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G = C23.201C24  order 128 = 27

54th central extension by C23 of C24

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

Aliases: C23.201C24, C24.193C23, C22.402+ 1+4, C22.252- 1+4, (C4×D4)⋊20C4, C42⋊4C4⋊10C2, C42.178(C2×C4), C23.83(C22×C4), C22.92(C23×C4), C23.7Q8⋊15C2, C4.14(C42⋊C2), C23.34D4⋊11C2, (C22×C4).466C23, C24.C22⋊3C2, (C2×C42).409C22, (C23×C4).290C22, (C22×D4).476C22, C23.65C23⋊11C2, C2.12(C22.11C24), C2.C42.38C22, C24.3C22.24C2, C2.2(C22.34C24), C2.2(C22.36C24), C2.9(C23.33C23), (C4×C4⋊C4)⋊24C2, (C2×C4×D4).30C2, C4⋊C4.204(C2×C4), (C2×D4).212(C2×C4), C22⋊C4.58(C2×C4), C22.86(C2×C4○D4), (C2×C4).644(C4○D4), (C2×C4⋊C4).175C22, (C22×C4).302(C2×C4), (C2×C4).224(C22×C4), C2.23(C2×C42⋊C2), (C2×C22⋊C4).26C22, SmallGroup(128,1051)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series
C1 — C22 — C23.201C24
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C23×C4 — C2×C4×D4 — C23.201C24
Lower central
C1 — C22 — C23.201C24
Upper central
C1 — C23 — C23.201C24
Jennings
C1 — C23 — C23.201C24

Generators and relations for C23.201C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=f2=1, d2=c, e2=a, g2=b, ab=ba, ac=ca, ede-1=gdg-1=ad=da, fef=ae=ea, af=fa, ag=ga, bc=cb, fdf=bd=db, be=eb, bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, cg=gc, eg=ge, fg=gf >

Subgroups: 492 in 270 conjugacy classes, 140 normal (20 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, D4, C23, C23, C23, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C24, C2.C42, C2.C42, C2×C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C4×D4, C23×C4, C22×D4, C42⋊4C4, C4×C4⋊C4, C23.7Q8, C23.34D4, C24.C22, C23.65C23, C24.3C22, C2×C4×D4, C23.201C24
Quotients: C1, C2, C4, C22, C2×C4, C23, C22×C4, C4○D4, C24, C42⋊C2, C23×C4, C2×C4○D4, 2+ 1+4, 2- 1+4, C2×C42⋊C2, C22.11C24, C23.33C23, C22.34C24, C22.36C24, C23.201C24

Smallest permutation representation of C23.201C24
►On 64 points
Generators in S64
(1 11)(2 12)(3 9)(4 10)(5 38)(6 39)(7 40)(8 37)(13 41)(14 42)(15 43)(16 44)(17 45)(18 46)(19 47)(20 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 63)(34 64)(35 61)(36 62)

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

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

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

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

44 conjugacy classes

class 1 2A···2G2H2I2J2K4A···4L4M···4AF
order12···222224···44···4
size11···144442···24···4

44 irreducible representations

dim1111111111244
type++++++++++-
imageC1C2C2C2C2C2C2C2C2C4C4○D42+ 1+42- 1+4
kernelC23.201C24C42⋊4C4C4×C4⋊C4C23.7Q8C23.34D4C24.C22C23.65C23C24.3C22C2×C4×D4C4×D4C2×C4C22C22
# reps11122422116831

Matrix representation of C23.201C24 ►in GL8(𝔽5)

10000000
01000000
00100000
00010000
00004000
00000400
00000040
00000004
,
40000000
04000000
00400000
00040000
00004000
00000400
00000040
00000004
,
40000000
04000000
00100000
00010000
00001000
00000100
00000010
00000001
,
13000000
14000000
00420000
00010000
00004310
00000302
00000414
00000102
,
40000000
04000000
00100000
00010000
00004100
00003100
00004313
00002014
,
40000000
41000000
00100000
00140000
00001400
00000400
00002242
00000301
,
20000000
02000000
00200000
00020000
00003000
00000300
00001220
00000102

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

C23.201C24 in GAP, Magma, Sage, TeX 

C_2^3._{201}C_2^4
 
% in TeX
 
G:=Group("C2^3.201C2^4");
 
// GroupNames label
 
G:=SmallGroup(128,1051);
 
// by ID
 
G=gap.SmallGroup(128,1051);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,120,758,219,675,136]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=f^2=1,d^2=c,e^2=a,g^2=b,a*b=b*a,a*c=c*a,e*d*e^-1=g*d*g^-1=a*d=d*a,f*e*f=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,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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