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## G = C24.53(C2×C4)  order 128 = 27

### 18th 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 — C24.53(C2×C4)
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C2×C42 — C2×C42⋊C2 — C24.53(C2×C4)
 Lower central C1 — C23 — C24.53(C2×C4)
 Upper central C1 — C22×C4 — C24.53(C2×C4)
 Jennings C1 — C2 — C2 — C22×C4 — C24.53(C2×C4)

Generators and relations for C24.53(C2×C4)
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e2=b, f4=d, faf-1=ab=ba, ac=ca, eae-1=ad=da, bc=cb, bd=db, be=eb, bf=fb, cd=dc, fef-1=ce=ec, cf=fc, de=ed, df=fd >

Subgroups: 260 in 146 conjugacy classes, 60 normal (10 characteristic)
C1, C2, C2 [×6], C2 [×2], C4 [×4], C4 [×6], C22, C22 [×6], C22 [×10], C8 [×4], C2×C4 [×8], C2×C4 [×22], C23, C23 [×2], C23 [×6], C42 [×4], C22⋊C4 [×4], C4⋊C4 [×4], C2×C8 [×12], C22×C4 [×2], C22×C4 [×8], C22×C4 [×4], C24, C22⋊C8 [×4], C2×C42 [×2], C2×C22⋊C4 [×2], C2×C4⋊C4 [×2], C42⋊C2 [×4], C22×C8 [×4], C23×C4, C22.7C42 [×4], C2×C22⋊C8 [×2], C2×C42⋊C2, C24.53(C2×C4)
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×2], C4○D4 [×4], C2×C22⋊C4, C42⋊C2 [×2], C22.D4 [×4], C8○D4 [×4], C23.34D4, (C22×C8)⋊C2 [×2], C42.7C22 [×4], C24.53(C2×C4)

Smallest permutation representation of C24.53(C2×C4)
On 64 points
Generators in S64
(2 24)(4 18)(6 20)(8 22)(9 46)(10 14)(11 48)(12 16)(13 42)(15 44)(26 50)(28 52)(30 54)(32 56)(33 37)(34 58)(35 39)(36 60)(38 62)(40 64)(41 45)(43 47)(57 61)(59 63)
(1 23)(2 24)(3 17)(4 18)(5 19)(6 20)(7 21)(8 22)(9 42)(10 43)(11 44)(12 45)(13 46)(14 47)(15 48)(16 41)(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 38)(10 39)(11 40)(12 33)(13 34)(14 35)(15 36)(16 37)(17 25)(18 26)(19 27)(20 28)(21 29)(22 30)(23 31)(24 32)(41 57)(42 58)(43 59)(44 60)(45 61)(46 62)(47 63)(48 64)
(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 14 23 47)(2 36 24 64)(3 16 17 41)(4 38 18 58)(5 10 19 43)(6 40 20 60)(7 12 21 45)(8 34 22 62)(9 26 42 50)(11 28 44 52)(13 30 46 54)(15 32 48 56)(25 57 49 37)(27 59 51 39)(29 61 53 33)(31 63 55 35)
(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)| (2,24)(4,18)(6,20)(8,22)(9,46)(10,14)(11,48)(12,16)(13,42)(15,44)(26,50)(28,52)(30,54)(32,56)(33,37)(34,58)(35,39)(36,60)(38,62)(40,64)(41,45)(43,47)(57,61)(59,63), (1,23)(2,24)(3,17)(4,18)(5,19)(6,20)(7,21)(8,22)(9,42)(10,43)(11,44)(12,45)(13,46)(14,47)(15,48)(16,41)(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)(9,38)(10,39)(11,40)(12,33)(13,34)(14,35)(15,36)(16,37)(17,25)(18,26)(19,27)(20,28)(21,29)(22,30)(23,31)(24,32)(41,57)(42,58)(43,59)(44,60)(45,61)(46,62)(47,63)(48,64), (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), (1,14,23,47)(2,36,24,64)(3,16,17,41)(4,38,18,58)(5,10,19,43)(6,40,20,60)(7,12,21,45)(8,34,22,62)(9,26,42,50)(11,28,44,52)(13,30,46,54)(15,32,48,56)(25,57,49,37)(27,59,51,39)(29,61,53,33)(31,63,55,35), (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( (2,24)(4,18)(6,20)(8,22)(9,46)(10,14)(11,48)(12,16)(13,42)(15,44)(26,50)(28,52)(30,54)(32,56)(33,37)(34,58)(35,39)(36,60)(38,62)(40,64)(41,45)(43,47)(57,61)(59,63), (1,23)(2,24)(3,17)(4,18)(5,19)(6,20)(7,21)(8,22)(9,42)(10,43)(11,44)(12,45)(13,46)(14,47)(15,48)(16,41)(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)(9,38)(10,39)(11,40)(12,33)(13,34)(14,35)(15,36)(16,37)(17,25)(18,26)(19,27)(20,28)(21,29)(22,30)(23,31)(24,32)(41,57)(42,58)(43,59)(44,60)(45,61)(46,62)(47,63)(48,64), (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), (1,14,23,47)(2,36,24,64)(3,16,17,41)(4,38,18,58)(5,10,19,43)(6,40,20,60)(7,12,21,45)(8,34,22,62)(9,26,42,50)(11,28,44,52)(13,30,46,54)(15,32,48,56)(25,57,49,37)(27,59,51,39)(29,61,53,33)(31,63,55,35), (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([(2,24),(4,18),(6,20),(8,22),(9,46),(10,14),(11,48),(12,16),(13,42),(15,44),(26,50),(28,52),(30,54),(32,56),(33,37),(34,58),(35,39),(36,60),(38,62),(40,64),(41,45),(43,47),(57,61),(59,63)], [(1,23),(2,24),(3,17),(4,18),(5,19),(6,20),(7,21),(8,22),(9,42),(10,43),(11,44),(12,45),(13,46),(14,47),(15,48),(16,41),(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),(9,38),(10,39),(11,40),(12,33),(13,34),(14,35),(15,36),(16,37),(17,25),(18,26),(19,27),(20,28),(21,29),(22,30),(23,31),(24,32),(41,57),(42,58),(43,59),(44,60),(45,61),(46,62),(47,63),(48,64)], [(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)], [(1,14,23,47),(2,36,24,64),(3,16,17,41),(4,38,18,58),(5,10,19,43),(6,40,20,60),(7,12,21,45),(8,34,22,62),(9,26,42,50),(11,28,44,52),(13,30,46,54),(15,32,48,56),(25,57,49,37),(27,59,51,39),(29,61,53,33),(31,63,55,35)], [(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 2I 4A ··· 4H 4I ··· 4R 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 1 2 2 2 type + + + + + image C1 C2 C2 C2 C4 C4 D4 C4○D4 C8○D4 kernel C24.53(C2×C4) C22.7C42 C2×C22⋊C8 C2×C42⋊C2 C2×C22⋊C4 C2×C4⋊C4 C22×C4 C2×C4 C22 # reps 1 4 2 1 4 4 4 8 16

Matrix representation of C24.53(C2×C4) in GL6(𝔽17)

 1 0 0 0 0 0 9 16 0 0 0 0 0 0 1 0 0 0 0 0 0 16 0 0 0 0 0 0 1 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 1 0 0 0 0 0 0 1
,
 16 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 1 0 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
,
 13 0 0 0 0 0 15 4 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0
,
 8 2 0 0 0 0 10 9 0 0 0 0 0 0 8 0 0 0 0 0 0 9 0 0 0 0 0 0 9 0 0 0 0 0 0 9

G:=sub<GL(6,GF(17))| [1,9,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[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],[13,15,0,0,0,0,0,4,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[8,10,0,0,0,0,2,9,0,0,0,0,0,0,8,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,9] >;

C24.53(C2×C4) in GAP, Magma, Sage, TeX

C_2^4._{53}(C_2\times C_4)
% in TeX

G:=Group("C2^4.53(C2xC4)");
// GroupNames label

G:=SmallGroup(128,550);
// by ID

G=gap.SmallGroup(128,550);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,422,723,58,124]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^2=b,f^4=d,f*a*f^-1=a*b=b*a,a*c=c*a,e*a*e^-1=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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