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G = C24.223C23order 128 = 27

63rd non-split extension by C24 of C23 acting via C23/C2=C22

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

Aliases: C24.223C23, C23.256C24, C22.872+ 1+4, C22.D410C4, (C23×C4).61C22, (C2×C42).27C22, C23.25(C22×C4), C22.147(C23×C4), (C22×C4).1255C23, C24.C2224C2, C23.23D4.11C2, (C22×D4).111C22, C23.63C2323C2, C2.9(C22.45C24), C2.36(C22.11C24), C24.3C22.30C2, C2.C42.483C22, C2.5(C22.53C24), C2.11(C22.47C24), (C4×C4⋊C4)⋊50C2, C4⋊C432(C2×C4), C2.43(C4×C4○D4), C22⋊C418(C2×C4), (C4×C22⋊C4)⋊12C2, (C22×C4)⋊14(C2×C4), (C2×D4).132(C2×C4), (C2×C4).53(C22×C4), (C2×C4).853(C4○D4), (C2×C4⋊C4).832C22, C22.141(C2×C4○D4), (C2×C22⋊C4).39C22, (C2×C22.D4).8C2, SmallGroup(128,1106)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C24.223C23
C1C2C22C23C22×C4C23×C4C4×C22⋊C4 — C24.223C23
C1C22 — C24.223C23
C1C23 — C24.223C23
C1C23 — C24.223C23

Generators and relations for C24.223C23
 G = < a,b,c,d,e,f,g | a2=b2=c2=1, d2=e2=c, f2=g2=ba=ab, ac=ca, ede-1=ad=da, geg-1=ae=ea, af=fa, ag=ga, bc=cb, fdf-1=bd=db, be=eb, bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, cg=gc, dg=gd, ef=fe, fg=gf >

Subgroups: 492 in 276 conjugacy classes, 140 normal (22 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×22], C22 [×3], C22 [×4], C22 [×20], C2×C4 [×18], C2×C4 [×38], D4 [×4], C23, C23 [×4], C23 [×12], C42 [×9], C22⋊C4 [×12], C22⋊C4 [×13], C4⋊C4 [×8], C4⋊C4 [×5], C22×C4, C22×C4 [×16], C22×C4 [×7], C2×D4 [×4], C2×D4 [×2], C24 [×2], C2.C42 [×10], C2×C42, C2×C42 [×6], C2×C22⋊C4 [×2], C2×C22⋊C4 [×8], C2×C4⋊C4, C2×C4⋊C4 [×4], C22.D4 [×8], C23×C4 [×2], C22×D4, C4×C22⋊C4 [×4], C4×C4⋊C4 [×2], C23.23D4, C23.63C23 [×2], C24.C22 [×4], C24.3C22, C2×C22.D4, C24.223C23
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], C22×C4 [×14], C4○D4 [×8], C24, C23×C4, C2×C4○D4 [×4], 2+ 1+4 [×2], C4×C4○D4 [×2], C22.11C24, C22.45C24, C22.47C24 [×2], C22.53C24, C24.223C23

Smallest permutation representation of C24.223C23
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 47 3 45)(2 20 4 18)(5 16 7 14)(6 41 8 43)(9 19 11 17)(10 48 12 46)(13 37 15 39)(21 31 23 29)(22 60 24 58)(25 33 27 35)(26 64 28 62)(30 50 32 52)(34 56 36 54)(38 44 40 42)(49 59 51 57)(53 63 55 61)
(1 41 23 27)(2 56 24 14)(3 43 21 25)(4 54 22 16)(5 20 36 58)(6 31 33 45)(7 18 34 60)(8 29 35 47)(9 13 51 55)(10 28 52 42)(11 15 49 53)(12 26 50 44)(17 39 59 63)(19 37 57 61)(30 38 48 62)(32 40 46 64)
(1 27 23 41)(2 28 24 42)(3 25 21 43)(4 26 22 44)(5 20 36 58)(6 17 33 59)(7 18 34 60)(8 19 35 57)(9 55 51 13)(10 56 52 14)(11 53 49 15)(12 54 50 16)(29 37 47 61)(30 38 48 62)(31 39 45 63)(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), (1,47,3,45)(2,20,4,18)(5,16,7,14)(6,41,8,43)(9,19,11,17)(10,48,12,46)(13,37,15,39)(21,31,23,29)(22,60,24,58)(25,33,27,35)(26,64,28,62)(30,50,32,52)(34,56,36,54)(38,44,40,42)(49,59,51,57)(53,63,55,61), (1,41,23,27)(2,56,24,14)(3,43,21,25)(4,54,22,16)(5,20,36,58)(6,31,33,45)(7,18,34,60)(8,29,35,47)(9,13,51,55)(10,28,52,42)(11,15,49,53)(12,26,50,44)(17,39,59,63)(19,37,57,61)(30,38,48,62)(32,40,46,64), (1,27,23,41)(2,28,24,42)(3,25,21,43)(4,26,22,44)(5,20,36,58)(6,17,33,59)(7,18,34,60)(8,19,35,57)(9,55,51,13)(10,56,52,14)(11,53,49,15)(12,54,50,16)(29,37,47,61)(30,38,48,62)(31,39,45,63)(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), (1,47,3,45)(2,20,4,18)(5,16,7,14)(6,41,8,43)(9,19,11,17)(10,48,12,46)(13,37,15,39)(21,31,23,29)(22,60,24,58)(25,33,27,35)(26,64,28,62)(30,50,32,52)(34,56,36,54)(38,44,40,42)(49,59,51,57)(53,63,55,61), (1,41,23,27)(2,56,24,14)(3,43,21,25)(4,54,22,16)(5,20,36,58)(6,31,33,45)(7,18,34,60)(8,29,35,47)(9,13,51,55)(10,28,52,42)(11,15,49,53)(12,26,50,44)(17,39,59,63)(19,37,57,61)(30,38,48,62)(32,40,46,64), (1,27,23,41)(2,28,24,42)(3,25,21,43)(4,26,22,44)(5,20,36,58)(6,17,33,59)(7,18,34,60)(8,19,35,57)(9,55,51,13)(10,56,52,14)(11,53,49,15)(12,54,50,16)(29,37,47,61)(30,38,48,62)(31,39,45,63)(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)], [(1,47,3,45),(2,20,4,18),(5,16,7,14),(6,41,8,43),(9,19,11,17),(10,48,12,46),(13,37,15,39),(21,31,23,29),(22,60,24,58),(25,33,27,35),(26,64,28,62),(30,50,32,52),(34,56,36,54),(38,44,40,42),(49,59,51,57),(53,63,55,61)], [(1,41,23,27),(2,56,24,14),(3,43,21,25),(4,54,22,16),(5,20,36,58),(6,31,33,45),(7,18,34,60),(8,29,35,47),(9,13,51,55),(10,28,52,42),(11,15,49,53),(12,26,50,44),(17,39,59,63),(19,37,57,61),(30,38,48,62),(32,40,46,64)], [(1,27,23,41),(2,28,24,42),(3,25,21,43),(4,26,22,44),(5,20,36,58),(6,17,33,59),(7,18,34,60),(8,19,35,57),(9,55,51,13),(10,56,52,14),(11,53,49,15),(12,54,50,16),(29,37,47,61),(30,38,48,62),(31,39,45,63),(32,40,46,64)])

50 conjugacy classes

class 1 2A···2G2H2I2J2K4A···4X4Y···4AL
order12···222224···44···4
size11···144442···24···4

50 irreducible representations

dim11111111124
type+++++++++
imageC1C2C2C2C2C2C2C2C4C4○D42+ 1+4
kernelC24.223C23C4×C22⋊C4C4×C4⋊C4C23.23D4C23.63C23C24.C22C24.3C22C2×C22.D4C22.D4C2×C4C22
# reps1421241116162

Matrix representation of C24.223C23 in GL6(𝔽5)

400000
040000
004000
000400
000010
000001
,
100000
010000
001000
000100
000040
000004
,
100000
010000
004000
000400
000010
000001
,
100000
040000
003000
001200
000043
000001
,
010000
100000
002300
000300
000040
000004
,
200000
020000
002000
000200
000030
000022
,
300000
020000
003000
001200
000020
000002

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

C24.223C23 in GAP, Magma, Sage, TeX

C_2^4._{223}C_2^3
% in TeX

G:=Group("C2^4.223C2^3");
// GroupNames label

G:=SmallGroup(128,1106);
// by ID

G=gap.SmallGroup(128,1106);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,456,758,268,346,192]);
// Polycyclic

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

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