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G = C24.448C23order 128 = 27

288th non-split extension by C24 of C23 acting via C23/C2=C22

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

Aliases: C24.448C23, C23.680C24, C22.4532+ 1+4, (C2×D4).19Q8, C23.42(C2×Q8), C2.60(D43Q8), C23.Q884C2, C23.4Q861C2, (C2×C42).709C22, (C22×C4).213C23, (C23×C4).173C22, C23.8Q8134C2, C23.7Q8109C2, C2.17(C232Q8), C22.158(C22×Q8), C23.23D4.71C2, (C22×D4).277C22, C24.3C22.73C2, C23.83C23116C2, C23.63C23181C2, C23.65C23152C2, C23.81C23123C2, C2.100(C22.32C24), C2.32(C22.54C24), C2.C42.384C22, C2.61(C22.34C24), C2.109(C22.47C24), (C2×C4).82(C2×Q8), (C2×C4).227(C4○D4), (C2×C4⋊C4).490C22, C22.541(C2×C4○D4), (C2×C22⋊C4).316C22, SmallGroup(128,1512)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C24.448C23
C1C2C22C23C22×C4C23×C4C23.8Q8 — C24.448C23
C1C23 — C24.448C23
C1C23 — C24.448C23
C1C23 — C24.448C23

Generators and relations for C24.448C23
 G = < a,b,c,d,e,f,g | a2=b2=c2=f2=1, d2=abc, e2=a, g2=ba=ab, ac=ca, ede-1=ad=da, geg-1=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, gdg-1=abd, fg=gf >

Subgroups: 484 in 232 conjugacy classes, 96 normal (82 characteristic)
C1, C2 [×7], C2 [×4], C4 [×16], C22 [×7], C22 [×20], C2×C4 [×6], C2×C4 [×44], D4 [×4], C23, C23 [×4], C23 [×12], C42, C22⋊C4 [×13], C4⋊C4 [×13], C22×C4 [×13], C22×C4 [×7], C2×D4 [×4], C2×D4 [×2], C24 [×2], C2.C42 [×10], C2×C42, C2×C22⋊C4 [×10], C2×C4⋊C4 [×11], C23×C4 [×2], C22×D4, C23.7Q8, C23.8Q8 [×3], C23.23D4 [×2], C23.63C23, C23.65C23, C24.3C22, C23.Q8 [×3], C23.81C23, C23.4Q8, C23.83C23, C24.448C23
Quotients: C1, C2 [×15], C22 [×35], Q8 [×4], C23 [×15], C2×Q8 [×6], C4○D4 [×4], C24, C22×Q8, C2×C4○D4 [×2], 2+ 1+4 [×4], C22.32C24, C22.34C24, C232Q8, C22.47C24, D43Q8 [×2], C22.54C24, C24.448C23

Smallest permutation representation of C24.448C23
On 64 points
Generators in S64
(1 61)(2 62)(3 63)(4 64)(5 53)(6 54)(7 55)(8 56)(9 14)(10 15)(11 16)(12 13)(17 21)(18 22)(19 23)(20 24)(25 30)(26 31)(27 32)(28 29)(33 57)(34 58)(35 59)(36 60)(37 43)(38 44)(39 41)(40 42)(45 50)(46 51)(47 52)(48 49)
(1 58)(2 59)(3 60)(4 57)(5 31)(6 32)(7 29)(8 30)(9 41)(10 42)(11 43)(12 44)(13 38)(14 39)(15 40)(16 37)(17 52)(18 49)(19 50)(20 51)(21 47)(22 48)(23 45)(24 46)(25 56)(26 53)(27 54)(28 55)(33 64)(34 61)(35 62)(36 63)
(1 36)(2 33)(3 34)(4 35)(5 28)(6 25)(7 26)(8 27)(9 37)(10 38)(11 39)(12 40)(13 42)(14 43)(15 44)(16 41)(17 45)(18 46)(19 47)(20 48)(21 50)(22 51)(23 52)(24 49)(29 53)(30 54)(31 55)(32 56)(57 62)(58 63)(59 64)(60 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 46 61 51)(2 52 62 47)(3 48 63 49)(4 50 64 45)(5 15 53 10)(6 11 54 16)(7 13 55 12)(8 9 56 14)(17 35 21 59)(18 60 22 36)(19 33 23 57)(20 58 24 34)(25 39 30 41)(26 42 31 40)(27 37 32 43)(28 44 29 38)
(1 11)(2 44)(3 9)(4 42)(5 47)(6 22)(7 45)(8 24)(10 57)(12 59)(13 35)(14 63)(15 33)(16 61)(17 26)(18 54)(19 28)(20 56)(21 31)(23 29)(25 51)(27 49)(30 46)(32 48)(34 37)(36 39)(38 62)(40 64)(41 60)(43 58)(50 55)(52 53)
(1 31 34 53)(2 54 35 32)(3 29 36 55)(4 56 33 30)(5 61 26 58)(6 59 27 62)(7 63 28 60)(8 57 25 64)(9 23 39 50)(10 51 40 24)(11 21 37 52)(12 49 38 22)(13 48 44 18)(14 19 41 45)(15 46 42 20)(16 17 43 47)

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

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

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

32 conjugacy classes

class 1 2A···2G2H2I2J2K4A···4N4O···4T
order12···222224···44···4
size11···144444···48···8

32 irreducible representations

dim11111111111224
type+++++++++++-+
imageC1C2C2C2C2C2C2C2C2C2C2Q8C4○D42+ 1+4
kernelC24.448C23C23.7Q8C23.8Q8C23.23D4C23.63C23C23.65C23C24.3C22C23.Q8C23.81C23C23.4Q8C23.83C23C2×D4C2×C4C22
# reps11321113111484

Matrix representation of C24.448C23 in GL6(𝔽5)

100000
010000
004000
000400
000010
000001
,
100000
010000
001000
000100
000040
000004
,
400000
040000
001000
000100
000010
000001
,
200000
020000
004300
001100
000020
000043
,
020000
300000
002400
000300
000040
000004
,
010000
100000
004000
000400
000014
000004
,
100000
010000
002000
003300
000023
000003

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

C24.448C23 in GAP, Magma, Sage, TeX

C_2^4._{448}C_2^3
% in TeX

G:=Group("C2^4.448C2^3");
// GroupNames label

G:=SmallGroup(128,1512);
// by ID

G=gap.SmallGroup(128,1512);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,784,253,120,758,723,1571,346,192]);
// Polycyclic

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

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