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## G = C24.135D4order 128 = 27

### 4th non-split extension by C24 of D4 acting via D4/C4=C2

p-group, metabelian, nilpotent (class 3), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — C24.135D4
 Chief series C1 — C2 — C22 — C23 — C22×C4 — C23×C4 — C23×C8 — C24.135D4
 Lower central C1 — C2 — C2×C4 — C24.135D4
 Upper central C1 — C23 — C23×C4 — C24.135D4
 Jennings C1 — C2 — C2 — C22×C4 — C24.135D4

Generators and relations for C24.135D4
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e4=f2=d, faf-1=ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=ce3 >

Subgroups: 356 in 190 conjugacy classes, 72 normal (22 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×2], C4 [×2], C4 [×8], C22 [×3], C22 [×8], C22 [×12], C8 [×4], C2×C4 [×2], C2×C4 [×6], C2×C4 [×24], Q8 [×6], C23, C23 [×6], C23 [×4], C22⋊C4 [×6], C4⋊C4 [×2], C4⋊C4 [×9], C2×C8 [×4], C2×C8 [×12], C22×C4 [×2], C22×C4 [×4], C22×C4 [×8], C2×Q8 [×2], C2×Q8 [×5], C24, C2.C42, Q8⋊C4 [×4], C2×C22⋊C4 [×2], C2×C4⋊C4, C2×C4⋊C4 [×2], C2×C4⋊C4, C22⋊Q8 [×4], C22⋊Q8 [×2], C22×C8 [×2], C22×C8 [×6], C23×C4, C22×Q8, C22.4Q16 [×2], C23.7Q8, C2×Q8⋊C4 [×2], C2×C22⋊Q8, C23×C8, C24.135D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×8], C23, C22⋊C4 [×4], SD16 [×2], Q16 [×2], C22×C4, C2×D4 [×4], C4○D4 [×2], Q8⋊C4 [×4], C2×C22⋊C4, C4×D4 [×2], C22≀C2, C4⋊D4 [×2], C22.D4, C2×SD16, C2×Q16, C4○D8 [×2], C23.23D4, C2×Q8⋊C4, C23.24D4, C88D4 [×2], C8.18D4 [×2], C24.135D4

Smallest permutation representation of C24.135D4
On 64 points
Generators in S64
(1 44)(2 45)(3 46)(4 47)(5 48)(6 41)(7 42)(8 43)(9 22)(10 23)(11 24)(12 17)(13 18)(14 19)(15 20)(16 21)(25 62)(26 63)(27 64)(28 57)(29 58)(30 59)(31 60)(32 61)(33 53)(34 54)(35 55)(36 56)(37 49)(38 50)(39 51)(40 52)
(1 39)(2 40)(3 33)(4 34)(5 35)(6 36)(7 37)(8 38)(9 61)(10 62)(11 63)(12 64)(13 57)(14 58)(15 59)(16 60)(17 27)(18 28)(19 29)(20 30)(21 31)(22 32)(23 25)(24 26)(41 56)(42 49)(43 50)(44 51)(45 52)(46 53)(47 54)(48 55)
(1 44)(2 45)(3 46)(4 47)(5 48)(6 41)(7 42)(8 43)(9 32)(10 25)(11 26)(12 27)(13 28)(14 29)(15 30)(16 31)(17 64)(18 57)(19 58)(20 59)(21 60)(22 61)(23 62)(24 63)(33 53)(34 54)(35 55)(36 56)(37 49)(38 50)(39 51)(40 52)
(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 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 57 5 61)(2 21 6 17)(3 63 7 59)(4 19 8 23)(9 39 13 35)(10 54 14 50)(11 37 15 33)(12 52 16 56)(18 48 22 44)(20 46 24 42)(25 34 29 38)(26 49 30 53)(27 40 31 36)(28 55 32 51)(41 64 45 60)(43 62 47 58)

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

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

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

44 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 4A ··· 4H 4I ··· 4P 8A ··· 8P order 1 2 ··· 2 2 2 2 2 4 ··· 4 4 ··· 4 8 ··· 8 size 1 1 ··· 1 2 2 2 2 2 ··· 2 8 ··· 8 2 ··· 2

44 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 type + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C4 D4 D4 D4 C4○D4 SD16 Q16 C4○D8 kernel C24.135D4 C22.4Q16 C23.7Q8 C2×Q8⋊C4 C2×C22⋊Q8 C23×C8 C22⋊Q8 C2×C8 C22×C4 C24 C2×C4 C23 C23 C22 # reps 1 2 1 2 1 1 8 4 3 1 4 4 4 8

Matrix representation of C24.135D4 in GL5(𝔽17)

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

G:=sub<GL(5,GF(17))| [1,0,0,0,0,0,16,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,1],[1,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,16],[16,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,16],[1,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,1,0,0,0,0,0,1],[4,0,0,0,0,0,2,0,0,0,0,0,9,0,0,0,0,0,13,0,0,0,0,0,13],[1,0,0,0,0,0,0,15,0,0,0,9,0,0,0,0,0,0,0,4,0,0,0,13,0] >;

C24.135D4 in GAP, Magma, Sage, TeX

C_2^4._{135}D_4
% in TeX

G:=Group("C2^4.135D4");
// GroupNames label

G:=SmallGroup(128,624);
// by ID

G=gap.SmallGroup(128,624);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,288,422,2019,248]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^4=f^2=d,f*a*f^-1=a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=c*e^3>;
// generators/relations

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