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G = C24.389C23order 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, C2.46D42, C4⋊C411D4, (C2×D4)⋊17D4, C232D439C2, C23.208(C2×D4), C2.58(D46D4), C2.28(Q86D4), C23.Q853C2, C23.8Q899C2, C23.10D478C2, C2.41(C233D4), (C2×C42).639C22, (C23×C4).450C22, (C22×C4).178C23, C22.391(C22×D4), C24.3C2276C2, (C22×D4).221C22, C2.60(C22.32C24), C23.65C23116C2, 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

C1C23 — C24.389C23
C1C2C22C23C24C23×C4C2×C22.D4 — C24.389C23
C1C23 — C24.389C23
C1C23 — C24.389C23
C1C23 — C24.389C23

Generators and relations for C24.389C23
 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 >

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], C232D4 [×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, C233D4, C22.31C24, C22.32C24, D42, D46D4, Q86D4, C22.56C24, C24.389C23

Smallest permutation representation of C24.389C23
On 64 points
Generators in S64
(1 49)(2 50)(3 51)(4 52)(5 56)(6 53)(7 54)(8 55)(9 21)(10 22)(11 23)(12 24)(13 47)(14 48)(15 45)(16 46)(17 43)(18 44)(19 41)(20 42)(25 38)(26 39)(27 40)(28 37)(29 61)(30 62)(31 63)(32 64)(33 60)(34 57)(35 58)(36 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 9)(2 10)(3 11)(4 12)(5 39)(6 40)(7 37)(8 38)(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 55)(26 56)(27 53)(28 54)(29 59)(30 60)(31 57)(32 58)(33 62)(34 63)(35 64)(36 61)
(1 53)(2 54)(3 55)(4 56)(5 52)(6 49)(7 50)(8 51)(9 27)(10 28)(11 25)(12 26)(13 31)(14 32)(15 29)(16 30)(17 36)(18 33)(19 34)(20 35)(21 40)(22 37)(23 38)(24 39)(41 57)(42 58)(43 59)(44 60)(45 61)(46 62)(47 63)(48 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 47 3 45)(2 64 4 62)(5 44 7 42)(6 57 8 59)(9 19 11 17)(10 35 12 33)(13 23 15 21)(14 39 16 37)(18 28 20 26)(22 32 24 30)(25 36 27 34)(29 40 31 38)(41 51 43 49)(46 54 48 56)(50 58 52 60)(53 63 55 61)
(1 45 11 19)(2 48 12 18)(3 47 9 17)(4 46 10 20)(5 58 37 30)(6 57 38 29)(7 60 39 32)(8 59 40 31)(13 51 43 21)(14 50 44 24)(15 49 41 23)(16 52 42 22)(25 34 53 61)(26 33 54 64)(27 36 55 63)(28 35 56 62)

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

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

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

32 conjugacy classes

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

32 irreducible representations

dim11111111122244
type++++++++++++-
imageC1C2C2C2C2C2C2C2C2D4D4C4○D42+ 1+42- 1+4
kernelC24.389C23C23.8Q8C23.65C23C24.3C22C232D4C23.10D4C23.Q8C2×C4⋊D4C2×C22.D4C4⋊C4C2×D4C2×C4C22C22
# reps12122222244431

Matrix representation of C24.389C23 in 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] >;

C24.389C23 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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