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## G = C2×C22.34C24order 128 = 27

### Direct product of C2 and C22.34C24

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22 — C2×C22.34C24
 Chief series C1 — C2 — C22 — C23 — C22×C4 — C23×C4 — C2×C42⋊C2 — C2×C22.34C24
 Lower central C1 — C22 — C2×C22.34C24
 Upper central C1 — C23 — C2×C22.34C24
 Jennings C1 — C22 — C2×C22.34C24

Generators and relations for C2×C22.34C24
G = < a,b,c,d,e,f,g | a2=b2=c2=d2=f2=1, e2=c, g2=b, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, ede-1=gdg-1=bd=db, fef=be=eb, bf=fb, bg=gb, fdf=cd=dc, ce=ec, cf=fc, cg=gc, eg=ge, fg=gf >

Subgroups: 1068 in 630 conjugacy classes, 396 normal (14 characteristic)
C1, C2, C2 [×6], C2 [×10], C4 [×4], C4 [×18], C22, C22 [×6], C22 [×50], C2×C4 [×24], C2×C4 [×42], D4 [×48], C23, C23 [×10], C23 [×30], C42 [×8], C22⋊C4 [×40], C4⋊C4 [×32], C22×C4 [×2], C22×C4 [×28], C22×C4 [×12], C2×D4 [×40], C2×D4 [×24], C24, C24 [×4], C2×C42 [×2], C2×C22⋊C4 [×10], C2×C4⋊C4 [×8], C42⋊C2 [×8], C4×D4 [×16], C4⋊D4 [×48], C22.D4 [×32], C42.C2 [×8], C41D4 [×8], C23×C4, C23×C4 [×4], C22×D4 [×10], C2×C42⋊C2, C2×C4×D4 [×2], C2×C4⋊D4 [×6], C2×C22.D4 [×4], C2×C42.C2, C2×C41D4, C22.34C24 [×16], C2×C22.34C24
Quotients: C1, C2 [×31], C22 [×155], C23 [×155], C4○D4 [×4], C24 [×31], C2×C4○D4 [×6], 2+ 1+4 [×4], C25, C22.34C24 [×4], C22×C4○D4, C2×2+ 1+4 [×2], C2×C22.34C24

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

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

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

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

44 conjugacy classes

 class 1 2A ··· 2G 2H ··· 2Q 4A ··· 4L 4M ··· 4Z order 1 2 ··· 2 2 ··· 2 4 ··· 4 4 ··· 4 size 1 1 ··· 1 4 ··· 4 2 ··· 2 4 ··· 4

44 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 4 type + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C4○D4 2+ 1+4 kernel C2×C22.34C24 C2×C42⋊C2 C2×C4×D4 C2×C4⋊D4 C2×C22.D4 C2×C42.C2 C2×C4⋊1D4 C22.34C24 C2×C4 C22 # reps 1 1 2 6 4 1 1 16 8 4

Matrix representation of C2×C22.34C24 in GL8(𝔽5)

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

G:=sub<GL(8,GF(5))| [4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[2,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0] >;

C2×C22.34C24 in GAP, Magma, Sage, TeX

C_2\times C_2^2._{34}C_2^4
% in TeX

G:=Group("C2xC2^2.34C2^4");
// GroupNames label

G:=SmallGroup(128,2184);
// by ID

G=gap.SmallGroup(128,2184);
# by ID

G:=PCGroup([7,-2,2,2,2,2,-2,2,477,456,1430,387,1123,136]);
// Polycyclic

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

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