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G = C23.574C24order 128 = 27

291st central stem extension by C23 of C24

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

Aliases: C24.59C23, C23.574C24, C22.3482+ 1+4, C22.2592- 1+4, C2.38D42, C4⋊C4.115D4, C22⋊C4.8D4, C23⋊Q838C2, C23.203(C2×D4), C2.56(D46D4), C2.42(Q85D4), C23.4Q841C2, C23.Q848C2, C23.8Q894C2, C23.11D474C2, (C23×C4).444C22, (C2×C42).634C22, (C22×C4).863C23, C22.383(C22×D4), C23.10D4.38C2, (C22×D4).215C22, (C22×Q8).173C22, C23.78C2337C2, C24.C22118C2, C23.65C23114C2, C23.63C23125C2, C2.C42.285C22, C2.7(C22.57C24), C2.68(C22.36C24), C2.52(C23.38C23), (C2×C4⋊Q8)⋊20C2, (C2×C4).84(C2×D4), (C2×C22⋊Q8)⋊33C2, (C2×C4).189(C4○D4), (C2×C4⋊C4).392C22, (C2×C4.4D4).30C2, C22.440(C2×C4○D4), (C2×C22⋊C4).245C22, (C2×C22.D4).22C2, SmallGroup(128,1406)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C23.574C24
C1C2C22C23C24C23×C4C2×C22.D4 — C23.574C24
C1C23 — C23.574C24
C1C23 — C23.574C24
C1C23 — C23.574C24

Generators and relations for C23.574C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=g2=1, d2=e2=f2=a, ab=ba, 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: 548 in 281 conjugacy classes, 104 normal (82 characteristic)
C1, C2 [×7], C2 [×3], C4 [×19], C22 [×7], C22 [×17], C2×C4 [×12], C2×C4 [×37], D4 [×4], Q8 [×8], C23, C23 [×2], C23 [×13], C42 [×4], C22⋊C4 [×4], C22⋊C4 [×19], C4⋊C4 [×4], C4⋊C4 [×17], C22×C4 [×13], C22×C4 [×5], C2×D4 [×5], C2×Q8 [×10], C24 [×2], C2.C42 [×8], C2×C42 [×2], C2×C22⋊C4 [×11], C2×C4⋊C4 [×10], C22⋊Q8 [×4], C22.D4 [×4], C4.4D4 [×4], C4⋊Q8 [×4], C23×C4, C22×D4, C22×Q8 [×2], C23.8Q8, C23.63C23, C24.C22 [×2], C23.65C23, C23⋊Q8, C23.10D4, C23.78C23, C23.Q8, C23.11D4, C23.4Q8, C2×C22⋊Q8, C2×C22.D4, C2×C4.4D4, C2×C4⋊Q8, C23.574C24
Quotients: C1, C2 [×15], C22 [×35], D4 [×8], C23 [×15], C2×D4 [×12], C4○D4 [×2], C24, C22×D4 [×2], C2×C4○D4, 2+ 1+4, 2- 1+4 [×3], C23.38C23 [×2], C22.36C24, D42, D46D4, Q85D4, C22.57C24, C23.574C24

Smallest permutation representation of C23.574C24
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 9)(2 10)(3 11)(4 12)(5 39)(6 40)(7 37)(8 38)(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 55)(26 56)(27 53)(28 54)(29 59)(30 60)(31 57)(32 58)(33 62)(34 63)(35 64)(36 61)
(1 53)(2 54)(3 55)(4 56)(5 52)(6 49)(7 50)(8 51)(9 27)(10 28)(11 25)(12 26)(13 31)(14 32)(15 29)(16 30)(17 36)(18 33)(19 34)(20 35)(21 40)(22 37)(23 38)(24 39)(41 57)(42 58)(43 59)(44 60)(45 61)(46 62)(47 63)(48 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 41 3 43)(2 44 4 42)(5 48 7 46)(6 47 8 45)(9 13 11 15)(10 16 12 14)(17 40 19 38)(18 39 20 37)(21 34 23 36)(22 33 24 35)(25 29 27 31)(26 32 28 30)(49 63 51 61)(50 62 52 64)(53 57 55 59)(54 60 56 58)
(1 47 3 45)(2 20 4 18)(5 30 7 32)(6 57 8 59)(9 19 11 17)(10 48 12 46)(13 23 15 21)(14 52 16 50)(22 42 24 44)(25 36 27 34)(26 62 28 64)(29 40 31 38)(33 54 35 56)(37 58 39 60)(41 51 43 49)(53 63 55 61)
(1 41)(2 16)(3 43)(4 14)(5 33)(6 61)(7 35)(8 63)(9 13)(10 44)(11 15)(12 42)(17 21)(18 52)(19 23)(20 50)(22 48)(24 46)(25 29)(26 58)(27 31)(28 60)(30 54)(32 56)(34 38)(36 40)(37 64)(39 62)(45 49)(47 51)(53 57)(55 59)

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

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

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

32 conjugacy classes

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

32 irreducible representations

dim11111111111111122244
type++++++++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2C2C2C2C2C2D4D4C4○D42+ 1+42- 1+4
kernelC23.574C24C23.8Q8C23.63C23C24.C22C23.65C23C23⋊Q8C23.10D4C23.78C23C23.Q8C23.11D4C23.4Q8C2×C22⋊Q8C2×C22.D4C2×C4.4D4C2×C4⋊Q8C22⋊C4C4⋊C4C2×C4C22C22
# reps11121111111111144413

Matrix representation of C23.574C24 in GL6(𝔽5)

100000
010000
001000
000100
000040
000004
,
400000
040000
001000
000100
000010
000001
,
100000
010000
004000
000400
000010
000001
,
410000
010000
001000
000100
000020
000003
,
400000
040000
004000
000100
000004
000010
,
400000
310000
000100
001000
000030
000003
,
400000
310000
004000
000400
000001
000010

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

C23.574C24 in GAP, Magma, Sage, TeX

C_2^3._{574}C_2^4
% in TeX

G:=Group("C2^3.574C2^4");
// GroupNames label

G:=SmallGroup(128,1406);
// by ID

G=gap.SmallGroup(128,1406);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,112,253,120,758,723,100,1571,346]);
// Polycyclic

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