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G = C23.535C24order 128 = 27

252nd central stem extension by C23 of C24

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

Aliases: C23.535C24, C24.372C23, C22.3112+ 1+4, (C22×C4)⋊37D4, C232D427C2, C23.198(C2×D4), C23.67(C4○D4), C23.4Q831C2, C23.34D443C2, C23.10D461C2, C23.23D471C2, C2.28(C233D4), (C2×C42).612C22, (C23×C4).433C22, (C22×C4).145C23, C22.360(C22×D4), (C22×D4).543C22, C23.83C2361C2, C24.C22106C2, C2.86(C22.19C24), C2.41(C22.32C24), C2.43(C22.29C24), C2.C42.260C22, C2.28(C22.34C24), (C2×C4×D4)⋊54C2, (C2×C4⋊D4)⋊24C2, (C2×C4).394(C2×D4), (C2×C4).170(C4○D4), (C2×C4⋊C4).362C22, C22.407(C2×C4○D4), (C2×C22⋊C4).223C22, SmallGroup(128,1367)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C23.535C24
C1C2C22C23C22×C4C2×C22⋊C4C23.23D4 — C23.535C24
C1C23 — C23.535C24
C1C23 — C23.535C24
C1C23 — C23.535C24

Generators and relations for C23.535C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=e2=f2=1, g2=cb=bc, eae=ab=ba, faf=ac=ca, ad=da, ag=ga, bd=db, be=eb, gfg-1=bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, cg=gc, fef=de=ed, df=fd, dg=gd, eg=ge >

Subgroups: 708 in 310 conjugacy classes, 96 normal (34 characteristic)
C1, C2 [×3], C2 [×4], C2 [×6], C4 [×14], C22 [×3], C22 [×4], C22 [×34], C2×C4 [×6], C2×C4 [×38], D4 [×28], C23, C23 [×4], C23 [×26], C42 [×2], C22⋊C4 [×20], C4⋊C4 [×6], C22×C4 [×5], C22×C4 [×10], C22×C4 [×6], C2×D4 [×26], C24 [×2], C24 [×2], C2.C42 [×2], C2.C42 [×6], C2×C42, C2×C22⋊C4 [×2], C2×C22⋊C4 [×12], C2×C4⋊C4 [×2], C2×C4⋊C4 [×2], C4×D4 [×4], C4⋊D4 [×4], C23×C4 [×2], C22×D4 [×2], C22×D4 [×4], C23.34D4, C23.23D4 [×2], C24.C22 [×2], C232D4, C232D4 [×2], C23.10D4, C23.10D4 [×2], C23.4Q8, C23.83C23, C2×C4×D4, C2×C4⋊D4, C23.535C24
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 [×4], C22.19C24, C233D4, C22.29C24, C22.32C24 [×2], C22.34C24 [×2], C23.535C24

Smallest permutation representation of C23.535C24
On 64 points
Generators in S64
(1 46)(2 47)(3 48)(4 45)(5 44)(6 41)(7 42)(8 43)(9 34)(10 35)(11 36)(12 33)(13 38)(14 39)(15 40)(16 37)(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 25)(2 26)(3 27)(4 28)(5 61)(6 62)(7 63)(8 64)(9 15)(10 16)(11 13)(12 14)(17 23)(18 24)(19 21)(20 22)(29 44)(30 41)(31 42)(32 43)(33 39)(34 40)(35 37)(36 38)(45 60)(46 57)(47 58)(48 59)(49 55)(50 56)(51 53)(52 54)
(1 27)(2 28)(3 25)(4 26)(5 63)(6 64)(7 61)(8 62)(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)(29 42)(30 43)(31 44)(32 41)(33 37)(34 38)(35 39)(36 40)(45 58)(46 59)(47 60)(48 57)(49 53)(50 54)(51 55)(52 56)
(1 23)(2 24)(3 21)(4 22)(5 11)(6 12)(7 9)(8 10)(13 61)(14 62)(15 63)(16 64)(17 25)(18 26)(19 27)(20 28)(29 38)(30 39)(31 40)(32 37)(33 41)(34 42)(35 43)(36 44)(45 54)(46 55)(47 56)(48 53)(49 57)(50 58)(51 59)(52 60)
(1 30)(2 31)(3 32)(4 29)(5 45)(6 46)(7 47)(8 48)(9 56)(10 53)(11 54)(12 55)(13 52)(14 49)(15 50)(16 51)(17 33)(18 34)(19 35)(20 36)(21 37)(22 38)(23 39)(24 40)(25 41)(26 42)(27 43)(28 44)(57 62)(58 63)(59 64)(60 61)
(2 26)(4 28)(5 9)(6 16)(7 11)(8 14)(10 62)(12 64)(13 63)(15 61)(18 24)(20 22)(29 36)(30 39)(31 34)(32 37)(33 41)(35 43)(38 44)(40 42)(45 47)(46 59)(48 57)(49 53)(50 52)(51 55)(54 56)(58 60)
(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)| (1,46)(2,47)(3,48)(4,45)(5,44)(6,41)(7,42)(8,43)(9,34)(10,35)(11,36)(12,33)(13,38)(14,39)(15,40)(16,37)(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), (1,25)(2,26)(3,27)(4,28)(5,61)(6,62)(7,63)(8,64)(9,15)(10,16)(11,13)(12,14)(17,23)(18,24)(19,21)(20,22)(29,44)(30,41)(31,42)(32,43)(33,39)(34,40)(35,37)(36,38)(45,60)(46,57)(47,58)(48,59)(49,55)(50,56)(51,53)(52,54), (1,27)(2,28)(3,25)(4,26)(5,63)(6,64)(7,61)(8,62)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(29,42)(30,43)(31,44)(32,41)(33,37)(34,38)(35,39)(36,40)(45,58)(46,59)(47,60)(48,57)(49,53)(50,54)(51,55)(52,56), (1,23)(2,24)(3,21)(4,22)(5,11)(6,12)(7,9)(8,10)(13,61)(14,62)(15,63)(16,64)(17,25)(18,26)(19,27)(20,28)(29,38)(30,39)(31,40)(32,37)(33,41)(34,42)(35,43)(36,44)(45,54)(46,55)(47,56)(48,53)(49,57)(50,58)(51,59)(52,60), (1,30)(2,31)(3,32)(4,29)(5,45)(6,46)(7,47)(8,48)(9,56)(10,53)(11,54)(12,55)(13,52)(14,49)(15,50)(16,51)(17,33)(18,34)(19,35)(20,36)(21,37)(22,38)(23,39)(24,40)(25,41)(26,42)(27,43)(28,44)(57,62)(58,63)(59,64)(60,61), (2,26)(4,28)(5,9)(6,16)(7,11)(8,14)(10,62)(12,64)(13,63)(15,61)(18,24)(20,22)(29,36)(30,39)(31,34)(32,37)(33,41)(35,43)(38,44)(40,42)(45,47)(46,59)(48,57)(49,53)(50,52)(51,55)(54,56)(58,60), (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( (1,46)(2,47)(3,48)(4,45)(5,44)(6,41)(7,42)(8,43)(9,34)(10,35)(11,36)(12,33)(13,38)(14,39)(15,40)(16,37)(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), (1,25)(2,26)(3,27)(4,28)(5,61)(6,62)(7,63)(8,64)(9,15)(10,16)(11,13)(12,14)(17,23)(18,24)(19,21)(20,22)(29,44)(30,41)(31,42)(32,43)(33,39)(34,40)(35,37)(36,38)(45,60)(46,57)(47,58)(48,59)(49,55)(50,56)(51,53)(52,54), (1,27)(2,28)(3,25)(4,26)(5,63)(6,64)(7,61)(8,62)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(29,42)(30,43)(31,44)(32,41)(33,37)(34,38)(35,39)(36,40)(45,58)(46,59)(47,60)(48,57)(49,53)(50,54)(51,55)(52,56), (1,23)(2,24)(3,21)(4,22)(5,11)(6,12)(7,9)(8,10)(13,61)(14,62)(15,63)(16,64)(17,25)(18,26)(19,27)(20,28)(29,38)(30,39)(31,40)(32,37)(33,41)(34,42)(35,43)(36,44)(45,54)(46,55)(47,56)(48,53)(49,57)(50,58)(51,59)(52,60), (1,30)(2,31)(3,32)(4,29)(5,45)(6,46)(7,47)(8,48)(9,56)(10,53)(11,54)(12,55)(13,52)(14,49)(15,50)(16,51)(17,33)(18,34)(19,35)(20,36)(21,37)(22,38)(23,39)(24,40)(25,41)(26,42)(27,43)(28,44)(57,62)(58,63)(59,64)(60,61), (2,26)(4,28)(5,9)(6,16)(7,11)(8,14)(10,62)(12,64)(13,63)(15,61)(18,24)(20,22)(29,36)(30,39)(31,34)(32,37)(33,41)(35,43)(38,44)(40,42)(45,47)(46,59)(48,57)(49,53)(50,52)(51,55)(54,56)(58,60), (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([(1,46),(2,47),(3,48),(4,45),(5,44),(6,41),(7,42),(8,43),(9,34),(10,35),(11,36),(12,33),(13,38),(14,39),(15,40),(16,37),(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)], [(1,25),(2,26),(3,27),(4,28),(5,61),(6,62),(7,63),(8,64),(9,15),(10,16),(11,13),(12,14),(17,23),(18,24),(19,21),(20,22),(29,44),(30,41),(31,42),(32,43),(33,39),(34,40),(35,37),(36,38),(45,60),(46,57),(47,58),(48,59),(49,55),(50,56),(51,53),(52,54)], [(1,27),(2,28),(3,25),(4,26),(5,63),(6,64),(7,61),(8,62),(9,13),(10,14),(11,15),(12,16),(17,21),(18,22),(19,23),(20,24),(29,42),(30,43),(31,44),(32,41),(33,37),(34,38),(35,39),(36,40),(45,58),(46,59),(47,60),(48,57),(49,53),(50,54),(51,55),(52,56)], [(1,23),(2,24),(3,21),(4,22),(5,11),(6,12),(7,9),(8,10),(13,61),(14,62),(15,63),(16,64),(17,25),(18,26),(19,27),(20,28),(29,38),(30,39),(31,40),(32,37),(33,41),(34,42),(35,43),(36,44),(45,54),(46,55),(47,56),(48,53),(49,57),(50,58),(51,59),(52,60)], [(1,30),(2,31),(3,32),(4,29),(5,45),(6,46),(7,47),(8,48),(9,56),(10,53),(11,54),(12,55),(13,52),(14,49),(15,50),(16,51),(17,33),(18,34),(19,35),(20,36),(21,37),(22,38),(23,39),(24,40),(25,41),(26,42),(27,43),(28,44),(57,62),(58,63),(59,64),(60,61)], [(2,26),(4,28),(5,9),(6,16),(7,11),(8,14),(10,62),(12,64),(13,63),(15,61),(18,24),(20,22),(29,36),(30,39),(31,34),(32,37),(33,41),(35,43),(38,44),(40,42),(45,47),(46,59),(48,57),(49,53),(50,52),(51,55),(54,56),(58,60)], [(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···2G2H2I2J2K2L2M4A4B4C4D4E···4L4M···4R
order12···222222244444···44···4
size11···144448822224···48···8

32 irreducible representations

dim11111111112224
type++++++++++++
imageC1C2C2C2C2C2C2C2C2C2D4C4○D4C4○D42+ 1+4
kernelC23.535C24C23.34D4C23.23D4C24.C22C232D4C23.10D4C23.4Q8C23.83C23C2×C4×D4C2×C4⋊D4C22×C4C2×C4C23C22
# reps11223311114444

Matrix representation of C23.535C24 in GL8(𝔽5)

31000000
22000000
00100000
00010000
00003232
00000030
00000200
00001402
,
10000000
01000000
00100000
00010000
00004000
00000400
00000040
00000004
,
40000000
04000000
00100000
00010000
00004000
00000400
00000040
00000004
,
10000000
01000000
00400000
00040000
00001000
00000100
00000010
00000001
,
40000000
04000000
00130000
00040000
00000010
00001414
00001000
00000001
,
10000000
44000000
00100000
00140000
00001000
00000400
00000010
00002024
,
30000000
03000000
00400000
00040000
00000100
00001000
00001414
00000004

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

C23.535C24 in GAP, Magma, Sage, TeX

C_2^3._{535}C_2^4
% in TeX

G:=Group("C2^3.535C2^4");
// GroupNames label

G:=SmallGroup(128,1367);
// by ID

G=gap.SmallGroup(128,1367);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,253,758,723,185,192]);
// 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=c*b=b*c,e*a*e=a*b=b*a,f*a*f=a*c=c*a,a*d=d*a,a*g=g*a,b*d=d*b,b*e=e*b,g*f*g^-1=b*f=f*b,b*g=g*b,c*d=d*c,c*e=e*c,c*f=f*c,c*g=g*c,f*e*f=d*e=e*d,d*f=f*d,d*g=g*d,e*g=g*e>;
// generators/relations

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