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G = C24.244C23order 128 = 27

84th non-split extension by C24 of C23 acting via C23/C2=C22

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

Aliases: C24.244C23, C23.307C24, C22.892- 1+4, C22.1242+ 1+4, (C2×D4)⋊42D4, (C2×Q8)⋊30D4, (C22×C4)⋊20D4, C4.79C22≀C2, C232(C4○D4), C232D45C2, C2.6(Q86D4), C2.8(Q85D4), C23.150(C2×D4), C2.14(D45D4), C2.10(D46D4), C23.10D45C2, C23.8Q824C2, C23.7Q830C2, C23.23D424C2, (C23×C4).327C22, (C22×C4).788C23, (C2×C42).460C22, C22.187(C22×D4), C24.3C2225C2, (C22×D4).500C22, (C22×Q8).414C22, C23.67C2327C2, C2.18(C22.19C24), C2.C42.75C22, C2.6(C22.31C24), (C2×C4×D4)⋊19C2, (C2×C4)⋊2(C4○D4), (C2×C4⋊D4)⋊2C2, (C2×C22⋊Q8)⋊2C2, (C2×C4).303(C2×D4), (C22×C4○D4)⋊3C2, C2.14(C2×C22≀C2), (C2×C4⋊C4).202C22, C22.186(C2×C4○D4), (C2×C22⋊C4).106C22, SmallGroup(128,1139)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C24.244C23
C1C2C22C23C22×C4C23×C4C2×C4×D4 — C24.244C23
C1C23 — C24.244C23
C1C23 — C24.244C23
C1C23 — C24.244C23

Generators and relations for C24.244C23
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=e2=f2=1, g2=a, ab=ba, ac=ca, ede=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: 884 in 454 conjugacy classes, 120 normal (42 characteristic)
C1, C2 [×7], C2 [×8], C4 [×4], C4 [×14], C22 [×7], C22 [×40], C2×C4 [×14], C2×C4 [×54], D4 [×40], Q8 [×8], C23, C23 [×8], C23 [×24], C42 [×2], C22⋊C4 [×20], C4⋊C4 [×10], C22×C4 [×5], C22×C4 [×10], C22×C4 [×22], C2×D4 [×4], C2×D4 [×32], C2×Q8 [×4], C2×Q8 [×4], C4○D4 [×32], C24 [×2], C24 [×2], C2.C42 [×6], C2×C42, C2×C22⋊C4 [×2], C2×C22⋊C4 [×10], C2×C4⋊C4 [×3], C2×C4⋊C4 [×2], C4×D4 [×4], C4⋊D4 [×4], C22⋊Q8 [×4], C23×C4 [×2], C23×C4 [×2], C22×D4 [×2], C22×D4 [×4], C22×Q8, C2×C4○D4 [×12], C23.7Q8, C23.8Q8 [×2], C23.23D4 [×2], C24.3C22, C23.67C23, C232D4 [×2], C23.10D4 [×2], C2×C4×D4, C2×C4⋊D4, C2×C22⋊Q8, C22×C4○D4, C24.244C23
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- 1+4, C2×C22≀C2, C22.19C24, C22.31C24, D45D4, D46D4, Q85D4, Q86D4, C24.244C23

Smallest permutation representation of C24.244C23
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 17)(2 18)(3 19)(4 20)(5 15)(6 16)(7 13)(8 14)(9 64)(10 61)(11 62)(12 63)(21 27)(22 28)(23 25)(24 26)(29 33)(30 34)(31 35)(32 36)(37 43)(38 44)(39 41)(40 42)(45 49)(46 50)(47 51)(48 52)(53 59)(54 60)(55 57)(56 58)
(1 25)(2 26)(3 27)(4 28)(5 64)(6 61)(7 62)(8 63)(9 15)(10 16)(11 13)(12 14)(17 23)(18 24)(19 21)(20 22)(29 41)(30 42)(31 43)(32 44)(33 39)(34 40)(35 37)(36 38)(45 57)(46 58)(47 59)(48 60)(49 55)(50 56)(51 53)(52 54)
(1 4)(2 3)(5 14)(6 13)(7 16)(8 15)(9 63)(10 62)(11 61)(12 64)(17 20)(18 19)(21 24)(22 23)(25 28)(26 27)(29 36)(30 35)(31 34)(32 33)(37 42)(38 41)(39 44)(40 43)(45 46)(47 48)(49 50)(51 52)(53 54)(55 56)(57 58)(59 60)
(1 45)(2 46)(3 47)(4 48)(5 44)(6 41)(7 42)(8 43)(9 36)(10 33)(11 34)(12 35)(13 40)(14 37)(15 38)(16 39)(17 49)(18 50)(19 51)(20 52)(21 53)(22 54)(23 55)(24 56)(25 57)(26 58)(27 59)(28 60)(29 61)(30 62)(31 63)(32 64)
(1 29)(2 30)(3 31)(4 32)(5 48)(6 45)(7 46)(8 47)(9 54)(10 55)(11 56)(12 53)(13 50)(14 51)(15 52)(16 49)(17 33)(18 34)(19 35)(20 36)(21 37)(22 38)(23 39)(24 40)(25 41)(26 42)(27 43)(28 44)(57 61)(58 62)(59 63)(60 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)

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

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

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

38 conjugacy classes

class 1 2A···2G2H···2O4A···4H4I···4R4S4T4U4V
order12···22···24···44···44444
size11···14···42···24···48888

38 irreducible representations

dim1111111111112222244
type++++++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2C2C2D4D4D4C4○D4C4○D42+ 1+42- 1+4
kernelC24.244C23C23.7Q8C23.8Q8C23.23D4C24.3C22C23.67C23C232D4C23.10D4C2×C4×D4C2×C4⋊D4C2×C22⋊Q8C22×C4○D4C22×C4C2×D4C2×Q8C2×C4C23C22C22
# reps1122112211114444411

Matrix representation of C24.244C23 in GL6(𝔽5)

100000
010000
001000
000100
000040
000004
,
100000
010000
004000
000400
000010
000001
,
400000
040000
004000
000400
000010
000001
,
400000
040000
001000
000400
000043
000001
,
030000
200000
004000
000100
000024
000033
,
010000
100000
000400
004000
000040
000004
,
400000
040000
001000
000100
000043
000011

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

C24.244C23 in GAP, Magma, Sage, TeX

C_2^4._{244}C_2^3
% in TeX

G:=Group("C2^4.244C2^3");
// GroupNames label

G:=SmallGroup(128,1139);
// by ID

G=gap.SmallGroup(128,1139);
# by ID

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

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