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G = C24.218C23  order 128 = 27

58th non-split extension by C24 of C23 acting via C23/C2=C22

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

Aliases: C24.218C23, C23.246C24, C22.792+ 1+4, C4⋊D4⋊21C4, C22⋊C4⋊45D4, C22.19(C4×D4), C23.420(C2×D4), C2.10(D4⋊5D4), (C2×C42).25C22, C23.20(C22×C4), C23.324(C4○D4), C23.23D4⋊15C2, (C23×C4).311C22, C22.137(C23×C4), C22.117(C22×D4), C24.C22⋊21C2, (C22×C4).1252C23, (C22×D4).489C22, C23.63C23⋊20C2, C2.31(C22.11C24), C2.C42.64C22, C2.8(C22.47C24), (C2×C4×D4)⋊14C2, C2.40(C2×C4×D4), C4⋊C4⋊16(C2×C4), (C2×D4)⋊20(C2×C4), C2.36(C4×C4○D4), (C4×C22⋊C4)⋊43C2, C22⋊C4⋊31(C2×C4), (C22×C4)⋊37(C2×C4), (C2×C4).892(C2×D4), (C2×C4⋊D4).20C2, (C2×C4).45(C22×C4), (C2×C4).723(C4○D4), (C2×C4⋊C4).828C22, C22.131(C2×C4○D4), (C2×C2.C42)⋊22C2, (C2×C22⋊C4).557C22, C22⋊C4○3(C2.C42), SmallGroup(128,1096)

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

Derived series
C1 — C22 — C24.218C23
Chief series
C1 — C2 — C22 — C23 — C24 — C23×C4 — C4×C22⋊C4 — C24.218C23
Lower central
C1 — C22 — C24.218C23
Upper central
C1 — C23 — C24.218C23
Jennings
C1 — C23 — C24.218C23

Generators and relations for C24.218C23
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=g2=1, e2=d, f2=b, eae-1=gag=ab=ba, faf-1=ac=ca, ad=da, bc=cb, bd=db, fef-1=geg=be=eb, gfg=bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, cg=gc, de=ed, df=fd, dg=gd >

Subgroups: 668 in 358 conjugacy classes, 152 normal (22 characteristic)
C1, C2, C2, C2, C4, C22, 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×C22⋊C4, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C4×D4, C4⋊D4, C23×C4, C23×C4, C22×D4, C22×D4, C2×C2.C42, C4×C22⋊C4, C4×C22⋊C4, C23.23D4, C23.63C23, C24.C22, C2×C4×D4, C2×C4⋊D4, C24.218C23
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, C22.47C24, C24.218C23

Smallest permutation representation of C24.218C23
►On 64 points
Generators in S64
(1 61)(2 39)(3 63)(4 37)(5 19)(6 49)(7 17)(8 51)(9 60)(10 28)(11 58)(12 26)(13 62)(14 40)(15 64)(16 38)(18 45)(20 47)(21 56)(22 32)(23 54)(24 30)(25 36)(27 34)(29 44)(31 42)(33 59)(35 57)(41 55)(43 53)(46 52)(48 50)

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

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

50 conjugacy classes

class 1 2A···2G2H2I2J2K2L2M2N2O4A···4T4U···4AH
order12···2222222224···44···4
size11···1222244442···24···4

50 irreducible representations

dim1111111112224
type++++++++++
imageC1C2C2C2C2C2C2C2C4D4C4○D4C4○D42+ 1+4
kernelC24.218C23C2×C2.C42C4×C22⋊C4C23.23D4C23.63C23C24.C22C2×C4×D4C2×C4⋊D4C4⋊D4C22⋊C4C2×C4C23C22
# reps11342221164842

Matrix representation of C24.218C23 ►in GL5(𝔽5)

10000
00400
04000
00024
00033
,
10000
01000
00100
00040
00004
,
10000
04000
00400
00010
00001
,
40000
01000
00100
00010
00001
,
20000
04000
00400
00010
00044
,
10000
01000
00400
00043
00011
,
40000
01000
00100
00043
00001

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

C24.218C23 in GAP, Magma, Sage, TeX 

C_2^4._{218}C_2^3
 
% in TeX
 
G:=Group("C2^4.218C2^3");
 
// GroupNames label
 
G:=SmallGroup(128,1096);
 
// by ID
 
G=gap.SmallGroup(128,1096);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,456,758,268,346]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=g^2=1,e^2=d,f^2=b,e*a*e^-1=g*a*g=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=b*e=e*b,g*f*g=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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