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G = C24.217C23  order 128 = 27

57th non-split extension by C24 of C23 acting via C23/C2=C22

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

Aliases: C24.217C23, C23.245C24, C22.782+ 1+4, C4⋊C4⋊51D4, C4.75(C4×D4), C4⋊D4⋊20C4, C2.9(D4⋊5D4), C2.4(Q8⋊6D4), C23.19(C22×C4), (C2×C42).24C22, C23.23D4⋊14C2, C22.136(C23×C4), (C22×C4).767C23, (C23×C4).310C22, C22.116(C22×D4), C24.C22⋊20C2, C24.3C22⋊21C2, (C22×D4).488C22, C23.65C23⋊28C2, C2.30(C22.11C24), C2.C42.570C22, C2.3(C22.49C24), C2.7(C22.47C24), (C2×C4×D4)⋊13C2, (C4×C4⋊C4)⋊45C2, C2.39(C2×C4×D4), C4⋊C4⋊31(C2×C4), (C2×D4)⋊19(C2×C4), C2.35(C4×C4○D4), C22⋊C4⋊30(C2×C4), (C4×C22⋊C4)⋊42C2, (C22×C4)⋊36(C2×C4), (C2×C4).891(C2×D4), (C2×C4⋊D4).19C2, (C2×C4).44(C22×C4), (C2×C42⋊C2)⋊15C2, (C2×C4).801(C4○D4), (C2×C4⋊C4).977C22, C4⋊C4○3(C2.C42), C22.130(C2×C4○D4), (C2×C22⋊C4).37C22, C2.C42○(C2×C4⋊C4), SmallGroup(128,1095)

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

Derived series
C1 — C22 — C24.217C23
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C23×C4 — C2×C42⋊C2 — C24.217C23
Lower central
C1 — C22 — C24.217C23
Upper central
C1 — C23 — C24.217C23
Jennings
C1 — C23 — C24.217C23

Generators and relations for C24.217C23
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=g2=1, e2=db=bd, f2=b, gag=ab=ba, faf-1=ac=ca, ad=da, ae=ea, bc=cb, geg=be=eb, bf=fb, bg=gb, cd=dc, ce=ec, gfg=cf=fc, cg=gc, de=ed, df=fd, dg=gd, ef=fe >

Subgroups: 636 in 344 conjugacy classes, 152 normal (42 characteristic)
C1, C2, C2, C4, C4, C22, C22, C2×C4, C2×C4, D4, C23, C23, C23, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C24, C24, C2.C42, C2.C42, C2×C42, C2×C42, C2×C22⋊C4, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C42⋊C2, C4×D4, C4⋊D4, C23×C4, C23×C4, C22×D4, C22×D4, C4×C22⋊C4, C4×C22⋊C4, C4×C4⋊C4, C23.23D4, C24.C22, C23.65C23, C24.3C22, C24.3C22, C2×C42⋊C2, C2×C4×D4, C2×C4⋊D4, C24.217C23
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22×C4, C2×D4, C4○D4, C24, C4×D4, C23×C4, C22×D4, C2×C4○D4, 2+ 1+4, C2×C4×D4, C4×C4○D4, C22.11C24, D4⋊5D4, Q8⋊6D4, C22.47C24, C22.49C24, C24.217C23

Smallest permutation representation of C24.217C23
►On 64 points
Generators in S64
(1 59)(2 60)(3 57)(4 58)(5 30)(6 31)(7 32)(8 29)(9 46)(10 47)(11 48)(12 45)(13 44)(14 41)(15 42)(16 43)(17 35)(18 36)(19 33)(20 34)(21 49)(22 50)(23 51)(24 52)(25 53)(26 54)(27 55)(28 56)(37 63)(38 64)(39 61)(40 62)

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

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

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

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

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

50 conjugacy classes

class 1 2A···2G2H···2M4A···4X4Y···4AJ
order12···22···24···44···4
size11···14···42···24···4

50 irreducible representations

dim11111111111224
type++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C4D4C4○D42+ 1+4
kernelC24.217C23C4×C22⋊C4C4×C4⋊C4C23.23D4C24.C22C23.65C23C24.3C22C2×C42⋊C2C2×C4×D4C2×C4⋊D4C4⋊D4C4⋊C4C2×C4C22
# reps1312213111164122

Matrix representation of C24.217C23 ►in GL5(𝔽5)

10000
00200
03000
00044
00001
,
10000
04000
00400
00010
00001
,
10000
01000
00100
00040
00004
,
40000
01000
00100
00010
00001
,
20000
00400
01000
00010
00001
,
10000
03000
00300
00010
00034
,
40000
00400
04000
00044
00001

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

C24.217C23 in GAP, Magma, Sage, TeX 

C_2^4._{217}C_2^3
 
% in TeX
 
G:=Group("C2^4.217C2^3");
 
// GroupNames label
 
G:=SmallGroup(128,1095);
 
// by ID
 
G=gap.SmallGroup(128,1095);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,568,758,100,346,80]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=g^2=1,e^2=d*b=b*d,f^2=b,g*a*g=a*b=b*a,f*a*f^-1=a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,g*e*g=b*e=e*b,b*f=f*b,b*g=g*b,c*d=d*c,c*e=e*c,g*f*g=c*f=f*c,c*g=g*c,d*e=e*d,d*f=f*d,d*g=g*d,e*f=f*e>;
 
// generators/relations
 

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