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## G = C24.51(C2×C4)  order 128 = 27

### 16th non-split extension by C24 of C2×C4 acting via C2×C4/C2=C22

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C23 — C24.51(C2×C4)
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C23×C4 — C22×C4○D4 — C24.51(C2×C4)
 Lower central C1 — C23 — C24.51(C2×C4)
 Upper central C1 — C22×C4 — C24.51(C2×C4)
 Jennings C1 — C2 — C2 — C22×C4 — C24.51(C2×C4)

Generators and relations for C24.51(C2×C4)
G = < a,b,c,d,e,f | a2=b2=c2=d2=e2=1, f4=d, faf-1=ab=ba, ac=ca, eae=ad=da, bc=cb, bd=db, be=eb, bf=fb, cd=dc, fef-1=ce=ec, cf=fc, de=ed, df=fd >

Subgroups: 580 in 306 conjugacy classes, 76 normal (7 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, D4, Q8, C23, C23, C23, C2×C8, C22×C4, C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, C24, C22⋊C8, C22×C8, C23×C4, C22×D4, C22×Q8, C2×C4○D4, C2×C22⋊C8, C22×C4○D4, C24.51(C2×C4)
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C2×C22⋊C4, C22≀C2, C8○D4, C243C4, (C22×C8)⋊C2, C24.51(C2×C4)

Smallest permutation representation of C24.51(C2×C4)
On 64 points
Generators in S64
(1 16)(2 45)(3 10)(4 47)(5 12)(6 41)(7 14)(8 43)(9 18)(11 20)(13 22)(15 24)(17 44)(19 46)(21 48)(23 42)(25 35)(26 64)(27 37)(28 58)(29 39)(30 60)(31 33)(32 62)(34 56)(36 50)(38 52)(40 54)(49 63)(51 57)(53 59)(55 61)
(1 17)(2 18)(3 19)(4 20)(5 21)(6 22)(7 23)(8 24)(9 45)(10 46)(11 47)(12 48)(13 41)(14 42)(15 43)(16 44)(25 49)(26 50)(27 51)(28 52)(29 53)(30 54)(31 55)(32 56)(33 61)(34 62)(35 63)(36 64)(37 57)(38 58)(39 59)(40 60)
(1 55)(2 56)(3 49)(4 50)(5 51)(6 52)(7 53)(8 54)(9 62)(10 63)(11 64)(12 57)(13 58)(14 59)(15 60)(16 61)(17 31)(18 32)(19 25)(20 26)(21 27)(22 28)(23 29)(24 30)(33 44)(34 45)(35 46)(36 47)(37 48)(38 41)(39 42)(40 43)
(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)(25 29)(26 30)(27 31)(28 32)(33 37)(34 38)(35 39)(36 40)(41 45)(42 46)(43 47)(44 48)(49 53)(50 54)(51 55)(52 56)(57 61)(58 62)(59 63)(60 64)
(1 10)(2 64)(3 12)(4 58)(5 14)(6 60)(7 16)(8 62)(9 54)(11 56)(13 50)(15 52)(17 46)(18 36)(19 48)(20 38)(21 42)(22 40)(23 44)(24 34)(25 37)(26 41)(27 39)(28 43)(29 33)(30 45)(31 35)(32 47)(49 57)(51 59)(53 61)(55 63)
(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)| (1,16)(2,45)(3,10)(4,47)(5,12)(6,41)(7,14)(8,43)(9,18)(11,20)(13,22)(15,24)(17,44)(19,46)(21,48)(23,42)(25,35)(26,64)(27,37)(28,58)(29,39)(30,60)(31,33)(32,62)(34,56)(36,50)(38,52)(40,54)(49,63)(51,57)(53,59)(55,61), (1,17)(2,18)(3,19)(4,20)(5,21)(6,22)(7,23)(8,24)(9,45)(10,46)(11,47)(12,48)(13,41)(14,42)(15,43)(16,44)(25,49)(26,50)(27,51)(28,52)(29,53)(30,54)(31,55)(32,56)(33,61)(34,62)(35,63)(36,64)(37,57)(38,58)(39,59)(40,60), (1,55)(2,56)(3,49)(4,50)(5,51)(6,52)(7,53)(8,54)(9,62)(10,63)(11,64)(12,57)(13,58)(14,59)(15,60)(16,61)(17,31)(18,32)(19,25)(20,26)(21,27)(22,28)(23,29)(24,30)(33,44)(34,45)(35,46)(36,47)(37,48)(38,41)(39,42)(40,43), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32)(33,37)(34,38)(35,39)(36,40)(41,45)(42,46)(43,47)(44,48)(49,53)(50,54)(51,55)(52,56)(57,61)(58,62)(59,63)(60,64), (1,10)(2,64)(3,12)(4,58)(5,14)(6,60)(7,16)(8,62)(9,54)(11,56)(13,50)(15,52)(17,46)(18,36)(19,48)(20,38)(21,42)(22,40)(23,44)(24,34)(25,37)(26,41)(27,39)(28,43)(29,33)(30,45)(31,35)(32,47)(49,57)(51,59)(53,61)(55,63), (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( (1,16)(2,45)(3,10)(4,47)(5,12)(6,41)(7,14)(8,43)(9,18)(11,20)(13,22)(15,24)(17,44)(19,46)(21,48)(23,42)(25,35)(26,64)(27,37)(28,58)(29,39)(30,60)(31,33)(32,62)(34,56)(36,50)(38,52)(40,54)(49,63)(51,57)(53,59)(55,61), (1,17)(2,18)(3,19)(4,20)(5,21)(6,22)(7,23)(8,24)(9,45)(10,46)(11,47)(12,48)(13,41)(14,42)(15,43)(16,44)(25,49)(26,50)(27,51)(28,52)(29,53)(30,54)(31,55)(32,56)(33,61)(34,62)(35,63)(36,64)(37,57)(38,58)(39,59)(40,60), (1,55)(2,56)(3,49)(4,50)(5,51)(6,52)(7,53)(8,54)(9,62)(10,63)(11,64)(12,57)(13,58)(14,59)(15,60)(16,61)(17,31)(18,32)(19,25)(20,26)(21,27)(22,28)(23,29)(24,30)(33,44)(34,45)(35,46)(36,47)(37,48)(38,41)(39,42)(40,43), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32)(33,37)(34,38)(35,39)(36,40)(41,45)(42,46)(43,47)(44,48)(49,53)(50,54)(51,55)(52,56)(57,61)(58,62)(59,63)(60,64), (1,10)(2,64)(3,12)(4,58)(5,14)(6,60)(7,16)(8,62)(9,54)(11,56)(13,50)(15,52)(17,46)(18,36)(19,48)(20,38)(21,42)(22,40)(23,44)(24,34)(25,37)(26,41)(27,39)(28,43)(29,33)(30,45)(31,35)(32,47)(49,57)(51,59)(53,61)(55,63), (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([[(1,16),(2,45),(3,10),(4,47),(5,12),(6,41),(7,14),(8,43),(9,18),(11,20),(13,22),(15,24),(17,44),(19,46),(21,48),(23,42),(25,35),(26,64),(27,37),(28,58),(29,39),(30,60),(31,33),(32,62),(34,56),(36,50),(38,52),(40,54),(49,63),(51,57),(53,59),(55,61)], [(1,17),(2,18),(3,19),(4,20),(5,21),(6,22),(7,23),(8,24),(9,45),(10,46),(11,47),(12,48),(13,41),(14,42),(15,43),(16,44),(25,49),(26,50),(27,51),(28,52),(29,53),(30,54),(31,55),(32,56),(33,61),(34,62),(35,63),(36,64),(37,57),(38,58),(39,59),(40,60)], [(1,55),(2,56),(3,49),(4,50),(5,51),(6,52),(7,53),(8,54),(9,62),(10,63),(11,64),(12,57),(13,58),(14,59),(15,60),(16,61),(17,31),(18,32),(19,25),(20,26),(21,27),(22,28),(23,29),(24,30),(33,44),(34,45),(35,46),(36,47),(37,48),(38,41),(39,42),(40,43)], [(1,5),(2,6),(3,7),(4,8),(9,13),(10,14),(11,15),(12,16),(17,21),(18,22),(19,23),(20,24),(25,29),(26,30),(27,31),(28,32),(33,37),(34,38),(35,39),(36,40),(41,45),(42,46),(43,47),(44,48),(49,53),(50,54),(51,55),(52,56),(57,61),(58,62),(59,63),(60,64)], [(1,10),(2,64),(3,12),(4,58),(5,14),(6,60),(7,16),(8,62),(9,54),(11,56),(13,50),(15,52),(17,46),(18,36),(19,48),(20,38),(21,42),(22,40),(23,44),(24,34),(25,37),(26,41),(27,39),(28,43),(29,33),(30,45),(31,35),(32,47),(49,57),(51,59),(53,61),(55,63)], [(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)]])

44 conjugacy classes

 class 1 2A ··· 2G 2H ··· 2M 4A ··· 4H 4I ··· 4N 8A ··· 8P order 1 2 ··· 2 2 ··· 2 4 ··· 4 4 ··· 4 8 ··· 8 size 1 1 ··· 1 4 ··· 4 1 ··· 1 4 ··· 4 4 ··· 4

44 irreducible representations

 dim 1 1 1 1 1 2 2 type + + + + image C1 C2 C2 C4 C4 D4 C8○D4 kernel C24.51(C2×C4) C2×C22⋊C8 C22×C4○D4 C22×D4 C22×Q8 C22×C4 C22 # reps 1 6 1 6 2 12 16

Matrix representation of C24.51(C2×C4) in GL6(𝔽17)

 16 0 0 0 0 0 0 16 0 0 0 0 0 0 4 9 0 0 0 0 4 13 0 0 0 0 0 0 16 2 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 0 16
,
 16 0 0 0 0 0 0 16 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 16 0 0 0 0 0 0 16
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 2 16 0 0 0 0 0 0 1 15 0 0 0 0 0 16 0 0 0 0 0 0 1 15 0 0 0 0 0 16
,
 13 4 0 0 0 0 0 4 0 0 0 0 0 0 15 4 0 0 0 0 0 2 0 0 0 0 0 0 9 6 0 0 0 0 9 8

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,4,4,0,0,0,0,9,13,0,0,0,0,0,0,16,0,0,0,0,0,2,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,2,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,15,16,0,0,0,0,0,0,1,0,0,0,0,0,15,16],[13,0,0,0,0,0,4,4,0,0,0,0,0,0,15,0,0,0,0,0,4,2,0,0,0,0,0,0,9,9,0,0,0,0,6,8] >;

C24.51(C2×C4) in GAP, Magma, Sage, TeX

C_2^4._{51}(C_2\times C_4)
% in TeX

G:=Group("C2^4.51(C2xC4)");
// GroupNames label

G:=SmallGroup(128,512);
// by ID

G=gap.SmallGroup(128,512);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,422,723,124]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=e^2=1,f^4=d,f*a*f^-1=a*b=b*a,a*c=c*a,e*a*e=a*d=d*a,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,c*d=d*c,f*e*f^-1=c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d>;
// generators/relations

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