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G = C24.229C23  order 128 = 27

69th non-split extension by C24 of C23 acting via C23/C2=C22

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

Aliases: C24.229C23, C23.262C24, C22.932+ (1+4), C4⋊1D4⋊21C4, C42⋊32(C2×C4), C42⋊5C4⋊8C2, (C23×C4).63C22, C23.30(C22×C4), C23.23D4⋊19C2, C22.153(C23×C4), (C22×C4).490C23, (C2×C42).450C22, (C22×D4).116C22, C2.4(C22.54C24), C2.42(C22.11C24), C2.C42.70C22, (C2×D4)⋊23(C2×C4), (C2×C4⋊1D4).14C2, (C2×C4).238(C22×C4), (C2×C22⋊C4).44C22, SmallGroup(128,1112)

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

Derived series
C1 — C22 — C24.229C23
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C2×C4⋊1D4 — C24.229C23
Lower central
C1 — C22 — C24.229C23
Upper central
C1 — C23 — C24.229C23
Jennings
C1 — C23 — C24.229C23

Subgroups: 732 in 324 conjugacy classes, 132 normal (6 characteristic)
C1, C2 [×7], C2 [×8], C4 [×14], C22, C22 [×6], C22 [×40], C2×C4 [×6], C2×C4 [×46], D4 [×24], C23, C23 [×8], C23 [×24], C42 [×4], C22⋊C4 [×12], C22×C4 [×11], C22×C4 [×12], C2×D4 [×24], C2×D4 [×12], C24 [×4], C2.C42 [×12], C2×C42, C2×C22⋊C4 [×12], C4⋊1D4 [×8], C23×C4 [×4], C22×D4 [×6], C42⋊5C4 [×2], C23.23D4 [×12], C2×C4⋊1D4, C24.229C23

Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], C22×C4 [×14], C24, C23×C4, 2+ (1+4) [×6], C22.11C24 [×3], C22.54C24 [×4], C24.229C23

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

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

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

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

Matrix representation ►G ⊆ GL9(𝔽5)

100000000
040000000
041000000
020100000
000040000
000001412
000000010
000000100
000000004
,
100000000
040000000
004000000
000400000
000040000
000001000
000000100
000000010
000000001
,
100000000
010000000
001000000
000100000
000010000
000004000
000000400
000000040
000000004
,
400000000
040000000
004000000
000400000
000040000
000001000
000000100
000000010
000000001
,
200000000
030300000
000410000
000200000
004200000
000004000
000000100
000000010
000000144
,
100000000
013000000
014000000
002010000
032400000
000000010
000001412
000004000
000001401
,
400000000
013000000
014000000
032040000
002100000
000000100
000001000
000001412
000000004

G:=sub<GL(9,GF(5))| [1,0,0,0,0,0,0,0,0,0,4,4,2,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,1,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,2,0,0,4],[1,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1],[2,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,3,4,2,2,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,1,0,1,0,0,0,0,0,0,0,1,4,0,0,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,0,0,0,1,1,0,3,0,0,0,0,0,3,4,2,2,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,4,1,0,0,0,0,0,0,4,0,4,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,2,0,1],[4,0,0,0,0,0,0,0,0,0,1,1,3,0,0,0,0,0,0,3,4,2,2,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,1,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,4] >;
 

38 conjugacy classes

class 1 2A···2G2H···2O4A···4V
order12···22···24···4
size11···14···44···4

38 irreducible representations

dim111114
type+++++
imageC1C2C2C2C42+ (1+4)
kernelC24.229C23C42⋊5C4C23.23D4C2×C4⋊1D4C4⋊1D4C22
# reps12121166

In GAP, Magma, Sage, TeX 

C_2^4._{229}C_2^3
 
% in TeX
 
G:=Group("C2^4.229C2^3");
 
// GroupNames label
 
G:=SmallGroup(128,1112);
 
// by ID
 
G=gap.SmallGroup(128,1112);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,758,555,268,1571,346,80]);
 
// 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,e*a*e^-1=g*a*g^-1=a*b=b*a,a*c=c*a,a*d=d*a,f*a*f^-1=a*b*c,b*d=d*b,b*e=e*b,b*f=f*b,b*g=g*b,c*d=d*c,f*e*f^-1=c*e=e*c,c*f=f*c,c*g=g*c,d*e=e*d,d*f=f*d,d*g=g*d,g*e*g^-1=b*c*e,f*g=g*f>;
 
// generators/relations
 

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