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G = C24.547C23  order 128 = 27

28th non-split extension by C24 of C23 acting via C23/C22=C2

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

Aliases: C24.547C23, C23.200C24, C22.242- 1+4, C22.392+ 1+4, (C4×D4)⋊19C4, C42⋊17(C2×C4), C42⋊5C4⋊3C2, C42⋊8C4⋊12C2, C23.8Q8⋊4C2, (C23×C4).44C22, C22.91(C23×C4), (C2×C42).12C22, C23.7Q8⋊14C2, C23.222(C4○D4), C23.34D4⋊10C2, C23.124(C22×C4), C24.C22⋊2C2, (C22×C4).465C23, C23.23D4.3C2, C23.63C23⋊3C2, C2.2(C22.32C24), C22.3(C42⋊C2), (C22×D4).475C22, C2.11(C22.11C24), C2.C42.37C22, C2.8(C23.33C23), C2.2(C22.33C24), C4⋊C4⋊41(C2×C4), (C2×C4×D4).29C2, (C4×C22⋊C4)⋊7C2, C22⋊C4⋊38(C2×C4), (C22×C4)⋊22(C2×C4), (C2×D4).211(C2×C4), C22.85(C2×C4○D4), (C2×C4⋊C4).174C22, (C2×C2.C42)⋊9C2, (C2×C4).223(C22×C4), C2.22(C2×C42⋊C2), (C2×C22⋊C4).25C22, SmallGroup(128,1050)

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

Derived series
C1 — C22 — C24.547C23
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C23×C4 — C2×C4×D4 — C24.547C23
Lower central
C1 — C22 — C24.547C23
Upper central
C1 — C23 — C24.547C23
Jennings
C1 — C23 — C24.547C23

Generators and relations for C24.547C23
 G = < a,b,c,d,e,f,g | a2=b2=c2=e2=f2=1, d2=c, g2=b, ab=ba, ac=ca, ede=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: 524 in 284 conjugacy classes, 140 normal (34 characteristic)
C1, C2, C2, C2, 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×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C4×D4, C23×C4, C23×C4, C22×D4, C2×C2.C42, C4×C22⋊C4, C23.7Q8, C23.34D4, C42⋊8C4, C42⋊5C4, C23.8Q8, C23.23D4, C23.63C23, C24.C22, C2×C4×D4, C24.547C23
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, C22.32C24, C22.33C24, C24.547C23

Smallest permutation representation of C24.547C23
►On 64 points
Generators in S64
(1 9)(2 10)(3 11)(4 12)(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 62)(6 63)(7 64)(8 61)(9 23)(10 24)(11 21)(12 22)(13 27)(14 28)(15 25)(16 26)(17 31)(18 32)(19 29)(20 30)(33 39)(34 40)(35 37)(36 38)(41 55)(42 56)(43 53)(44 54)(45 59)(46 60)(47 57)(48 58)

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

(2 52)(4 50)(5 36)(6 39)(7 34)(8 37)(10 24)(12 22)(14 28)(16 26)(17 45)(18 60)(19 47)(20 58)(29 57)(30 48)(31 59)(32 46)(33 63)(35 61)(38 62)(40 64)(42 56)(44 54)

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

44 conjugacy classes

class 1 2A···2G2H2I2J2K2L2M4A···4H4I···4AD
order12···22222224···44···4
size11···12222442···24···4

44 irreducible representations

dim1111111111111244
type+++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2C2C2C4C4○D42+ 1+42- 1+4
kernelC24.547C23C2×C2.C42C4×C22⋊C4C23.7Q8C23.34D4C42⋊8C4C42⋊5C4C23.8Q8C23.23D4C23.63C23C24.C22C2×C4×D4C4×D4C23C22C22
# reps11111112222116831

Matrix representation of C24.547C23 ►in GL8(𝔽5)

10000000
01000000
00100000
00010000
00004000
00000400
00000040
00000004
,
40000000
04000000
00400000
00040000
00001000
00000100
00000010
00000001
,
10000000
01000000
00400000
00040000
00004000
00000400
00000040
00000004
,
01000000
10000000
00220000
00030000
00003320
00004433
00004203
00000003
,
40000000
04000000
00400000
00040000
00000100
00001000
00004411
00002204
,
10000000
04000000
00100000
00340000
00001000
00000400
00000310
00004234
,
30000000
03000000
00300000
00030000
00001000
00000100
00002240
00002204

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

C24.547C23 in GAP, Magma, Sage, TeX 

C_2^4._{547}C_2^3
 
% in TeX
 
G:=Group("C2^4.547C2^3");
 
// GroupNames label
 
G:=SmallGroup(128,1050);
 
// by ID
 
G=gap.SmallGroup(128,1050);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,758,219,675,136]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=e^2=f^2=1,d^2=c,g^2=b,a*b=b*a,a*c=c*a,e*d*e=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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