GL6(𝔽5)

100000
010000
001000
000100
000040
000004
,
400000
040000
001000
000100
000040
000004
,
100000
010000
004000
000400
000010
000001
,
020000
300000
004000
000400
000030
000022
,
100000
010000
000100
004000
000031
000002
,
010000
100000
004000
000100
000012
000004
,
200000
020000
001000
000100
000024
000003

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

C24.327C23 in GAP, Magma, Sage, TeX

C_2^4._{327}C_2^3
% in TeX

G:=Group("C2^4.327C2^3");
// GroupNames label

G:=SmallGroup(128,1286);
// by ID

G=gap.SmallGroup(128,1286);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,112,253,568,758,723,675,136]);
// Polycyclic

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