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G = C23.530C24  order 128 = 27

247th central stem extension by C23 of C24

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

Aliases: C23.530C24, C24.369C23, C22.3072+ 1+4, C22.2252- 1+4, (C22×C4)⋊36D4, C23.197(C2×D4), C23.64(C4○D4), C23⋊2D4.17C2, C23.8Q8⋊85C2, C23.7Q8⋊80C2, C23.Q8⋊39C2, C23.10D4⋊60C2, C23.11D4⋊59C2, C23.23D4⋊69C2, C2.27(C23⋊3D4), (C23×C4).432C22, (C22×C4).140C23, (C2×C42).607C22, C22.355(C22×D4), (C22×D4).540C22, C23.81C23⋊62C2, C24.C22⋊104C2, C2.81(C22.19C24), C2.39(C22.32C24), C23.63C23⋊112C2, C2.C42.255C22, C2.27(C22.34C24), C2.39(C22.33C24), C2.48(C22.36C24), C2.28(C22.31C24), (C2×C4×D4)⋊52C2, (C2×C4).389(C2×D4), (C2×C4).167(C4○D4), (C2×C4⋊C4).358C22, C22.402(C2×C4○D4), (C2×C22.D4)⋊26C2, (C2×C22⋊C4).219C22, SmallGroup(128,1362)

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

Derived series
C1 — C23 — C23.530C24
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C2×C22⋊C4 — C23.23D4 — C23.530C24
Lower central
C1 — C23 — C23.530C24
Upper central
C1 — C23 — C23.530C24
Jennings
C1 — C23 — C23.530C24

Generators and relations for C23.530C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=e2=f2=1, g2=b, ab=ba, ac=ca, ede=ad=da, ae=ea, gfg-1=af=fa, ag=ga, bc=cb, fdf=bd=db, be=eb, bf=fb, bg=gb, cd=dc, fef=ce=ec, cf=fc, cg=gc, dg=gd, eg=ge >

Subgroups: 596 in 281 conjugacy classes, 96 normal (82 characteristic)
C1, C2, C2, C4, C22, C22, C2×C4, C2×C4, D4, C23, C23, C23, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C24, C2.C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C4×D4, C22.D4, C23×C4, C22×D4, C23.7Q8, C23.8Q8, C23.23D4, C23.63C23, C24.C22, C23⋊2D4, C23.10D4, C23.Q8, C23.11D4, C23.81C23, C2×C4×D4, C2×C22.D4, C23.530C24
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C22×D4, C2×C4○D4, 2+ 1+4, 2- 1+4, C22.19C24, C23⋊3D4, C22.31C24, C22.32C24, C22.33C24, C22.34C24, C22.36C24, C23.530C24

Smallest permutation representation of C23.530C24
►On 64 points
Generators in S64
(1 5)(2 6)(3 7)(4 8)(9 62)(10 63)(11 64)(12 61)(13 17)(14 18)(15 19)(16 20)(21 28)(22 25)(23 26)(24 27)(29 33)(30 34)(31 35)(32 36)(37 44)(38 41)(39 42)(40 43)(45 49)(46 50)(47 51)(48 52)(53 60)(54 57)(55 58)(56 59)

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

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

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

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

32 conjugacy classes

class 1 2A···2G2H2I2J2K2L4A4B4C4D4E···4L4M···4S
order12···22222244444···44···4
size11···14444822224···48···8

32 irreducible representations

dim111111111111122244
type+++++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2C2C2C2D4C4○D4C4○D42+ 1+42- 1+4
kernelC23.530C24C23.7Q8C23.8Q8C23.23D4C23.63C23C24.C22C23⋊2D4C23.10D4C23.Q8C23.11D4C23.81C23C2×C4×D4C2×C22.D4C22×C4C2×C4C23C22C22
# reps111111132111144431

Matrix representation of C23.530C24 ►in GL8(𝔽5)

10000000
01000000
00100000
00010000
00004000
00000400
00000040
00000004
,
40000000
04000000
00100000
00010000
00001000
00000100
00000010
00000001
,
10000000
01000000
00400000
00040000
00001000
00000100
00000010
00000001
,
32000000
12000000
00400000
00040000
00000010
00001143
00001000
00000004
,
10000000
01000000
00010000
00100000
00004000
00000400
00000010
00004401
,
10000000
24000000
00100000
00040000
00001000
00000400
00000010
00001044
,
20000000
02000000
00400000
00040000
00000100
00001000
00001143
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],[4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[3,1,0,0,0,0,0,0,2,2,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,4,0,0,0,0,0,0,0,3,0,4],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,4,0,0,4,0,0,0,0,0,4,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,2,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,4,0,0,0,0,0,0,0,4],[2,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,1,0,1,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,3,1] >;
 

C23.530C24 in GAP, Magma, Sage, TeX 

C_2^3._{530}C_2^4
 
% in TeX
 
G:=Group("C2^3.530C2^4");
 
// GroupNames label
 
G:=SmallGroup(128,1362);
 
// by ID
 
G=gap.SmallGroup(128,1362);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,253,758,723,185,136]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=e^2=f^2=1,g^2=b,a*b=b*a,a*c=c*a,e*d*e=a*d=d*a,a*e=e*a,g*f*g^-1=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,d*g=g*d,e*g=g*e>;
 
// generators/relations
 

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