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G = C23.600C24order 128 = 27

317th central stem extension by C23 of C24

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

Aliases: C24.62C23, C23.600C24, C22.2782- 1+4, C22.3742+ 1+4, C22⋊C4.17D4, C23⋊Q846C2, C23.215(C2×D4), C2.105(D45D4), C23.Q862C2, C23.34D449C2, C23.11D487C2, (C2×C42).652C22, (C22×C4).877C23, (C23×C4).462C22, C22.409(C22×D4), C23.23D4.53C2, C23.10D4.43C2, (C22×D4).236C22, (C22×Q8).186C22, C23.78C2346C2, C24.C22131C2, C23.67C2382C2, C23.63C23136C2, C2.C42.306C22, C2.55(C22.50C24), C2.16(C22.56C24), C2.61(C23.38C23), C2.74(C22.36C24), (C2×C4).102(C2×D4), (C2×C22⋊Q8)⋊41C2, (C2×C4).427(C4○D4), (C2×C4⋊C4).413C22, (C2×C4.4D4).31C2, C22.462(C2×C4○D4), (C2×C22⋊C4).266C22, SmallGroup(128,1432)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C23.600C24
C1C2C22C23C24C23×C4C23.34D4 — C23.600C24
C1C23 — C23.600C24
C1C23 — C23.600C24
C1C23 — C23.600C24

Generators and relations for C23.600C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=g2=1, d2=b, e2=ba=ab, f2=a, ac=ca, ede-1=ad=da, geg=ae=ea, af=fa, ag=ga, bc=cb, fdf-1=bd=db, be=eb, bf=fb, bg=gb, cd=dc, fef-1=ce=ec, cf=fc, cg=gc, gdg=abd, fg=gf >

Subgroups: 516 in 253 conjugacy classes, 96 normal (82 characteristic)
C1, C2 [×7], C2 [×3], C4 [×17], C22 [×7], C22 [×17], C2×C4 [×8], C2×C4 [×39], D4 [×4], Q8 [×8], C23, C23 [×2], C23 [×13], C42 [×3], C22⋊C4 [×4], C22⋊C4 [×15], C4⋊C4 [×10], C22×C4 [×13], C22×C4 [×5], C2×D4 [×5], C2×Q8 [×10], C24 [×2], C2.C42 [×12], C2×C42 [×2], C2×C22⋊C4 [×11], C2×C4⋊C4 [×6], C22⋊Q8 [×4], C4.4D4 [×4], C23×C4, C22×D4, C22×Q8 [×2], C23.34D4, C23.23D4, C23.63C23, C24.C22 [×2], C23.67C23 [×2], C23⋊Q8, C23.10D4, C23.78C23, C23.Q8, C23.11D4 [×2], C2×C22⋊Q8, C2×C4.4D4, C23.600C24
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×4], C24, C22×D4, C2×C4○D4 [×2], 2+ 1+4 [×2], 2- 1+4 [×2], C23.38C23, C22.36C24 [×2], D45D4 [×2], C22.50C24, C22.56C24, C23.600C24

Smallest permutation representation of C23.600C24
On 64 points
Generators in S64
(1 37)(2 38)(3 39)(4 40)(5 11)(6 12)(7 9)(8 10)(13 41)(14 42)(15 43)(16 44)(17 46)(18 47)(19 48)(20 45)(21 50)(22 51)(23 52)(24 49)(25 56)(26 53)(27 54)(28 55)(29 60)(30 57)(31 58)(32 59)(33 64)(34 61)(35 62)(36 63)
(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 49)(2 50)(3 51)(4 52)(5 28)(6 25)(7 26)(8 27)(9 53)(10 54)(11 55)(12 56)(13 31)(14 32)(15 29)(16 30)(17 34)(18 35)(19 36)(20 33)(21 38)(22 39)(23 40)(24 37)(41 58)(42 59)(43 60)(44 57)(45 64)(46 61)(47 62)(48 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 40 39 2)(3 38 37 4)(5 12 9 8)(6 7 10 11)(13 32 43 57)(14 60 44 31)(15 30 41 59)(16 58 42 29)(17 64 48 35)(18 34 45 63)(19 62 46 33)(20 36 47 61)(21 24 52 51)(22 50 49 23)(25 26 54 55)(27 28 56 53)
(1 32 37 59)(2 31 38 58)(3 30 39 57)(4 29 40 60)(5 61 11 34)(6 64 12 33)(7 63 9 36)(8 62 10 35)(13 21 41 50)(14 24 42 49)(15 23 43 52)(16 22 44 51)(17 28 46 55)(18 27 47 54)(19 26 48 53)(20 25 45 56)
(1 11)(2 8)(3 9)(4 6)(5 37)(7 39)(10 38)(12 40)(13 47)(14 17)(15 45)(16 19)(18 41)(20 43)(21 54)(22 26)(23 56)(24 28)(25 52)(27 50)(29 64)(30 36)(31 62)(32 34)(33 60)(35 58)(42 46)(44 48)(49 55)(51 53)(57 63)(59 61)

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

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

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

32 conjugacy classes

class 1 2A···2G2H2I2J4A···4P4Q···4U
order12···22224···44···4
size11···14484···48···8

32 irreducible representations

dim11111111111112244
type+++++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2C2C2C2D4C4○D42+ 1+42- 1+4
kernelC23.600C24C23.34D4C23.23D4C23.63C23C24.C22C23.67C23C23⋊Q8C23.10D4C23.78C23C23.Q8C23.11D4C2×C22⋊Q8C2×C4.4D4C22⋊C4C2×C4C22C22
# reps11112211112114822

Matrix representation of C23.600C24 in GL6(𝔽5)

100000
010000
004000
000400
000010
000001
,
100000
010000
001000
000100
000040
000004
,
400000
040000
001000
000100
000010
000001
,
100000
010000
004300
000100
000030
000002
,
100000
040000
004300
001100
000020
000002
,
040000
400000
002000
000200
000004
000040
,
400000
040000
001000
004400
000001
000010

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

C23.600C24 in GAP, Magma, Sage, TeX

C_2^3._{600}C_2^4
% in TeX

G:=Group("C2^3.600C2^4");
// GroupNames label

G:=SmallGroup(128,1432);
// by ID

G=gap.SmallGroup(128,1432);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,224,253,344,758,723,100,1571,346]);
// Polycyclic

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

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