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G = C23.535C24  order 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, C23⋊2D4⋊27C2, C23.198(C2×D4), C23.67(C4○D4), C23.4Q8⋊31C2, C23.34D4⋊43C2, C23.10D4⋊61C2, C23.23D4⋊71C2, C2.28(C23⋊3D4), (C2×C42).612C22, (C23×C4).433C22, (C22×C4).145C23, C22.360(C22×D4), (C22×D4).543C22, C23.83C23⋊61C2, C24.C22⋊106C2, 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

Derived series
C1 — C23 — C23.535C24
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C2×C22⋊C4 — C23.23D4 — C23.535C24
Lower central
C1 — C23 — C23.535C24
Upper central
C1 — C23 — C23.535C24
Jennings
C1 — C23 — 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, C2, C2, C4, C22, C22, C22, C2×C4, C2×C4, D4, C23, C23, C23, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C24, C24, C2.C42, C2.C42, C2×C42, C2×C22⋊C4, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C4×D4, C4⋊D4, C23×C4, C22×D4, C22×D4, C23.34D4, C23.23D4, C24.C22, C23⋊2D4, C23⋊2D4, C23.10D4, C23.10D4, C23.4Q8, C23.83C23, C2×C4×D4, C2×C4⋊D4, C23.535C24
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C22×D4, C2×C4○D4, 2+ 1+4, C22.19C24, C23⋊3D4, C22.29C24, C22.32C24, C22.34C24, 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 17)(2 18)(3 19)(4 20)(5 13)(6 14)(7 15)(8 16)(9 63)(10 64)(11 61)(12 62)(21 27)(22 28)(23 25)(24 26)(29 36)(30 33)(31 34)(32 35)(37 43)(38 44)(39 41)(40 42)(45 52)(46 49)(47 50)(48 51)(53 59)(54 60)(55 57)(56 58)

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

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

(2 18)(4 20)(5 9)(6 64)(7 11)(8 62)(10 14)(12 16)(13 63)(15 61)(22 28)(24 26)(29 44)(30 39)(31 42)(32 37)(33 41)(34 40)(35 43)(36 38)(45 47)(46 51)(48 49)(50 52)(53 57)(54 56)(55 59)(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), (1,17)(2,18)(3,19)(4,20)(5,13)(6,14)(7,15)(8,16)(9,63)(10,64)(11,61)(12,62)(21,27)(22,28)(23,25)(24,26)(29,36)(30,33)(31,34)(32,35)(37,43)(38,44)(39,41)(40,42)(45,52)(46,49)(47,50)(48,51)(53,59)(54,60)(55,57)(56,58), (1,19)(2,20)(3,17)(4,18)(5,15)(6,16)(7,13)(8,14)(9,61)(10,62)(11,63)(12,64)(21,25)(22,26)(23,27)(24,28)(29,34)(30,35)(31,36)(32,33)(37,41)(38,42)(39,43)(40,44)(45,50)(46,51)(47,52)(48,49)(53,57)(54,58)(55,59)(56,60), (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)(5,54)(6,55)(7,56)(8,53)(9,47)(10,48)(11,45)(12,46)(13,60)(14,57)(15,58)(16,59)(17,33)(18,34)(19,35)(20,36)(21,37)(22,38)(23,39)(24,40)(25,41)(26,42)(27,43)(28,44)(49,62)(50,63)(51,64)(52,61), (2,18)(4,20)(5,9)(6,64)(7,11)(8,62)(10,14)(12,16)(13,63)(15,61)(22,28)(24,26)(29,44)(30,39)(31,42)(32,37)(33,41)(34,40)(35,43)(36,38)(45,47)(46,51)(48,49)(50,52)(53,57)(54,56)(55,59)(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), (1,17)(2,18)(3,19)(4,20)(5,13)(6,14)(7,15)(8,16)(9,63)(10,64)(11,61)(12,62)(21,27)(22,28)(23,25)(24,26)(29,36)(30,33)(31,34)(32,35)(37,43)(38,44)(39,41)(40,42)(45,52)(46,49)(47,50)(48,51)(53,59)(54,60)(55,57)(56,58), (1,19)(2,20)(3,17)(4,18)(5,15)(6,16)(7,13)(8,14)(9,61)(10,62)(11,63)(12,64)(21,25)(22,26)(23,27)(24,28)(29,34)(30,35)(31,36)(32,33)(37,41)(38,42)(39,43)(40,44)(45,50)(46,51)(47,52)(48,49)(53,57)(54,58)(55,59)(56,60), (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)(5,54)(6,55)(7,56)(8,53)(9,47)(10,48)(11,45)(12,46)(13,60)(14,57)(15,58)(16,59)(17,33)(18,34)(19,35)(20,36)(21,37)(22,38)(23,39)(24,40)(25,41)(26,42)(27,43)(28,44)(49,62)(50,63)(51,64)(52,61), (2,18)(4,20)(5,9)(6,64)(7,11)(8,62)(10,14)(12,16)(13,63)(15,61)(22,28)(24,26)(29,44)(30,39)(31,42)(32,37)(33,41)(34,40)(35,43)(36,38)(45,47)(46,51)(48,49)(50,52)(53,57)(54,56)(55,59)(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)], [(1,17),(2,18),(3,19),(4,20),(5,13),(6,14),(7,15),(8,16),(9,63),(10,64),(11,61),(12,62),(21,27),(22,28),(23,25),(24,26),(29,36),(30,33),(31,34),(32,35),(37,43),(38,44),(39,41),(40,42),(45,52),(46,49),(47,50),(48,51),(53,59),(54,60),(55,57),(56,58)], [(1,19),(2,20),(3,17),(4,18),(5,15),(6,16),(7,13),(8,14),(9,61),(10,62),(11,63),(12,64),(21,25),(22,26),(23,27),(24,28),(29,34),(30,35),(31,36),(32,33),(37,41),(38,42),(39,43),(40,44),(45,50),(46,51),(47,52),(48,49),(53,57),(54,58),(55,59),(56,60)], [(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),(5,54),(6,55),(7,56),(8,53),(9,47),(10,48),(11,45),(12,46),(13,60),(14,57),(15,58),(16,59),(17,33),(18,34),(19,35),(20,36),(21,37),(22,38),(23,39),(24,40),(25,41),(26,42),(27,43),(28,44),(49,62),(50,63),(51,64),(52,61)], [(2,18),(4,20),(5,9),(6,64),(7,11),(8,62),(10,14),(12,16),(13,63),(15,61),(22,28),(24,26),(29,44),(30,39),(31,42),(32,37),(33,41),(34,40),(35,43),(36,38),(45,47),(46,51),(48,49),(50,52),(53,57),(54,56),(55,59),(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.C22C23⋊2D4C23.10D4C23.4Q8C23.83C23C2×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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