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G = C24.212C23  order 128 = 27

52nd non-split extension by C24 of C23 acting via C23/C2=C22

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

Aliases: C24.212C23, C23.239C24, C22.732+ 1+4, (C4×D4)⋊26C4, C42⋊8C4⋊19C2, C42.191(C2×C4), C23.34D4⋊3C2, (C2×C42).21C22, (C23×C4).54C22, C23.93(C22×C4), C4.43(C42⋊C2), C22.130(C23×C4), C24.C22⋊16C2, (C22×C4).1251C23, (C22×D4).484C22, C2.28(C22.11C24), C24.3C22.28C2, C2.C42.61C22, C2.5(C22.47C24), C2.2(C22.53C24), (C4×C4⋊C4)⋊40C2, (C2×C4×D4).38C2, C2.32(C4×C4○D4), C4⋊C4.241(C2×C4), (C4×C22⋊C4)⋊11C2, (C2×D4).217(C2×C4), C22⋊C4.62(C2×C4), (C2×C4).797(C4○D4), (C2×C4⋊C4).975C22, (C2×C4).494(C22×C4), (C22×C4).133(C2×C4), C2.36(C2×C42⋊C2), C22.124(C2×C4○D4), (C2×C22⋊C4).441C22, SmallGroup(128,1089)

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

Derived series
C1 — C22 — C24.212C23
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C2×C4×D4 — C24.212C23
Lower central
C1 — C22 — C24.212C23
Upper central
C1 — C23 — C24.212C23
Jennings
C1 — C23 — C24.212C23

Generators and relations for C24.212C23
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=1, e2=d, f2=cb=bc, g2=b, gag-1=ab=ba, ac=ca, ad=da, eae-1=abc, af=fa, bd=db, fef-1=be=eb, gfg-1=bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, cg=gc, de=ed, df=fd, dg=gd, eg=ge >

Subgroups: 492 in 280 conjugacy classes, 144 normal (20 characteristic)
C1, C2, C2, C2, C4, 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×C4⋊C4, C2×C4⋊C4, C4×D4, C23×C4, C22×D4, C4×C22⋊C4, C4×C4⋊C4, C4×C4⋊C4, C23.34D4, C42⋊8C4, C24.C22, C24.3C22, C2×C4×D4, C24.212C23
Quotients: C1, C2, C4, C22, C2×C4, C23, C22×C4, C4○D4, C24, C42⋊C2, C23×C4, C2×C4○D4, 2+ 1+4, C2×C42⋊C2, C4×C4○D4, C22.11C24, C22.47C24, C22.53C24, C24.212C23

Smallest permutation representation of C24.212C23
►On 64 points
Generators in S64
(1 41)(2 28)(3 43)(4 26)(5 30)(6 45)(7 32)(8 47)(9 13)(10 56)(11 15)(12 54)(14 52)(16 50)(17 39)(18 64)(19 37)(20 62)(21 25)(22 44)(23 27)(24 42)(29 35)(31 33)(34 46)(36 48)(38 58)(40 60)(49 53)(51 55)(57 61)(59 63)

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

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

50 conjugacy classes

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

50 irreducible representations

dim11111111124
type+++++++++
imageC1C2C2C2C2C2C2C2C4C4○D42+ 1+4
kernelC24.212C23C4×C22⋊C4C4×C4⋊C4C23.34D4C42⋊8C4C24.C22C24.3C22C2×C4×D4C4×D4C2×C4C22
# reps1232142116162

Matrix representation of C24.212C23 ►in GL5(𝔽5)

40000
01000
02400
00012
00004
,
10000
01000
00100
00040
00004
,
10000
04000
00400
00010
00001
,
40000
04000
00400
00010
00001
,
30000
02300
00300
00031
00022
,
10000
02000
00200
00024
00003
,
10000
01000
00100
00012
00044

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

C24.212C23 in GAP, Magma, Sage, TeX 

C_2^4._{212}C_2^3
 
% in TeX
 
G:=Group("C2^4.212C2^3");
 
// GroupNames label
 
G:=SmallGroup(128,1089);
 
// by ID
 
G=gap.SmallGroup(128,1089);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,344,758,268,346,80]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=1,e^2=d,f^2=c*b=b*c,g^2=b,g*a*g^-1=a*b=b*a,a*c=c*a,a*d=d*a,e*a*e^-1=a*b*c,a*f=f*a,b*d=d*b,f*e*f^-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,e*g=g*e>;
 
// generators/relations
 

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