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G = C23.430C24order 128 = 27

147th central stem extension by C23 of C24

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

Aliases: C23.430C24, C24.316C23, C22.2212+ 1+4, C22.1702- 1+4, C424C423C2, C23.47(C4○D4), (C22×C4).90C23, (C2×C42).59C22, (C23×C4).387C22, C23.7Q8.51C2, C23.8Q8.28C2, C23.11D4.17C2, C23.63C2381C2, C23.83C2335C2, C23.65C2383C2, C24.C22.29C2, C2.42(C22.45C24), C2.C42.176C22, C2.73(C23.36C23), C2.57(C22.46C24), C2.50(C22.47C24), C2.42(C22.36C24), (C4×C22⋊C4).60C2, (C2×C4).382(C4○D4), (C2×C4⋊C4).292C22, C22.307(C2×C4○D4), (C2×C22⋊C4).169C22, SmallGroup(128,1262)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C23.430C24
C1C2C22C23C22×C4C23×C4C4×C22⋊C4 — C23.430C24
C1C23 — C23.430C24
C1C23 — C23.430C24
C1C23 — C23.430C24

Generators and relations for C23.430C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=f2=1, d2=c, e2=b, g2=ba=ab, ac=ca, ede-1=gdg-1=ad=da, 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, eg=ge, fg=gf >

Subgroups: 372 in 206 conjugacy classes, 92 normal (82 characteristic)
C1, C2 [×7], C2 [×2], C4 [×18], C22 [×7], C22 [×10], C2×C4 [×8], C2×C4 [×42], C23, C23 [×2], C23 [×6], C42 [×7], C22⋊C4 [×11], C4⋊C4 [×13], C22×C4 [×14], C22×C4 [×5], C24, C2.C42 [×16], C2×C42 [×4], C2×C22⋊C4 [×6], C2×C4⋊C4 [×8], C23×C4, C424C4, C4×C22⋊C4, C23.7Q8, C23.8Q8, C23.63C23 [×4], C24.C22 [×2], C23.65C23, C23.11D4 [×2], C23.83C23 [×2], C23.430C24
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], C4○D4 [×10], C24, C2×C4○D4 [×5], 2+ 1+4, 2- 1+4, C23.36C23 [×2], C22.36C24, C22.45C24, C22.46C24 [×2], C22.47C24, C23.430C24

Smallest permutation representation of C23.430C24
On 64 points
Generators in S64
(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 59 51 45)(2 32 52 18)(3 57 49 47)(4 30 50 20)(5 16 62 26)(6 41 63 55)(7 14 64 28)(8 43 61 53)(9 31 23 17)(10 60 24 46)(11 29 21 19)(12 58 22 48)(13 33 27 39)(15 35 25 37)(34 56 40 42)(36 54 38 44)
(2 52)(4 50)(5 64)(6 8)(7 62)(10 24)(12 22)(14 28)(16 26)(17 19)(18 30)(20 32)(29 31)(33 35)(34 38)(36 40)(37 39)(42 56)(44 54)(45 47)(46 58)(48 60)(57 59)(61 63)
(1 27 23 41)(2 56 24 14)(3 25 21 43)(4 54 22 16)(5 20 36 58)(6 45 33 31)(7 18 34 60)(8 47 35 29)(9 55 51 13)(10 28 52 42)(11 53 49 15)(12 26 50 44)(17 63 59 39)(19 61 57 37)(30 38 48 62)(32 40 46 64)

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

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

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

38 conjugacy classes

class 1 2A···2G2H2I4A···4H4I···4X4Y4Z4AA4AB
order12···2224···44···44444
size11···1442···24···48888

38 irreducible representations

dim11111111112244
type+++++++++++-
imageC1C2C2C2C2C2C2C2C2C2C4○D4C4○D42+ 1+42- 1+4
kernelC23.430C24C424C4C4×C22⋊C4C23.7Q8C23.8Q8C23.63C23C24.C22C23.65C23C23.11D4C23.83C23C2×C4C23C22C22
# reps111114212216411

Matrix representation of C23.430C24 in GL6(𝔽5)

100000
010000
004000
000400
000010
000001
,
400000
040000
001000
000100
000010
000001
,
100000
010000
001000
000100
000040
000004
,
010000
100000
004200
000100
000030
000003
,
200000
020000
003400
003200
000001
000010
,
100000
040000
001000
000100
000010
000004
,
300000
030000
004200
004100
000040
000004

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

C23.430C24 in GAP, Magma, Sage, TeX

C_2^3._{430}C_2^4
% in TeX

G:=Group("C2^3.430C2^4");
// GroupNames label

G:=SmallGroup(128,1262);
// by ID

G=gap.SmallGroup(128,1262);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,232,758,723,675,192]);
// Polycyclic

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

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