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G = C24.414C23  order 128 = 27

254th non-split extension by C24 of C23 acting via C23/C2=C22

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

Aliases: C24.414C23, C23.615C24, C22.2902- (1+4), C22.3892+ (1+4), C22⋊C4.18D4, C23.72(C2×D4), C2.67(D4⋊6D4), C2.120(D4⋊5D4), C23.7Q8⋊97C2, C23.Q8⋊69C2, C23.11D4⋊96C2, (C23×C4).470C22, (C22×C4).193C23, (C2×C42).666C22, C23.8Q8⋊116C2, C22.424(C22×D4), C23.10D4.46C2, C23.23D4.55C2, (C22×D4).249C22, (C22×Q8).192C22, C24.C22⋊142C2, C23.67C23⋊87C2, C23.78C23⋊48C2, C2.73(C22.32C24), C23.65C23⋊128C2, C23.63C23⋊142C2, C2.C42.321C22, C2.51(C22.31C24), C2.18(C22.57C24), C2.80(C22.36C24), C2.56(C22.50C24), (C2×C4).426(C2×D4), (C2×C42⋊2C2)⋊23C2, (C2×C4).433(C4○D4), (C2×C4⋊C4).428C22, (C2×C4.4D4).32C2, C22.477(C2×C4○D4), (C2×C22⋊C4).280C22, SmallGroup(128,1447)

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

Derived series
C1 — C23 — C24.414C23
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C23.67C23 — C24.414C23
Lower central
C1 — C23 — C24.414C23
Upper central
C1 — C23 — C24.414C23
Jennings
C1 — C23 — C24.414C23

Subgroups: 500 in 249 conjugacy classes, 96 normal (82 characteristic)
C1, C2 [×7], C2 [×3], C4 [×17], C22 [×7], C22 [×17], C2×C4 [×8], C2×C4 [×39], D4 [×4], Q8 [×4], C23, C23 [×2], C23 [×13], C42 [×4], C22⋊C4 [×4], C22⋊C4 [×16], C4⋊C4 [×14], C22×C4 [×13], C22×C4 [×4], C2×D4 [×5], C2×Q8 [×5], C24 [×2], C2.C42 [×10], C2×C42 [×2], C2×C22⋊C4 [×11], C2×C4⋊C4 [×9], C4.4D4 [×4], C42⋊2C2 [×4], C23×C4, C22×D4, C22×Q8, C23.7Q8, C23.8Q8, C23.23D4, C23.63C23, C24.C22, C23.65C23, C23.67C23, C23.10D4, C23.78C23, C23.Q8 [×2], C23.11D4 [×2], C2×C4.4D4, C2×C42⋊2C2, C24.414C23

Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×4], C24, C22×D4, C2×C4○D4 [×2], 2+ (1+4) [×2], 2- (1+4) [×2], C22.31C24, C22.32C24, C22.36C24, D4⋊5D4, D4⋊6D4, C22.50C24, C22.57C24, C24.414C23

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

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

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

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

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

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

Matrix representation ►G ⊆ GL6(𝔽5)

120000
040000
004000
000100
000040
000004
,
400000
040000
001000
000100
000010
000001
,
100000
010000
004000
000400
000010
000001
,
100000
010000
001000
000100
000040
000004
,
120000
440000
004000
000400
000040
000001
,
300000
030000
000200
002000
000001
000010
,
300000
220000
000100
001000
000040
000004

G:=sub<GL(6,GF(5))| [1,0,0,0,0,0,2,4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[1,4,0,0,0,0,2,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,1],[3,0,0,0,0,0,0,3,0,0,0,0,0,0,0,2,0,0,0,0,2,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[3,2,0,0,0,0,0,2,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4] >;
 

32 conjugacy classes

class 1 2A···2G2H2I2J4A···4P4Q···4U
order12···22224···44···4
size11···14484···48···8

32 irreducible representations

dim111111111111112244
type++++++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2C2C2C2C2D4C4○D42+ (1+4)2- (1+4)
kernelC24.414C23C23.7Q8C23.8Q8C23.23D4C23.63C23C24.C22C23.65C23C23.67C23C23.10D4C23.78C23C23.Q8C23.11D4C2×C4.4D4C2×C42⋊2C2C22⋊C4C2×C4C22C22
# reps111111111122114822

In GAP, Magma, Sage, TeX 

C_2^4._{414}C_2^3
 
% in TeX
 
G:=Group("C2^4.414C2^3");
 
// GroupNames label
 
G:=SmallGroup(128,1447);
 
// by ID
 
G=gap.SmallGroup(128,1447);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,253,120,758,723,268,1571,346,80]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=1,e^2=g^2=b,f^2=c*b=b*c,e*a*e^-1=a*b=b*a,f*a*f^-1=a*c=c*a,a*d=d*a,g*a*g^-1=a*b*c,b*d=d*b,g*e*g^-1=b*e=e*b,b*f=f*b,b*g=g*b,c*d=d*c,c*e=e*c,c*f=f*c,c*g=g*c,f*e*f^-1=d*e=e*d,d*f=f*d,d*g=g*d,f*g=g*f>;
 
// generators/relations
 

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