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G = C24.220C23  order 128 = 27

60th non-split extension by C24 of C23 acting via C23/C2=C22

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

Aliases: C24.220C23, C23.249C24, C22.812+ 1+4, C22.602- 1+4, C4.99(C4×D4), C4⋊C4.397D4, C42⋊28(C2×C4), C4.4D4⋊24C4, C2.7(Q8⋊5D4), C2.11(D4⋊5D4), (C23×C4).58C22, C23.22(C22×C4), C23.8Q8⋊18C2, C22.140(C23×C4), (C2×C42).440C22, C22.120(C22×D4), C24.C22⋊22C2, (C22×C4).1254C23, (C22×D4).491C22, (C22×Q8).407C22, C23.67C23⋊22C2, C2.C42.65C22, C24.3C22.29C2, C2.7(C22.50C24), C2.4(C22.53C24), C2.35(C23.33C23), (C4×C4⋊C4)⋊47C2, (C2×C4×Q8)⋊10C2, C2.43(C2×C4×D4), (C2×C4×D4).40C2, (C2×Q8)⋊26(C2×C4), C2.37(C4×C4○D4), C22⋊C4⋊17(C2×C4), (C4×C22⋊C4)⋊44C2, (C2×C4).894(C2×D4), (C2×D4).170(C2×C4), (C2×C4).47(C22×C4), (C2×C4).892(C4○D4), (C2×C4⋊C4).979C22, C4⋊C4○4(C2.C42), (C2×C4.4D4).17C2, C22.134(C2×C4○D4), (C2×C22⋊C4).38C22, SmallGroup(128,1099)

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

Derived series
C1 — C22 — C24.220C23
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C4×C4⋊C4 — C24.220C23
Lower central
C1 — C22 — C24.220C23
Upper central
C1 — C23 — C24.220C23
Jennings
C1 — C23 — C24.220C23

Generators and relations for C24.220C23
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=1, e2=d, f2=g2=b, eae-1=gag-1=ab=ba, faf-1=ac=ca, ad=da, bc=cb, bd=db, fef-1=geg-1=be=eb, gfg-1=bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, cg=gc, de=ed, df=fd, dg=gd >

Subgroups: 540 in 316 conjugacy classes, 152 normal (42 characteristic)
C1, C2, C2, C4, C4, C22, C22, C2×C4, C2×C4, D4, Q8, C23, C23, C23, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C24, C2.C42, C2.C42, C2×C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C4×D4, C4×Q8, C4.4D4, C23×C4, C22×D4, C22×Q8, C4×C22⋊C4, C4×C4⋊C4, C23.8Q8, C24.C22, C24.3C22, C23.67C23, C2×C4×D4, C2×C4×Q8, C2×C4.4D4, C24.220C23
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, 2- 1+4, C2×C4×D4, C4×C4○D4, C23.33C23, D4⋊5D4, Q8⋊5D4, C22.50C24, C22.53C24, C24.220C23

Smallest permutation representation of C24.220C23
►On 64 points
Generators in S64
(1 10)(2 36)(3 12)(4 34)(5 60)(6 28)(7 58)(8 26)(9 15)(11 13)(14 33)(16 35)(17 54)(18 30)(19 56)(20 32)(21 64)(22 38)(23 62)(24 40)(25 48)(27 46)(29 50)(31 52)(37 42)(39 44)(41 63)(43 61)(45 59)(47 57)(49 53)(51 55)

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

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

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

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

50 conjugacy classes

class 1 2A···2G2H2I2J2K4A···4X4Y···4AL
order12···222224···44···4
size11···144442···24···4

50 irreducible representations

dim111111111112244
type++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2C4D4C4○D42+ 1+42- 1+4
kernelC24.220C23C4×C22⋊C4C4×C4⋊C4C23.8Q8C24.C22C24.3C22C23.67C23C2×C4×D4C2×C4×Q8C2×C4.4D4C4.4D4C4⋊C4C2×C4C22C22
# reps12224111111641211

Matrix representation of C24.220C23 ►in GL5(𝔽5)

10000
04000
00100
00001
00010
,
10000
01000
00100
00040
00004
,
10000
04000
00400
00010
00001
,
40000
04000
00400
00010
00001
,
30000
03000
00300
00010
00004
,
10000
00400
04000
00003
00030
,
40000
04000
00400
00001
00040

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

C24.220C23 in GAP, Magma, Sage, TeX 

C_2^4._{220}C_2^3
 
% in TeX
 
G:=Group("C2^4.220C2^3");
 
// GroupNames label
 
G:=SmallGroup(128,1099);
 
// by ID
 
G=gap.SmallGroup(128,1099);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,456,758,268,346,80]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=1,e^2=d,f^2=g^2=b,e*a*e^-1=g*a*g^-1=a*b=b*a,f*a*f^-1=a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,f*e*f^-1=g*e*g^-1=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,d*e=e*d,d*f=f*d,d*g=g*d>;
 
// generators/relations
 

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