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## G = C2×C22.56C24order 128 = 27

### Direct product of C2 and C22.56C24

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22 — C2×C22.56C24
 Chief series C1 — C2 — C22 — C23 — C24 — C23×C4 — C2×C22.D4 — C2×C22.56C24
 Lower central C1 — C22 — C2×C22.56C24
 Upper central C1 — C23 — C2×C22.56C24
 Jennings C1 — C22 — C2×C22.56C24

Generators and relations for C2×C22.56C24
G = < a,b,c,d,e,f,g | a2=b2=c2=d2=e2=g2=1, f2=b, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, ede=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: 956 in 576 conjugacy classes, 388 normal (8 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C2×C4, C2×C4, D4, Q8, C23, C23, C23, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C24, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C42.C2, C23×C4, C22×D4, C22×Q8, C2×C4⋊D4, C2×C22⋊Q8, C2×C22.D4, C2×C4.4D4, C2×C42.C2, C22.56C24, C2×C22.56C24
Quotients: C1, C2, C22, C23, C24, 2+ 1+4, 2- 1+4, C25, C22.56C24, C2×2+ 1+4, C2×2- 1+4, C2×C22.56C24

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

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

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

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

38 conjugacy classes

 class 1 2A ··· 2G 2H ··· 2O 4A ··· 4V order 1 2 ··· 2 2 ··· 2 4 ··· 4 size 1 1 ··· 1 4 ··· 4 4 ··· 4

38 irreducible representations

 dim 1 1 1 1 1 1 1 4 4 type + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 2+ 1+4 2- 1+4 kernel C2×C22.56C24 C2×C4⋊D4 C2×C22⋊Q8 C2×C22.D4 C2×C4.4D4 C2×C42.C2 C22.56C24 C22 C22 # reps 1 4 4 4 2 1 16 4 2

Matrix representation of C2×C22.56C24 in GL9(𝔽5)

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

G:=sub<GL(9,GF(5))| [4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,0,0,0,0,3,2,0,0,0,0,0,0,0,3,0,2,0,0,0,0,0,3,3,0,2,0,0,0,0,0,0,1,0,2,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,3,0,0,3,0,0,0,0,0,0,3,2,0],[1,0,0,0,0,0,0,0,0,0,0,4,1,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,1,4,0,0,0,0,0,0,0,0,3,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,4,0],[4,0,0,0,0,0,0,0,0,0,0,4,1,0,0,0,0,0,0,1,0,1,4,0,0,0,0,0,0,0,1,4,0,0,0,0,0,0,0,2,4,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,0,0,0,0,3,4,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,2],[4,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,0,0,0,0,1,0,4,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,0,0,0,0,4,2,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,4] >;

C2×C22.56C24 in GAP, Magma, Sage, TeX

C_2\times C_2^2._{56}C_2^4
% in TeX

G:=Group("C2xC2^2.56C2^4");
// GroupNames label

G:=SmallGroup(128,2259);
// by ID

G=gap.SmallGroup(128,2259);
# by ID

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

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