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G = C23.559C24order 128 = 27

276th central stem extension by C23 of C24

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

Aliases: C23.559C24, C24.594C23, C22.2492- 1+4, C22.3332+ 1+4, (C22×C4)⋊14Q8, C22.3(C4⋊Q8), (C22×C4).408D4, C23.372(C2×D4), C23.102(C2×Q8), C2.32(C233D4), (C23×C4).436C22, (C22×C4).858C23, C2.10(C232Q8), C22.371(C22×D4), C23.7Q8.61C2, C22.137(C22×Q8), (C22×Q8).166C22, C23.78C2332C2, C23.81C2371C2, C2.C42.273C22, C2.33(C22.31C24), C2.24(C23.41C23), C2.19(C2×C4⋊Q8), (C2×C4).405(C2×D4), (C2×C4).134(C2×Q8), (C2×C22⋊Q8).40C2, (C2×C4⋊C4).382C22, (C2×C22⋊C4).239C22, (C2×C2.C42).31C2, SmallGroup(128,1391)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C23.559C24
C1C2C22C23C24C2×C22⋊C4C23.7Q8 — C23.559C24
C1C23 — C23.559C24
C1C23 — C23.559C24
C1C23 — C23.559C24

Generators and relations for C23.559C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=g2=1, d2=f2=b, e2=c, ab=ba, ac=ca, ede-1=ad=da, ae=ea, gfg=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, dg=gd, eg=ge >

Subgroups: 500 in 262 conjugacy classes, 116 normal (12 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×20], C22 [×3], C22 [×8], C22 [×12], C2×C4 [×12], C2×C4 [×48], Q8 [×8], C23, C23 [×6], C23 [×4], C22⋊C4 [×8], C4⋊C4 [×20], C22×C4 [×26], C22×C4 [×6], C2×Q8 [×8], C24, C2.C42 [×12], C2×C22⋊C4 [×4], C2×C4⋊C4 [×14], C22⋊Q8 [×8], C23×C4, C23×C4 [×2], C22×Q8 [×2], C2×C2.C42, C23.7Q8 [×4], C23.78C23 [×4], C23.81C23 [×4], C2×C22⋊Q8 [×2], C23.559C24
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], Q8 [×8], C23 [×15], C2×D4 [×6], C2×Q8 [×12], C24, C4⋊Q8 [×4], C22×D4, C22×Q8 [×2], 2+ 1+4 [×3], 2- 1+4, C2×C4⋊Q8, C233D4, C22.31C24, C232Q8 [×2], C23.41C23 [×2], C23.559C24

Smallest permutation representation of C23.559C24
On 64 points
Generators in S64
(1 39)(2 40)(3 37)(4 38)(5 17)(6 18)(7 19)(8 20)(9 41)(10 42)(11 43)(12 44)(13 45)(14 46)(15 47)(16 48)(21 49)(22 50)(23 51)(24 52)(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 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 51)(2 52)(3 49)(4 50)(5 36)(6 33)(7 34)(8 35)(9 53)(10 54)(11 55)(12 56)(13 57)(14 58)(15 59)(16 60)(17 61)(18 62)(19 63)(20 64)(21 37)(22 38)(23 39)(24 40)(25 41)(26 42)(27 43)(28 44)(29 45)(30 46)(31 47)(32 48)
(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 59 51 15)(2 32 52 48)(3 57 49 13)(4 30 50 46)(5 41 36 25)(6 10 33 54)(7 43 34 27)(8 12 35 56)(9 61 53 17)(11 63 55 19)(14 38 58 22)(16 40 60 24)(18 42 62 26)(20 44 64 28)(21 45 37 29)(23 47 39 31)
(1 9 3 11)(2 12 4 10)(5 29 7 31)(6 32 8 30)(13 63 15 61)(14 62 16 64)(17 57 19 59)(18 60 20 58)(21 27 23 25)(22 26 24 28)(33 48 35 46)(34 47 36 45)(37 43 39 41)(38 42 40 44)(49 55 51 53)(50 54 52 56)
(1 51)(2 52)(3 49)(4 50)(5 61)(6 62)(7 63)(8 64)(9 25)(10 26)(11 27)(12 28)(13 57)(14 58)(15 59)(16 60)(17 36)(18 33)(19 34)(20 35)(21 37)(22 38)(23 39)(24 40)(29 45)(30 46)(31 47)(32 48)(41 53)(42 54)(43 55)(44 56)

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

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

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

32 conjugacy classes

class 1 2A···2G2H2I2J2K4A···4L4M···4T
order12···222224···44···4
size11···122224···48···8

32 irreducible representations

dim1111112244
type+++++++-+-
imageC1C2C2C2C2C2D4Q82+ 1+42- 1+4
kernelC23.559C24C2×C2.C42C23.7Q8C23.78C23C23.81C23C2×C22⋊Q8C22×C4C22×C4C22C22
# reps1144424831

Matrix representation of C23.559C24 in GL8(𝔽5)

10000000
01000000
00100000
00010000
00004000
00000400
00000040
00000004
,
40000000
04000000
00400000
00040000
00004000
00000400
00000040
00000004
,
10000000
01000000
00400000
00040000
00004000
00000400
00000040
00000004
,
20000000
03000000
00040000
00100000
00004020
00000102
00004010
00000404
,
10000000
01000000
00010000
00400000
00002010
00000301
00000030
00000002
,
04000000
10000000
00300000
00020000
00000100
00004000
00000004
00000010
,
40000000
04000000
00100000
00010000
00004000
00000100
00000040
00000001

G:=sub<GL(8,GF(5))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[2,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,0,4,0,0,0,0,0,0,1,0,4,0,0,0,0,2,0,1,0,0,0,0,0,0,2,0,4],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,1,0,3,0,0,0,0,0,0,1,0,2],[0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0],[4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1] >;

C23.559C24 in GAP, Magma, Sage, TeX

C_2^3._{559}C_2^4
% in TeX

G:=Group("C2^3.559C2^4");
// GroupNames label

G:=SmallGroup(128,1391);
// by ID

G=gap.SmallGroup(128,1391);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,224,253,456,758,723,184,185]);
// Polycyclic

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

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