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G = C24.254C23order 128 = 27

94th non-split extension by C24 of C23 acting via C23/C2=C22

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C23 — C24.254C23
 Chief series C1 — C2 — C22 — C23 — C22×C4 — C23×C4 — C4×C22⋊C4 — C24.254C23
 Lower central C1 — C23 — C24.254C23
 Upper central C1 — C23 — C24.254C23
 Jennings C1 — C23 — C24.254C23

Generators and relations for C24.254C23
G = < a,b,c,d,e,f,g | a2=b2=c2=d2=f2=1, e2=b, g2=a, ab=ba, 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: 612 in 298 conjugacy classes, 104 normal (42 characteristic)
C1, C2 [×7], C2 [×6], C4 [×4], C4 [×12], C22 [×7], C22 [×30], C2×C4 [×8], C2×C4 [×44], D4 [×16], C23, C23 [×6], C23 [×18], C42 [×4], C22⋊C4 [×18], C4⋊C4 [×8], C22×C4 [×6], C22×C4 [×6], C22×C4 [×16], C2×D4 [×4], C2×D4 [×14], C24, C24 [×2], C2.C42 [×2], C2.C42 [×8], C2×C42 [×2], C2×C22⋊C4 [×2], C2×C22⋊C4 [×10], C2×C4⋊C4 [×3], C2×C4⋊C4 [×2], C4×D4 [×4], C4⋊D4 [×4], C23×C4, C23×C4 [×2], C22×D4, C22×D4 [×2], C4×C22⋊C4, C23.7Q8, C23.34D4 [×2], C428C4, C23.23D4 [×2], C23.65C23, C24.3C22, C23.10D4 [×2], C23.11D4 [×2], C2×C4×D4, C2×C4⋊D4, C24.254C23
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×8], C24, C22.D4 [×4], C22×D4, C2×C4○D4 [×4], 2+ 1+4 [×2], C2×C22.D4, C23.36C23, C22.34C24, D45D4 [×2], C22.47C24, C22.49C24, C24.254C23

Smallest permutation representation of C24.254C23
On 64 points
Generators in S64
(1 39)(2 40)(3 37)(4 38)(5 17)(6 18)(7 19)(8 20)(9 41)(10 42)(11 43)(12 44)(13 45)(14 46)(15 47)(16 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 62)(34 63)(35 64)(36 61)
(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 51)(2 52)(3 49)(4 50)(5 36)(6 33)(7 34)(8 35)(9 53)(10 54)(11 55)(12 56)(13 57)(14 58)(15 59)(16 60)(17 61)(18 62)(19 63)(20 64)(21 37)(22 38)(23 39)(24 40)(25 41)(26 42)(27 43)(28 44)(29 45)(30 46)(31 47)(32 48)
(1 11)(2 44)(3 9)(4 42)(5 47)(6 16)(7 45)(8 14)(10 38)(12 40)(13 19)(15 17)(18 48)(20 46)(21 25)(22 54)(23 27)(24 56)(26 50)(28 52)(29 34)(30 64)(31 36)(32 62)(33 60)(35 58)(37 41)(39 43)(49 53)(51 55)(57 63)(59 61)
(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 15)(2 60)(3 13)(4 58)(5 41)(6 26)(7 43)(8 28)(9 17)(10 62)(11 19)(12 64)(14 50)(16 52)(18 54)(20 56)(21 29)(22 46)(23 31)(24 48)(25 36)(27 34)(30 38)(32 40)(33 42)(35 44)(37 45)(39 47)(49 57)(51 59)(53 61)(55 63)
(1 43 39 11)(2 44 40 12)(3 41 37 9)(4 42 38 10)(5 45 17 13)(6 46 18 14)(7 47 19 15)(8 48 20 16)(21 53 49 25)(22 54 50 26)(23 55 51 27)(24 56 52 28)(29 61 57 36)(30 62 58 33)(31 63 59 34)(32 64 60 35)

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

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

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

38 conjugacy classes

 class 1 2A ··· 2G 2H ··· 2M 4A ··· 4H 4I ··· 4T 4U 4V 4W 4X order 1 2 ··· 2 2 ··· 2 4 ··· 4 4 ··· 4 4 4 4 4 size 1 1 ··· 1 4 ··· 4 2 ··· 2 4 ··· 4 8 8 8 8

38 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 4 type + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 D4 C4○D4 C4○D4 2+ 1+4 kernel C24.254C23 C4×C22⋊C4 C23.7Q8 C23.34D4 C42⋊8C4 C23.23D4 C23.65C23 C24.3C22 C23.10D4 C23.11D4 C2×C4×D4 C2×C4⋊D4 C2×D4 C2×C4 C23 C22 # reps 1 1 1 2 1 2 1 1 2 2 1 1 4 12 4 2

Matrix representation of C24.254C23 in GL6(𝔽5)

 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
,
 1 0 0 0 0 0 0 4 0 0 0 0 0 0 1 2 0 0 0 0 0 4 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 2 0 0 0 0 0 0 2 0 0 0 0 0 0 3 1 0 0 0 0 2 2 0 0 0 0 0 0 0 1 0 0 0 0 1 0
,
 0 1 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 4 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 2 0 0 0 0 4 4 0 0 0 0 0 0 1 0 0 0 0 0 0 1

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

C24.254C23 in GAP, Magma, Sage, TeX

C_2^4._{254}C_2^3
% in TeX

G:=Group("C2^4.254C2^3");
// GroupNames label

G:=SmallGroup(128,1152);
// by ID

G=gap.SmallGroup(128,1152);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,253,232,758,723,675,80]);
// Polycyclic

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