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G = C24.389C23  order 128 = 27

229th non-split extension by C24 of C23 acting via C23/C2=C22

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

Aliases: C24.389C23, C23.582C24, C22.2652- (1+4), C22.3562+ (1+4), C4⋊C4⋊11D4, (C2×D4)⋊17D4, C2.46(D42), C23⋊2D4⋊39C2, C23.208(C2×D4), C2.58(D4⋊6D4), C2.28(Q8⋊6D4), C23.Q8⋊53C2, C23.8Q8⋊99C2, C23.10D4⋊78C2, C2.41(C23⋊3D4), (C2×C42).639C22, (C23×C4).450C22, (C22×C4).178C23, C22.391(C22×D4), C24.3C22⋊76C2, (C22×D4).221C22, C2.60(C22.32C24), C23.65C23⋊116C2, C2.C42.291C22, C2.10(C22.56C24), C2.41(C22.31C24), (C2×C4).90(C2×D4), (C2×C4⋊D4)⋊34C2, (C2×C4).190(C4○D4), (C2×C4⋊C4).398C22, C22.445(C2×C4○D4), (C2×C22.D4)⋊33C2, (C2×C22⋊C4).251C22, SmallGroup(128,1414)

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

Derived series
C1 — C23 — C24.389C23
Chief series
C1 — C2 — C22 — C23 — C24 — C23×C4 — C2×C22.D4 — C24.389C23
Lower central
C1 — C23 — C24.389C23
Upper central
C1 — C23 — C24.389C23
Jennings
C1 — C23 — C24.389C23

Subgroups: 740 in 334 conjugacy classes, 104 normal (30 characteristic)
C1, C2 [×7], C2 [×6], C4 [×16], C22 [×7], C22 [×34], C2×C4 [×10], C2×C4 [×36], D4 [×24], C23, C23 [×4], C23 [×26], C42, C22⋊C4 [×26], C4⋊C4 [×4], C4⋊C4 [×11], C22×C4 [×3], C22×C4 [×8], C22×C4 [×10], C2×D4 [×4], C2×D4 [×24], C24 [×4], C2.C42 [×4], C2×C42, C2×C22⋊C4 [×14], C2×C4⋊C4 [×4], C2×C4⋊C4 [×4], C4⋊D4 [×8], C22.D4 [×8], C23×C4 [×2], C22×D4 [×2], C22×D4 [×4], C23.8Q8 [×2], C23.65C23, C24.3C22 [×2], C23⋊2D4 [×2], C23.10D4 [×2], C23.Q8 [×2], C2×C4⋊D4 [×2], C2×C22.D4 [×2], C24.389C23

Quotients:
C1, C2 [×15], C22 [×35], D4 [×8], C23 [×15], C2×D4 [×12], C4○D4 [×2], C24, C22×D4 [×2], C2×C4○D4, 2+ (1+4) [×3], 2- (1+4), C23⋊3D4, C22.31C24, C22.32C24, D42, D4⋊6D4, Q8⋊6D4, C22.56C24, C24.389C23

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

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

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

Matrix representation ►G ⊆ GL6(𝔽5)

020000
300000
001000
000100
000010
000004
,
400000
040000
001000
000100
000010
000001
,
100000
010000
001000
000100
000040
000004
,
100000
010000
004000
000400
000010
000001
,
040000
100000
000100
001000
000040
000004
,
200000
020000
001000
000400
000001
000010
,
200000
030000
001000
000100
000004
000010

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

32 conjugacy classes

class 1 2A···2G2H2I2J2K2L2M4A···4N4O4P4Q4R
order12···22222224···44444
size11···14444884···48888

32 irreducible representations

dim11111111122244
type++++++++++++-
imageC1C2C2C2C2C2C2C2C2D4D4C4○D42+ (1+4)2- (1+4)
kernelC24.389C23C23.8Q8C23.65C23C24.3C22C23⋊2D4C23.10D4C23.Q8C2×C4⋊D4C2×C22.D4C4⋊C4C2×D4C2×C4C22C22
# reps12122222244431

In GAP, Magma, Sage, TeX 

C_2^4._{389}C_2^3
 
% in TeX
 
G:=Group("C2^4.389C2^3");
 
// GroupNames label
 
G:=SmallGroup(128,1414);
 
// by ID
 
G=gap.SmallGroup(128,1414);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,336,253,120,758,723,100,1571,346]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=1,e^2=f^2=b,g^2=c*b=b*c,a*b=b*a,f*a*f^-1=a*c=c*a,a*d=d*a,a*e=e*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,g*f*g^-1=c*f=f*c,c*g=g*c,f*e*f^-1=d*e=e*d,d*f=f*d,d*g=g*d>;
 
// generators/relations
 

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