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G = C23.530C24order 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), C232D4.17C2, C23.8Q885C2, C23.7Q880C2, C23.Q839C2, C23.10D460C2, C23.11D459C2, C23.23D469C2, C2.27(C233D4), (C23×C4).432C22, (C22×C4).140C23, (C2×C42).607C22, C22.355(C22×D4), (C22×D4).540C22, C23.81C2362C2, C24.C22104C2, C2.81(C22.19C24), C2.39(C22.32C24), C23.63C23112C2, 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

C1C23 — C23.530C24
C1C2C22C23C22×C4C2×C22⋊C4C23.23D4 — C23.530C24
C1C23 — C23.530C24
C1C23 — C23.530C24
C1C23 — 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 [×7], C2 [×5], C4 [×15], C22 [×7], C22 [×27], C2×C4 [×6], C2×C4 [×41], D4 [×16], C23, C23 [×4], C23 [×19], C42 [×2], C22⋊C4 [×20], C4⋊C4 [×12], C22×C4 [×12], C22×C4 [×4], C22×C4 [×6], C2×D4 [×14], C24 [×3], C2.C42 [×8], C2×C42, C2×C22⋊C4 [×13], C2×C4⋊C4 [×8], C4×D4 [×4], C22.D4 [×4], C23×C4 [×2], C22×D4 [×3], C23.7Q8, C23.8Q8, C23.23D4, C23.63C23, C24.C22, C232D4, C23.10D4 [×3], C23.Q8 [×2], C23.11D4, C23.81C23, C2×C4×D4, C2×C22.D4, C23.530C24
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 [×3], 2- 1+4, C22.19C24, C233D4, 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.C22C232D4C23.10D4C23.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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