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## G = C2×C22.57C24order 128 = 27

### Direct product of C2 and C22.57C24

direct product, p-group, metabelian, nilpotent (class 2), monomial, rational

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22 — C2×C22.57C24
 Chief series C1 — C2 — C22 — C23 — C24 — C23×C4 — C2×C22.D4 — C2×C22.57C24
 Lower central C1 — C22 — C2×C22.57C24
 Upper central C1 — C23 — C2×C22.57C24
 Jennings C1 — C22 — C2×C22.57C24

Generators and relations for C2×C22.57C24
G = < a,b,c,d,e,f,g | a2=b2=c2=g2=1, d2=e2=f2=b, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, ede-1=bd=db, geg=be=eb, bf=fb, bg=gb, fdf-1=cd=dc, ce=ec, cf=fc, cg=gc, gdg=bcd, fef-1=bce, fg=gf >

Subgroups: 732 in 512 conjugacy classes, 388 normal (10 characteristic)
C1, C2, C2 [×6], C2 [×4], C4 [×26], C22, C22 [×6], C22 [×20], C2×C4 [×26], C2×C4 [×34], D4 [×4], Q8 [×12], C23, C23 [×4], C23 [×12], C42 [×12], C22⋊C4 [×40], C4⋊C4 [×64], C22×C4, C22×C4 [×20], C22×C4 [×4], C2×D4 [×4], C2×D4 [×2], C2×Q8 [×12], C2×Q8 [×6], C24 [×2], C2×C42, C2×C42 [×2], C2×C22⋊C4 [×10], C2×C4⋊C4 [×16], C22⋊Q8 [×32], C22.D4 [×16], C4.4D4 [×8], C42.C2 [×16], C422C2 [×32], C4⋊Q8 [×16], C23×C4 [×2], C22×D4, C22×Q8, C22×Q8 [×2], C2×C22⋊Q8 [×4], C2×C22.D4 [×2], C2×C4.4D4, C2×C42.C2 [×2], C2×C422C2 [×4], C2×C4⋊Q8 [×2], C22.57C24 [×16], C2×C22.57C24
Quotients: C1, C2 [×31], C22 [×155], C23 [×155], C24 [×31], 2+ 1+4 [×2], 2- 1+4 [×4], C25, C22.57C24 [×4], C2×2+ 1+4, C2×2- 1+4 [×2], C2×C22.57C24

Smallest permutation representation of C2×C22.57C24
On 64 points
Generators in S64
(1 49)(2 50)(3 51)(4 52)(5 20)(6 17)(7 18)(8 19)(9 55)(10 56)(11 53)(12 54)(13 63)(14 64)(15 61)(16 62)(21 59)(22 60)(23 57)(24 58)(25 43)(26 44)(27 41)(28 42)(29 35)(30 36)(31 33)(32 34)(37 48)(38 45)(39 46)(40 47)
(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 19)(2 20)(3 17)(4 18)(5 50)(6 51)(7 52)(8 49)(9 29)(10 30)(11 31)(12 32)(13 40)(14 37)(15 38)(16 39)(21 25)(22 26)(23 27)(24 28)(33 53)(34 54)(35 55)(36 56)(41 57)(42 58)(43 59)(44 60)(45 61)(46 62)(47 63)(48 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 41 3 43)(2 44 4 42)(5 22 7 24)(6 21 8 23)(9 47 11 45)(10 46 12 48)(13 33 15 35)(14 36 16 34)(17 59 19 57)(18 58 20 60)(25 49 27 51)(26 52 28 50)(29 63 31 61)(30 62 32 64)(37 56 39 54)(38 55 40 53)
(1 33 3 35)(2 54 4 56)(5 32 7 30)(6 9 8 11)(10 50 12 52)(13 59 15 57)(14 44 16 42)(17 55 19 53)(18 36 20 34)(21 61 23 63)(22 46 24 48)(25 45 27 47)(26 62 28 64)(29 49 31 51)(37 60 39 58)(38 41 40 43)
(2 18)(4 20)(5 52)(7 50)(10 32)(12 30)(13 15)(14 37)(16 39)(21 23)(22 26)(24 28)(25 27)(34 56)(36 54)(38 40)(41 43)(42 58)(44 60)(45 47)(46 62)(48 64)(57 59)(61 63)

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

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

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

38 conjugacy classes

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

38 irreducible representations

 dim 1 1 1 1 1 1 1 1 4 4 type + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 C2 2+ 1+4 2- 1+4 kernel C2×C22.57C24 C2×C22⋊Q8 C2×C22.D4 C2×C4.4D4 C2×C42.C2 C2×C42⋊2C2 C2×C4⋊Q8 C22.57C24 C22 C22 # reps 1 4 2 1 2 4 2 16 2 4

Matrix representation of C2×C22.57C24 in GL12(𝔽5)

 4 0 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 4
,
 4 0 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 4
,
 4 0 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1
,
 2 0 0 0 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 0 2 3 2 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 0 3 2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 3 0 0
,
 0 0 4 0 0 0 0 0 0 0 0 0 4 1 4 2 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 4 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 1 4 3 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 1 1 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 4 0 0
,
 0 4 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 4 1 4 2 0 0 0 0 0 0 0 0 4 0 4 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 1 1 4 3 0 0 0 0 0 0 0 0 4 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 1 0
,
 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 1 4 0 4 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 1 1 0 4 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 4

G:=sub<GL(12,GF(5))| [4,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1],[2,0,0,0,0,0,0,0,0,0,0,0,0,3,0,2,0,0,0,0,0,0,0,0,0,0,3,3,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,3,0,3,0,0,0,0,0,0,0,0,0,0,3,2,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0],[0,4,1,1,0,0,0,0,0,0,0,0,0,1,0,4,0,0,0,0,0,0,0,0,4,4,0,0,0,0,0,0,0,0,0,0,0,2,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,1,4,1,0,0,0,0,0,0,0,0,0,1,0,1,0,0,0,0,0,0,0,0,1,4,0,0,0,0,0,0,0,0,0,0,0,3,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[0,1,4,4,0,0,0,0,0,0,0,0,4,0,1,0,0,0,0,0,0,0,0,0,0,0,4,4,0,0,0,0,0,0,0,0,0,0,2,1,0,0,0,0,0,0,0,0,0,0,0,0,0,4,1,4,0,0,0,0,0,0,0,0,1,0,1,0,0,0,0,0,0,0,0,0,0,0,4,1,0,0,0,0,0,0,0,0,0,0,3,1,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,4,0],[1,0,0,1,0,0,0,0,0,0,0,0,0,1,0,4,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,1,0,1,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4] >;

C2×C22.57C24 in GAP, Magma, Sage, TeX

C_2\times C_2^2._{57}C_2^4
% in TeX

G:=Group("C2xC2^2.57C2^4");
// GroupNames label

G:=SmallGroup(128,2260);
// by ID

G=gap.SmallGroup(128,2260);
# by ID

G:=PCGroup([7,-2,2,2,2,2,-2,2,224,477,232,1430,1059,184,2915,570]);
// Polycyclic

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

׿
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