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G = C24.549C23order 128 = 27

30th non-split extension by C24 of C23 acting via C23/C22=C2

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

Aliases: C24.549C23, C23.221C24, C22.412- 1+4, C22.592+ 1+4, D47(C22⋊C4), (C2×D4).340D4, C2.2(D45D4), C2.1(D46D4), C23.412(C2×D4), D42(C2.C42), (C23×C4).52C22, C23.7Q820C2, C23.34D414C2, C23.23D411C2, C22.112(C23×C4), (C2×C42).424C22, C23.127(C22×C4), C22.100(C22×D4), (C22×C4).1242C23, (C22×D4).608C22, (C22×Q8).401C22, C23.67C2318C2, C2.C42.55C22, C2.22(C23.33C23), (C2×C4×D4)⋊10C2, (C2×C4○D4)⋊15C4, (C2×D4)⋊42(C2×C4), (C2×Q8)⋊34(C2×C4), C2.23(C4×C4○D4), (C2×C4)⋊16(C4○D4), (C4×C22⋊C4)⋊37C2, (C22×C4)⋊29(C2×C4), C4.25(C2×C22⋊C4), (C2×C4).1066(C2×D4), (C22×C4○D4).8C2, C22.1(C2×C22⋊C4), (C2×C4⋊C4).815C22, (C2×C4).489(C22×C4), C22.106(C2×C4○D4), C2.17(C22×C22⋊C4), (C2×C2.C42)⋊18C2, (C2×D4)2(C2.C42), (C2×C22⋊C4).433C22, SmallGroup(128,1071)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C24.549C23
C1C2C22C23C24C23×C4C22×C4○D4 — C24.549C23
C1C22 — C24.549C23
C1C23 — C24.549C23
C1C23 — C24.549C23

Generators and relations for C24.549C23
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=f2=1, e2=d, g2=b, faf=ab=ba, ac=ca, ad=da, ae=ea, ag=ga, bc=cb, bd=db, fef=be=eb, bf=fb, bg=gb, cd=dc, geg-1=ce=ec, cf=fc, cg=gc, de=ed, df=fd, dg=gd, fg=gf >

Subgroups: 812 in 464 conjugacy classes, 184 normal (20 characteristic)
C1, C2 [×3], C2 [×4], C2 [×10], C4 [×4], C4 [×16], C22 [×3], C22 [×12], C22 [×34], C2×C4 [×16], C2×C4 [×68], D4 [×16], D4 [×16], Q8 [×8], C23, C23 [×14], C23 [×14], C42 [×4], C22⋊C4 [×14], C4⋊C4 [×4], C22×C4 [×2], C22×C4 [×22], C22×C4 [×32], C2×D4 [×20], C2×D4 [×8], C2×Q8 [×4], C2×Q8 [×4], C4○D4 [×32], C24, C24 [×2], C2.C42 [×2], C2.C42 [×10], C2×C42 [×2], C2×C22⋊C4 [×8], C2×C4⋊C4 [×2], C4×D4 [×8], C23×C4, C23×C4 [×6], C22×D4, C22×D4 [×2], C22×Q8, C2×C4○D4 [×8], C2×C4○D4 [×8], C2×C2.C42 [×2], C4×C22⋊C4, C23.7Q8, C23.34D4 [×2], C23.23D4 [×4], C23.67C23 [×2], C2×C4×D4 [×2], C22×C4○D4, C24.549C23
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], D4 [×8], C23 [×15], C22⋊C4 [×16], C22×C4 [×14], C2×D4 [×12], C4○D4 [×4], C24, C2×C22⋊C4 [×12], C23×C4, C22×D4 [×2], C2×C4○D4 [×2], 2+ 1+4, 2- 1+4, C22×C22⋊C4, C4×C4○D4, C23.33C23, D45D4 [×2], D46D4 [×2], C24.549C23

Smallest permutation representation of C24.549C23
On 64 points
Generators in S64
(1 41)(2 42)(3 43)(4 44)(5 48)(6 45)(7 46)(8 47)(9 13)(10 14)(11 15)(12 16)(17 39)(18 40)(19 37)(20 38)(21 25)(22 26)(23 27)(24 28)(29 35)(30 36)(31 33)(32 34)(49 53)(50 54)(51 55)(52 56)(57 61)(58 62)(59 63)(60 64)
(1 9)(2 10)(3 11)(4 12)(5 38)(6 39)(7 40)(8 37)(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 53)(26 54)(27 55)(28 56)(29 57)(30 58)(31 59)(32 60)(33 63)(34 64)(35 61)(36 62)
(1 51)(2 52)(3 49)(4 50)(5 62)(6 63)(7 64)(8 61)(9 23)(10 24)(11 21)(12 22)(13 27)(14 28)(15 25)(16 26)(17 31)(18 32)(19 29)(20 30)(33 39)(34 40)(35 37)(36 38)(41 55)(42 56)(43 53)(44 54)(45 59)(46 60)(47 57)(48 58)
(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 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 6)(2 40)(3 8)(4 38)(5 12)(7 10)(9 39)(11 37)(13 45)(14 18)(15 47)(16 20)(17 41)(19 43)(21 35)(22 62)(23 33)(24 64)(25 57)(26 30)(27 59)(28 32)(29 53)(31 55)(34 52)(36 50)(42 46)(44 48)(49 61)(51 63)(54 58)(56 60)
(1 17 9 45)(2 32 10 60)(3 19 11 47)(4 30 12 58)(5 54 38 26)(6 41 39 13)(7 56 40 28)(8 43 37 15)(14 64 42 34)(16 62 44 36)(18 24 46 52)(20 22 48 50)(21 57 49 29)(23 59 51 31)(25 61 53 35)(27 63 55 33)

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

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

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

50 conjugacy classes

class 1 2A···2G2H···2O2P2Q4A···4P4Q···4AF
order12···22···2224···44···4
size11···12···2442···24···4

50 irreducible representations

dim11111111112244
type+++++++++++-
imageC1C2C2C2C2C2C2C2C2C4D4C4○D42+ 1+42- 1+4
kernelC24.549C23C2×C2.C42C4×C22⋊C4C23.7Q8C23.34D4C23.23D4C23.67C23C2×C4×D4C22×C4○D4C2×C4○D4C2×D4C2×C4C22C22
# reps121124221168811

Matrix representation of C24.549C23 in GL6(𝔽5)

400000
040000
001000
000100
000044
000001
,
100000
010000
001000
000100
000040
000004
,
400000
040000
004000
000400
000010
000001
,
400000
040000
001000
000100
000040
000004
,
010000
400000
000100
001000
000033
000002
,
100000
010000
001000
000100
000033
000042
,
400000
010000
001000
000400
000030
000003

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

C24.549C23 in GAP, Magma, Sage, TeX

C_2^4._{549}C_2^3
% in TeX

G:=Group("C2^4.549C2^3");
// GroupNames label

G:=SmallGroup(128,1071);
// by ID

G=gap.SmallGroup(128,1071);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,758,100,346]);
// Polycyclic

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

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