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G = C23.364C24order 128 = 27

81st central stem extension by C23 of C24

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

Aliases: C23.364C24, C24.284C23, C22.1702+ 1+4, C2.25D42, C22⋊C424D4, C232D415C2, C23.173(C2×D4), C2.47(D45D4), C23.33(C4○D4), (C23×C4).89C22, C23.8Q851C2, C23.10D431C2, C23.23D446C2, (C22×C4).817C23, (C2×C42).507C22, C22.244(C22×D4), C24.C2248C2, C24.3C2242C2, (C22×D4).136C22, C23.81C2316C2, C2.36(C22.19C24), C2.C42.121C22, C2.20(C22.26C24), C2.23(C22.47C24), C2.13(C22.34C24), (C2×C4×D4)⋊39C2, (C2×C4).56(C2×D4), (C2×C4⋊D4)⋊13C2, (C4×C22⋊C4)⋊64C2, (C2×C4).114(C4○D4), (C2×C4⋊C4).850C22, C22.241(C2×C4○D4), (C2×C22.D4)⋊16C2, (C2×C22⋊C4).139C22, SmallGroup(128,1196)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C23.364C24
C1C2C22C23C22×C4C23×C4C4×C22⋊C4 — C23.364C24
C1C23 — C23.364C24
C1C23 — C23.364C24
C1C23 — C23.364C24

Generators and relations for C23.364C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=f2=1, e2=a, 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: 756 in 355 conjugacy classes, 108 normal (82 characteristic)
C1, C2 [×7], C2 [×7], C4 [×17], C22 [×7], C22 [×37], C2×C4 [×12], C2×C4 [×39], D4 [×28], C23, C23 [×6], C23 [×25], C42 [×3], C22⋊C4 [×8], C22⋊C4 [×19], C4⋊C4 [×10], C22×C4 [×11], C22×C4 [×15], C2×D4 [×31], C24 [×4], C2.C42 [×6], C2×C42 [×2], C2×C22⋊C4 [×13], C2×C4⋊C4 [×5], C4×D4 [×4], C4⋊D4 [×8], C22.D4 [×4], C23×C4 [×3], C22×D4 [×6], C4×C22⋊C4, C23.8Q8, C23.23D4 [×2], C24.C22 [×2], C24.3C22, C232D4 [×2], C23.10D4, C23.81C23, C2×C4×D4, C2×C4⋊D4 [×2], C2×C22.D4, C23.364C24
Quotients: C1, C2 [×15], C22 [×35], D4 [×8], C23 [×15], C2×D4 [×12], C4○D4 [×6], C24, C22×D4 [×2], C2×C4○D4 [×3], 2+ 1+4 [×2], C22.19C24, C22.26C24, C22.34C24, D42, D45D4 [×2], C22.47C24, C23.364C24

Smallest permutation representation of C23.364C24
On 64 points
Generators in S64
(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 41)(2 42)(3 43)(4 44)(5 20)(6 17)(7 18)(8 19)(9 57)(10 58)(11 59)(12 60)(13 25)(14 26)(15 27)(16 28)(21 34)(22 35)(23 36)(24 33)(29 45)(30 46)(31 47)(32 48)(37 56)(38 53)(39 54)(40 55)(49 61)(50 62)(51 63)(52 64)
(1 11)(2 12)(3 9)(4 10)(5 50)(6 51)(7 52)(8 49)(13 31)(14 32)(15 29)(16 30)(17 63)(18 64)(19 61)(20 62)(21 40)(22 37)(23 38)(24 39)(25 47)(26 48)(27 45)(28 46)(33 54)(34 55)(35 56)(36 53)(41 59)(42 60)(43 57)(44 58)
(1 34)(2 33)(3 36)(4 35)(5 46)(6 45)(7 48)(8 47)(9 53)(10 56)(11 55)(12 54)(13 61)(14 64)(15 63)(16 62)(17 29)(18 32)(19 31)(20 30)(21 41)(22 44)(23 43)(24 42)(25 49)(26 52)(27 51)(28 50)(37 58)(38 57)(39 60)(40 59)
(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 61)(2 20)(3 63)(4 18)(5 42)(6 57)(7 44)(8 59)(9 17)(10 64)(11 19)(12 62)(13 21)(14 37)(15 23)(16 39)(22 32)(24 30)(25 34)(26 56)(27 36)(28 54)(29 38)(31 40)(33 46)(35 48)(41 49)(43 51)(45 53)(47 55)(50 60)(52 58)
(1 15 43 25)(2 16 44 26)(3 13 41 27)(4 14 42 28)(5 54 18 37)(6 55 19 38)(7 56 20 39)(8 53 17 40)(9 31 59 45)(10 32 60 46)(11 29 57 47)(12 30 58 48)(21 49 36 63)(22 50 33 64)(23 51 34 61)(24 52 35 62)

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

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

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

38 conjugacy classes

class 1 2A···2G2H···2M2N4A···4H4I···4T4U4V4W
order12···22···224···44···4444
size11···14···482···24···4888

38 irreducible representations

dim1111111111112224
type++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2C2D4C4○D4C4○D42+ 1+4
kernelC23.364C24C4×C22⋊C4C23.8Q8C23.23D4C24.C22C24.3C22C232D4C23.10D4C23.81C23C2×C4×D4C2×C4⋊D4C2×C22.D4C22⋊C4C2×C4C23C22
# reps1112212111218842

Matrix representation of C23.364C24 in GL6(𝔽5)

400000
040000
001000
000100
000010
000001
,
100000
010000
004000
000400
000010
000001
,
100000
010000
001000
000100
000040
000004
,
040000
400000
003100
002200
000040
000004
,
010000
400000
004000
000400
000013
000004
,
100000
010000
004300
000100
000040
000041
,
010000
400000
002000
000200
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],[0,4,0,0,0,0,4,0,0,0,0,0,0,0,3,2,0,0,0,0,1,2,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[0,4,0,0,0,0,1,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,3,4],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,3,1,0,0,0,0,0,0,4,4,0,0,0,0,0,1],[0,4,0,0,0,0,1,0,0,0,0,0,0,0,2,0,0,0,0,0,0,2,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

C23.364C24 in GAP, Magma, Sage, TeX

C_2^3._{364}C_2^4
% in TeX

G:=Group("C2^3.364C2^4");
// GroupNames label

G:=SmallGroup(128,1196);
// by ID

G=gap.SmallGroup(128,1196);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,253,120,758,723,675,192]);
// Polycyclic

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