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G = C23.616C24order 128 = 27

333rd central stem extension by C23 of C24

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

Aliases: C24.69C23, C23.616C24, C22.3902+ 1+4, C22.2912- 1+4, C4⋊C4.123D4, C2.50(Q85D4), C2.121(D45D4), C23⋊Q8.20C2, (C22×C4).194C23, (C2×C42).667C22, C22.425(C22×D4), C23.Q8.28C2, C23.11D4.36C2, (C22×Q8).193C22, C23.67C2388C2, C23.83C2385C2, C23.81C2394C2, C24.C22.52C2, C23.63C23143C2, C2.C42.322C22, C2.63(C23.38C23), C2.43(C22.35C24), C2.81(C22.36C24), C2.19(C22.57C24), C2.57(C22.50C24), (C2×C4⋊Q8)⋊22C2, (C2×C4).113(C2×D4), (C2×C4).202(C4○D4), (C2×C4⋊C4).429C22, C22.478(C2×C4○D4), (C2×C422C2).14C2, (C2×C22⋊C4).281C22, SmallGroup(128,1448)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C23.616C24
C1C2C22C23C22×C4C2×C42C23.67C23 — C23.616C24
C1C23 — C23.616C24
C1C23 — C23.616C24
C1C23 — C23.616C24

Generators and relations for C23.616C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=1, d2=e2=g2=a, f2=ba=ab, ac=ca, ede-1=ad=da, geg-1=ae=ea, af=fa, ag=ga, bc=cb, fdf-1=bd=db, be=eb, bf=fb, bg=gb, cd=dc, fef-1=ce=ec, cf=fc, cg=gc, gdg-1=abd, fg=gf >

Subgroups: 420 in 225 conjugacy classes, 96 normal (82 characteristic)
C1, C2 [×7], C2, C4 [×19], C22 [×7], C22 [×7], C2×C4 [×10], C2×C4 [×37], Q8 [×8], C23, C23 [×7], C42 [×5], C22⋊C4 [×12], C4⋊C4 [×4], C4⋊C4 [×15], C22×C4 [×14], C2×Q8 [×10], C24, C2.C42 [×14], C2×C42 [×3], C2×C22⋊C4 [×7], C2×C4⋊C4 [×9], C422C2 [×4], C4⋊Q8 [×4], C22×Q8 [×2], C23.63C23 [×3], C24.C22 [×2], C23.67C23 [×2], C23⋊Q8 [×2], C23.Q8, C23.11D4, C23.81C23, C23.83C23, C2×C422C2, C2×C4⋊Q8, C23.616C24
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×4], C24, C22×D4, C2×C4○D4 [×2], 2+ 1+4, 2- 1+4 [×3], C23.38C23, C22.35C24, C22.36C24, D45D4, Q85D4, C22.50C24, C22.57C24, C23.616C24

Smallest permutation representation of C23.616C24
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 25)(2 26)(3 27)(4 28)(5 19)(6 20)(7 17)(8 18)(9 15)(10 16)(11 13)(12 14)(21 62)(22 63)(23 64)(24 61)(29 35)(30 36)(31 33)(32 34)(37 51)(38 52)(39 49)(40 50)(41 59)(42 60)(43 57)(44 58)(45 56)(46 53)(47 54)(48 55)
(1 31)(2 32)(3 29)(4 30)(5 53)(6 54)(7 55)(8 56)(9 37)(10 38)(11 39)(12 40)(13 49)(14 50)(15 51)(16 52)(17 48)(18 45)(19 46)(20 47)(21 42)(22 43)(23 44)(24 41)(25 33)(26 34)(27 35)(28 36)(57 63)(58 64)(59 61)(60 62)
(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 57 3 59)(2 60 4 58)(5 52 7 50)(6 51 8 49)(9 45 11 47)(10 48 12 46)(13 54 15 56)(14 53 16 55)(17 40 19 38)(18 39 20 37)(21 36 23 34)(22 35 24 33)(25 43 27 41)(26 42 28 44)(29 61 31 63)(30 64 32 62)
(1 28 27 2)(3 26 25 4)(5 45 17 54)(6 53 18 48)(7 47 19 56)(8 55 20 46)(9 10 13 14)(11 12 15 16)(21 43 64 59)(22 58 61 42)(23 41 62 57)(24 60 63 44)(29 34 33 30)(31 36 35 32)(37 38 49 50)(39 40 51 52)
(1 9 3 11)(2 14 4 16)(5 62 7 64)(6 24 8 22)(10 26 12 28)(13 25 15 27)(17 23 19 21)(18 63 20 61)(29 39 31 37)(30 52 32 50)(33 51 35 49)(34 40 36 38)(41 56 43 54)(42 48 44 46)(45 57 47 59)(53 60 55 58)

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

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

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

32 conjugacy classes

class 1 2A···2G2H4A···4R4S···4W
order12···224···44···4
size11···184···48···8

32 irreducible representations

dim111111111112244
type+++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2C2D4C4○D42+ 1+42- 1+4
kernelC23.616C24C23.63C23C24.C22C23.67C23C23⋊Q8C23.Q8C23.11D4C23.81C23C23.83C23C2×C422C2C2×C4⋊Q8C4⋊C4C2×C4C22C22
# reps132221111114813

Matrix representation of C23.616C24 in GL6(𝔽5)

100000
010000
004000
000400
000010
000001
,
100000
010000
001000
000100
000040
000004
,
400000
040000
001000
000100
000010
000001
,
400000
040000
003000
000200
000034
000032
,
420000
010000
000300
003000
000040
000004
,
400000
410000
002000
000200
000034
000002
,
400000
040000
000100
004000
000042
000001

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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,3,0,0,0,0,0,0,2,0,0,0,0,0,0,3,3,0,0,0,0,4,2],[4,0,0,0,0,0,2,1,0,0,0,0,0,0,0,3,0,0,0,0,3,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[4,4,0,0,0,0,0,1,0,0,0,0,0,0,2,0,0,0,0,0,0,2,0,0,0,0,0,0,3,0,0,0,0,0,4,2],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,4,0,0,0,0,1,0,0,0,0,0,0,0,4,0,0,0,0,0,2,1] >;

C23.616C24 in GAP, Magma, Sage, TeX

C_2^3._{616}C_2^4
% in TeX

G:=Group("C2^3.616C2^4");
// GroupNames label

G:=SmallGroup(128,1448);
// by ID

G=gap.SmallGroup(128,1448);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,112,253,120,758,723,268,1571,346,80]);
// Polycyclic

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

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