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G = C24.450C23order 128 = 27

290th non-split extension by C24 of C23 acting via C23/C2=C22

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

Aliases: C24.450C23, C23.684C24, C22.4572+ 1+4, (C2×D4)⋊7Q8, C23.43(C2×Q8), C23⋊Q856C2, C2.61(D43Q8), C23.Q886C2, (C23×C4).494C22, (C2×C42).711C22, (C22×C4).214C23, C23.8Q8137C2, C23.7Q8111C2, C2.18(C232Q8), C22.159(C22×Q8), C23.23D4.74C2, (C22×D4).280C22, (C22×Q8).218C22, C23.78C2359C2, C24.3C22.74C2, C23.83C23120C2, C23.63C23184C2, C2.102(C22.32C24), C2.33(C22.54C24), C2.C42.388C22, C2.45(C22.49C24), (C2×C4).83(C2×Q8), (C2×C4).470(C4○D4), (C2×C4⋊C4).494C22, C22.545(C2×C4○D4), (C2×C22⋊C4).320C22, SmallGroup(128,1516)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C24.450C23
C1C2C22C23C22×C4C23×C4C23.8Q8 — C24.450C23
C1C23 — C24.450C23
C1C23 — C24.450C23
C1C23 — C24.450C23

Generators and relations for C24.450C23
 G = < a,b,c,d,e,f,g | a2=b2=c2=f2=1, d2=ca=ac, e2=g2=ba=ab, ede-1=ad=da, geg-1=ae=ea, af=fa, ag=ga, bc=cb, fdf=bd=db, be=eb, bf=fb, bg=gb, cd=dc, fef=ce=ec, cf=fc, cg=gc, gdg-1=abd, fg=gf >

Subgroups: 500 in 236 conjugacy classes, 96 normal (22 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 [×4], C23, C23 [×4], C23 [×12], C42, C22⋊C4 [×12], C4⋊C4 [×10], C22×C4 [×3], C22×C4 [×10], C22×C4 [×8], C2×D4 [×4], C2×D4 [×2], C2×Q8 [×3], C24 [×2], C2.C42 [×2], C2.C42 [×10], C2×C42, C2×C22⋊C4 [×10], C2×C4⋊C4 [×2], C2×C4⋊C4 [×6], C23×C4 [×2], C22×D4, C22×Q8, C23.7Q8 [×2], C23.8Q8 [×2], C23.23D4 [×2], C23.63C23 [×2], C24.3C22, C23⋊Q8 [×2], C23.78C23, C23.Q8 [×2], C23.83C23, C24.450C23
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], C22.49C24, C22.54C24, C24.450C23

Smallest permutation representation of C24.450C23
On 64 points
Generators in S64
(1 36)(2 33)(3 34)(4 35)(5 55)(6 56)(7 53)(8 54)(9 16)(10 13)(11 14)(12 15)(17 24)(18 21)(19 22)(20 23)(25 32)(26 29)(27 30)(28 31)(37 44)(38 41)(39 42)(40 43)(45 52)(46 49)(47 50)(48 51)(57 62)(58 63)(59 64)(60 61)
(1 58)(2 59)(3 60)(4 57)(5 27)(6 28)(7 25)(8 26)(9 43)(10 44)(11 41)(12 42)(13 37)(14 38)(15 39)(16 40)(17 51)(18 52)(19 49)(20 50)(21 45)(22 46)(23 47)(24 48)(29 54)(30 55)(31 56)(32 53)(33 64)(34 61)(35 62)(36 63)
(1 34)(2 35)(3 36)(4 33)(5 53)(6 54)(7 55)(8 56)(9 14)(10 15)(11 16)(12 13)(17 22)(18 23)(19 24)(20 21)(25 30)(26 31)(27 32)(28 29)(37 42)(38 43)(39 44)(40 41)(45 50)(46 51)(47 52)(48 49)(57 64)(58 61)(59 62)(60 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 46 63 19)(2 50 64 23)(3 48 61 17)(4 52 62 21)(5 39 30 12)(6 43 31 16)(7 37 32 10)(8 41 29 14)(9 56 40 28)(11 54 38 26)(13 53 44 25)(15 55 42 27)(18 35 45 57)(20 33 47 59)(22 36 49 58)(24 34 51 60)
(1 11)(2 42)(3 9)(4 44)(5 21)(6 46)(7 23)(8 48)(10 57)(12 59)(13 62)(14 36)(15 64)(16 34)(17 29)(18 55)(19 31)(20 53)(22 28)(24 26)(25 47)(27 45)(30 52)(32 50)(33 39)(35 37)(38 63)(40 61)(41 58)(43 60)(49 56)(51 54)
(1 7 63 32)(2 29 64 8)(3 5 61 30)(4 31 62 6)(9 21 40 52)(10 49 37 22)(11 23 38 50)(12 51 39 24)(13 46 44 19)(14 20 41 47)(15 48 42 17)(16 18 43 45)(25 36 53 58)(26 59 54 33)(27 34 55 60)(28 57 56 35)

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

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

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

32 conjugacy classes

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

32 irreducible representations

dim1111111111224
type++++++++++-+
imageC1C2C2C2C2C2C2C2C2C2Q8C4○D42+ 1+4
kernelC24.450C23C23.7Q8C23.8Q8C23.23D4C23.63C23C24.3C22C23⋊Q8C23.78C23C23.Q8C23.83C23C2×D4C2×C4C22
# reps1222212121484

Matrix representation of C24.450C23 in GL6(𝔽5)

400000
040000
001000
000100
000010
000001
,
100000
010000
004000
000400
000010
000001
,
100000
010000
001000
000100
000040
000004
,
030000
300000
001000
000400
000020
000002
,
300000
020000
003000
000300
000002
000030
,
100000
010000
000300
002000
000001
000010
,
010000
400000
000100
004000
000010
000001

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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[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],[0,3,0,0,0,0,3,0,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,2,0,0,0,0,0,0,2],[3,0,0,0,0,0,0,2,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,0,3,0,0,0,0,2,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,2,0,0,0,0,3,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[0,4,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

C24.450C23 in GAP, Magma, Sage, TeX

C_2^4._{450}C_2^3
% in TeX

G:=Group("C2^4.450C2^3");
// GroupNames label

G:=SmallGroup(128,1516);
// by ID

G=gap.SmallGroup(128,1516);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,560,253,344,758,723,1571,346,192]);
// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=f^2=1,d^2=c*a=a*c,e^2=g^2=b*a=a*b,e*d*e^-1=a*d=d*a,g*e*g^-1=a*e=e*a,a*f=f*a,a*g=g*a,b*c=c*b,f*d*f=b*d=d*b,b*e=e*b,b*f=f*b,b*g=g*b,c*d=d*c,f*e*f=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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