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G = C24.263C23order 128 = 27

103rd non-split extension by C24 of C23 acting via C23/C2=C22

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

Aliases: C24.263C23, C23.331C24, C22.1432+ 1+4, (C2×D4)⋊45D4, (C2×Q8)⋊32D4, (C22×C4)⋊26D4, C4.28C22≀C2, C232D412C2, C23.160(C2×D4), C2.30(D45D4), C2.11(Q86D4), C23.23D436C2, (C22×C4).798C23, (C23×C4).344C22, (C2×C42).477C22, C22.211(C22×D4), C24.3C2235C2, (C22×D4).128C22, (C22×Q8).421C22, C23.67C2335C2, C2.12(C22.29C24), C2.C42.92C22, C2.15(C22.26C24), (C2×C4)⋊3(C4○D4), (C2×C41D4)⋊4C2, (C2×C4⋊D4)⋊10C2, (C4×C22⋊C4)⋊55C2, (C2×C4).316(C2×D4), (C2×C4.4D4)⋊6C2, (C22×C4○D4)⋊5C2, C2.19(C2×C22≀C2), (C2×C4⋊C4).217C22, C22.210(C2×C4○D4), (C2×C22⋊C4).121C22, SmallGroup(128,1163)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C24.263C23
C1C2C22C23C22×C4C2×C42C4×C22⋊C4 — C24.263C23
C1C23 — C24.263C23
C1C23 — C24.263C23
C1C23 — C24.263C23

Generators and relations for C24.263C23
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=e2=1, f2=b, g2=a, ab=ba, ac=ca, ede=gdg-1=ad=da, 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, eg=ge, fg=gf >

Subgroups: 996 in 476 conjugacy classes, 120 normal (26 characteristic)
C1, C2 [×3], C2 [×4], C2 [×8], C4 [×4], C4 [×14], C22 [×3], C22 [×4], C22 [×44], C2×C4 [×16], C2×C4 [×46], D4 [×52], Q8 [×8], C23, C23 [×6], C23 [×32], C42 [×4], C22⋊C4 [×20], C4⋊C4 [×2], C22×C4 [×2], C22×C4 [×12], C22×C4 [×16], C2×D4 [×4], C2×D4 [×50], C2×Q8 [×4], C2×Q8 [×4], C4○D4 [×32], C24, C24 [×4], C2.C42 [×6], C2×C42 [×2], C2×C22⋊C4 [×12], C2×C4⋊C4, C4⋊D4 [×4], C4.4D4 [×4], C41D4 [×4], C23×C4, C23×C4 [×2], C22×D4 [×2], C22×D4 [×8], C22×Q8, C2×C4○D4 [×12], C4×C22⋊C4, C23.23D4 [×4], C24.3C22, C23.67C23, C232D4 [×4], C2×C4⋊D4, C2×C4.4D4, C2×C41D4, C22×C4○D4, C24.263C23
Quotients: C1, C2 [×15], C22 [×35], D4 [×12], C23 [×15], C2×D4 [×18], C4○D4 [×4], C24, C22≀C2 [×4], C22×D4 [×3], C2×C4○D4 [×2], 2+ 1+4 [×2], C2×C22≀C2, C22.26C24, C22.29C24, D45D4 [×2], Q86D4 [×2], C24.263C23

Smallest permutation representation of C24.263C23
On 64 points
Generators in S64
(1 23)(2 24)(3 21)(4 22)(5 9)(6 10)(7 11)(8 12)(13 25)(14 26)(15 27)(16 28)(17 62)(18 63)(19 64)(20 61)(29 36)(30 33)(31 34)(32 35)(37 50)(38 51)(39 52)(40 49)(41 54)(42 55)(43 56)(44 53)(45 58)(46 59)(47 60)(48 57)
(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 27)(2 28)(3 25)(4 26)(5 36)(6 33)(7 34)(8 35)(9 29)(10 30)(11 31)(12 32)(13 21)(14 22)(15 23)(16 24)(17 44)(18 41)(19 42)(20 43)(37 46)(38 47)(39 48)(40 45)(49 58)(50 59)(51 60)(52 57)(53 62)(54 63)(55 64)(56 61)
(1 4)(2 3)(5 12)(6 11)(7 10)(8 9)(13 16)(14 15)(17 41)(18 44)(19 43)(20 42)(21 24)(22 23)(25 28)(26 27)(29 35)(30 34)(31 33)(32 36)(37 58)(38 57)(39 60)(40 59)(45 50)(46 49)(47 52)(48 51)(53 63)(54 62)(55 61)(56 64)
(1 59)(2 51)(3 57)(4 49)(5 20)(6 44)(7 18)(8 42)(9 61)(10 53)(11 63)(12 55)(13 39)(14 45)(15 37)(16 47)(17 33)(19 35)(21 48)(22 40)(23 46)(24 38)(25 52)(26 58)(27 50)(28 60)(29 56)(30 62)(31 54)(32 64)(34 41)(36 43)
(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 7 23 11)(2 8 24 12)(3 5 21 9)(4 6 22 10)(13 29 25 36)(14 30 26 33)(15 31 27 34)(16 32 28 35)(17 45 62 58)(18 46 63 59)(19 47 64 60)(20 48 61 57)(37 54 50 41)(38 55 51 42)(39 56 52 43)(40 53 49 44)

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

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

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

38 conjugacy classes

class 1 2A···2G2H···2M2N2O4A···4H4I···4T4U4V
order12···22···2224···44···444
size11···14···4882···24···488

38 irreducible representations

dim111111111122224
type++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2D4D4D4C4○D42+ 1+4
kernelC24.263C23C4×C22⋊C4C23.23D4C24.3C22C23.67C23C232D4C2×C4⋊D4C2×C4.4D4C2×C41D4C22×C4○D4C22×C4C2×D4C2×Q8C2×C4C22
# reps114114111144482

Matrix representation of C24.263C23 in GL6(𝔽5)

400000
040000
004000
000400
000010
000001
,
400000
040000
001000
000100
000040
000004
,
100000
010000
001000
000100
000040
000004
,
010000
100000
004000
000100
000001
000010
,
030000
200000
000300
002000
000001
000010
,
040000
100000
004000
000400
000001
000040
,
010000
400000
000100
004000
000010
000001

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

C24.263C23 in GAP, Magma, Sage, TeX

C_2^4._{263}C_2^3
% in TeX

G:=Group("C2^4.263C2^3");
// GroupNames label

G:=SmallGroup(128,1163);
// by ID

G=gap.SmallGroup(128,1163);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,253,758,723,184,675,80]);
// Polycyclic

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

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