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G = C24.434C23order 128 = 27

274th non-split extension by C24 of C23 acting via C23/C2=C22

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

Aliases: C24.434C23, C23.648C24, C22.4212+ 1+4, (C2×D4).18Q8, C23.39(C2×Q8), C2.55(D43Q8), C23.Q878C2, C23.4Q859C2, (C2×C42).689C22, (C23×C4).160C22, (C22×C4).206C23, C23.7Q8104C2, C23.8Q8125C2, C2.16(C232Q8), C22.153(C22×Q8), C23.23D4.65C2, (C22×D4).265C22, C24.3C22.69C2, C23.63C23164C2, C23.81C23111C2, C2.25(C22.54C24), C2.C42.352C22, C2.100(C22.45C24), C2.52(C22.34C24), (C2×C4).77(C2×Q8), (C2×C4).449(C4○D4), (C2×C4⋊C4).459C22, C22.509(C2×C4○D4), (C2×C22⋊C4).304C22, SmallGroup(128,1480)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C24.434C23
C1C2C22C23C22×C4C23×C4C23.7Q8 — C24.434C23
C1C23 — C24.434C23
C1C23 — C24.434C23
C1C23 — C24.434C23

Generators and relations for C24.434C23
 G = < a,b,c,d,e,f,g | a2=b2=c2=f2=1, d2=abc, e2=ba=ab, g2=a, ac=ca, 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: 484 in 232 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], C23, C23 [×4], C23 [×12], C42, C22⋊C4 [×12], C4⋊C4 [×13], C22×C4, C22×C4 [×12], C22×C4 [×8], C2×D4 [×4], C2×D4 [×2], C24 [×2], C2.C42 [×10], C2×C42, C2×C22⋊C4 [×2], C2×C22⋊C4 [×8], C2×C4⋊C4, C2×C4⋊C4 [×10], C23×C4 [×2], C22×D4, C23.7Q8 [×2], C23.8Q8 [×2], C23.23D4 [×2], C23.63C23 [×2], C24.3C22, C23.Q8 [×2], C23.81C23 [×2], C23.4Q8 [×2], C24.434C23
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], C22.34C24 [×2], C232Q8, C22.45C24, D43Q8 [×2], C22.54C24, C24.434C23

Smallest permutation representation of C24.434C23
On 64 points
Generators in S64
(1 61)(2 62)(3 63)(4 64)(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 32)(26 29)(27 30)(28 31)(33 57)(34 58)(35 59)(36 60)(37 44)(38 41)(39 42)(40 43)(45 50)(46 51)(47 52)(48 49)
(1 58)(2 59)(3 60)(4 57)(5 25)(6 26)(7 27)(8 28)(9 38)(10 39)(11 40)(12 37)(13 44)(14 41)(15 42)(16 43)(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 36)(2 33)(3 34)(4 35)(5 30)(6 31)(7 32)(8 29)(9 43)(10 44)(11 41)(12 42)(13 39)(14 40)(15 37)(16 38)(17 48)(18 45)(19 46)(20 47)(21 52)(22 49)(23 50)(24 51)(25 55)(26 56)(27 53)(28 54)(57 62)(58 63)(59 64)(60 61)
(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 22 34 51)(2 18 35 47)(3 24 36 49)(4 20 33 45)(5 13 32 37)(6 9 29 41)(7 15 30 39)(8 11 31 43)(10 27 42 55)(12 25 44 53)(14 26 38 54)(16 28 40 56)(17 58 46 61)(19 60 48 63)(21 57 50 64)(23 59 52 62)
(1 9)(2 39)(3 11)(4 37)(5 47)(6 24)(7 45)(8 22)(10 59)(12 57)(13 33)(14 61)(15 35)(16 63)(17 56)(18 32)(19 54)(20 30)(21 27)(23 25)(26 48)(28 46)(29 49)(31 51)(34 41)(36 43)(38 58)(40 60)(42 62)(44 64)(50 55)(52 53)
(1 25 61 32)(2 54 62 6)(3 27 63 30)(4 56 64 8)(5 34 53 58)(7 36 55 60)(9 23 14 18)(10 49 15 48)(11 21 16 20)(12 51 13 46)(17 44 22 37)(19 42 24 39)(26 59 29 35)(28 57 31 33)(38 47 41 52)(40 45 43 50)

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

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

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

32 conjugacy classes

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

32 irreducible representations

dim111111111224
type+++++++++-+
imageC1C2C2C2C2C2C2C2C2Q8C4○D42+ 1+4
kernelC24.434C23C23.7Q8C23.8Q8C23.23D4C23.63C23C24.3C22C23.Q8C23.81C23C23.4Q8C2×D4C2×C4C22
# reps122221222484

Matrix representation of C24.434C23 in GL6(𝔽5)

400000
040000
001000
000100
000010
000001
,
100000
010000
001000
000100
000040
000004
,
100000
010000
004000
000400
000010
000001
,
200000
030000
003000
000300
000020
000023
,
020000
200000
000200
003000
000030
000003
,
400000
040000
000100
001000
000042
000001
,
010000
400000
004000
000400
000042
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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[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],[2,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,2,2,0,0,0,0,0,3],[0,2,0,0,0,0,2,0,0,0,0,0,0,0,0,3,0,0,0,0,2,0,0,0,0,0,0,0,3,0,0,0,0,0,0,3],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,4,0,0,0,0,0,2,1],[0,4,0,0,0,0,1,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,2,1] >;

C24.434C23 in GAP, Magma, Sage, TeX

C_2^4._{434}C_2^3
% in TeX

G:=Group("C2^4.434C2^3");
// GroupNames label

G:=SmallGroup(128,1480);
// by ID

G=gap.SmallGroup(128,1480);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,784,253,344,758,723,1571,346,80]);
// Polycyclic

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