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G = C24.456C23order 128 = 27

296th non-split extension by C24 of C23 acting via C23/C2=C22

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

Aliases: C24.456C23, C23.704C24, C22.4772+ 1+4, (C2×D4)⋊8Q8, C23.46(C2×Q8), C23⋊Q858C2, C2.67(D43Q8), (C22×C4).220C23, (C23×C4).178C22, (C2×C42).724C22, C23.8Q8141C2, C2.20(C232Q8), C22.166(C22×Q8), C23.23D4.77C2, (C22×D4).289C22, (C22×Q8).225C22, C2.13(C24⋊C22), C24.3C22.78C2, C23.67C23104C2, C23.83C23128C2, C2.109(C22.32C24), C2.C42.408C22, C2.47(C22.53C24), (C2×C4).90(C2×Q8), (C2×C4).245(C4○D4), (C2×C4⋊C4).514C22, C22.565(C2×C4○D4), (C2×C22⋊C4).329C22, SmallGroup(128,1536)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C24.456C23
C1C2C22C23C22×C4C2×C42C24.3C22 — C24.456C23
C1C23 — C24.456C23
C1C23 — C24.456C23
C1C23 — C24.456C23

Generators and relations for C24.456C23
 G = < a,b,c,d,e,f,g | a2=b2=c2=e2=1, d2=abc, f2=cb=bc, g2=ba=ab, ac=ca, ede=ad=da, geg-1=ae=ea, af=fa, ag=ga, fdf-1=bd=db, be=eb, bf=fb, bg=gb, cd=dc, fef-1=ce=ec, cf=fc, cg=gc, gdg-1=abd, fg=gf >

Subgroups: 516 in 240 conjugacy classes, 96 normal (14 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×16], C22 [×3], C22 [×4], C22 [×20], C2×C4 [×6], C2×C4 [×44], D4 [×4], Q8 [×8], C23, C23 [×4], C23 [×12], C42, C22⋊C4 [×14], C4⋊C4 [×5], C22×C4, C22×C4 [×12], C22×C4 [×6], C2×D4 [×4], C2×D4 [×2], C2×Q8 [×8], C24 [×2], C2.C42 [×14], C2×C42, C2×C22⋊C4 [×10], C2×C4⋊C4, C2×C4⋊C4 [×4], C23×C4 [×2], C22×D4, C22×Q8 [×2], C23.8Q8 [×4], C23.23D4 [×2], C24.3C22, C23.67C23 [×2], C23⋊Q8 [×4], C23.83C23 [×2], C24.456C23
Quotients: C1, C2 [×15], C22 [×35], Q8 [×4], C23 [×15], C2×Q8 [×6], C4○D4 [×4], C24, C22×Q8, C2×C4○D4 [×2], 2+ 1+4 [×4], C22.32C24 [×2], C232Q8, D43Q8 [×2], C22.53C24, C24⋊C22, C24.456C23

Smallest permutation representation of C24.456C23
On 64 points
Generators in S64
(1 55)(2 56)(3 53)(4 54)(5 61)(6 62)(7 63)(8 64)(9 43)(10 44)(11 41)(12 42)(13 45)(14 46)(15 47)(16 48)(17 49)(18 50)(19 51)(20 52)(21 58)(22 59)(23 60)(24 57)(25 32)(26 29)(27 30)(28 31)(33 40)(34 37)(35 38)(36 39)
(1 44)(2 41)(3 42)(4 43)(5 52)(6 49)(7 50)(8 51)(9 54)(10 55)(11 56)(12 53)(13 31)(14 32)(15 29)(16 30)(17 62)(18 63)(19 64)(20 61)(21 39)(22 40)(23 37)(24 38)(25 46)(26 47)(27 48)(28 45)(33 59)(34 60)(35 57)(36 58)
(1 12)(2 9)(3 10)(4 11)(5 18)(6 19)(7 20)(8 17)(13 26)(14 27)(15 28)(16 25)(21 34)(22 35)(23 36)(24 33)(29 45)(30 46)(31 47)(32 48)(37 58)(38 59)(39 60)(40 57)(41 54)(42 55)(43 56)(44 53)(49 64)(50 61)(51 62)(52 63)
(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 39)(2 33)(3 37)(4 35)(5 15)(6 48)(7 13)(8 46)(9 24)(10 58)(11 22)(12 60)(14 64)(16 62)(17 30)(18 28)(19 32)(20 26)(21 44)(23 42)(25 51)(27 49)(29 52)(31 50)(34 53)(36 55)(38 54)(40 56)(41 59)(43 57)(45 63)(47 61)
(1 61 53 7)(2 17 54 51)(3 63 55 5)(4 19 56 49)(6 43 64 11)(8 41 62 9)(10 52 42 18)(12 50 44 20)(13 60 47 21)(14 35 48 40)(15 58 45 23)(16 33 46 38)(22 32 57 27)(24 30 59 25)(26 39 31 34)(28 37 29 36)
(1 45 10 31)(2 32 11 46)(3 47 12 29)(4 30 9 48)(5 60 20 37)(6 38 17 57)(7 58 18 39)(8 40 19 59)(13 44 28 55)(14 56 25 41)(15 42 26 53)(16 54 27 43)(21 50 36 63)(22 64 33 51)(23 52 34 61)(24 62 35 49)

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

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

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

32 conjugacy classes

class 1 2A···2G2H2I2J2K4A···4N4O···4T
order12···222224···44···4
size11···144444···48···8

32 irreducible representations

dim1111111224
type+++++++-+
imageC1C2C2C2C2C2C2Q8C4○D42+ 1+4
kernelC24.456C23C23.8Q8C23.23D4C24.3C22C23.67C23C23⋊Q8C23.83C23C2×D4C2×C4C22
# reps1421242484

Matrix representation of C24.456C23 in GL6(𝔽5)

400000
040000
001000
000100
000010
000001
,
100000
010000
001000
000100
000040
000004
,
100000
010000
004000
000400
000010
000001
,
200000
030000
002000
000200
000003
000030
,
010000
100000
000400
004000
000040
000004
,
100000
010000
000100
004000
000001
000040
,
010000
400000
004000
000400
000001
000040

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

C24.456C23 in GAP, Magma, Sage, TeX

C_2^4._{456}C_2^3
% in TeX

G:=Group("C2^4.456C2^3");
// GroupNames label

G:=SmallGroup(128,1536);
// by ID

G=gap.SmallGroup(128,1536);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,784,253,758,723,520,1571,346,192]);
// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=e^2=1,d^2=a*b*c,f^2=c*b=b*c,g^2=b*a=a*b,a*c=c*a,e*d*e=a*d=d*a,g*e*g^-1=a*e=e*a,a*f=f*a,a*g=g*a,f*d*f^-1=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,g*d*g^-1=a*b*d,f*g=g*f>;
// generators/relations

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