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## G = C2×C22.31C24order 128 = 27

### Direct product of C2 and C22.31C24

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22 — C2×C22.31C24
 Chief series C1 — C2 — C22 — C23 — C22×C4 — C23×C4 — C22×C4○D4 — C2×C22.31C24
 Lower central C1 — C22 — C2×C22.31C24
 Upper central C1 — C23 — C2×C22.31C24
 Jennings C1 — C22 — C2×C22.31C24

Generators and relations for C2×C22.31C24
G = < a,b,c,d,e,f,g | a2=b2=c2=d2=e2=f2=1, g2=b, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, ede=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: 1324 in 820 conjugacy classes, 436 normal (10 characteristic)
C1, C2 [×3], C2 [×4], C2 [×12], C4 [×8], C4 [×16], C22, C22 [×10], C22 [×52], C2×C4 [×44], C2×C4 [×56], D4 [×64], Q8 [×16], C23, C23 [×14], C23 [×28], C22⋊C4 [×32], C4⋊C4 [×32], C22×C4 [×46], C22×C4 [×20], C2×D4 [×40], C2×D4 [×32], C2×Q8 [×8], C2×Q8 [×8], C4○D4 [×64], C24, C24 [×4], C2×C22⋊C4 [×8], C2×C4⋊C4 [×16], C4⋊D4 [×64], C22⋊Q8 [×32], C23×C4, C23×C4 [×6], C22×D4 [×10], C22×Q8 [×2], C2×C4○D4 [×16], C2×C4○D4 [×16], C22×C4⋊C4, C2×C4⋊D4 [×8], C2×C22⋊Q8 [×4], C22.31C24 [×16], C22×C4○D4 [×2], C2×C22.31C24
Quotients: C1, C2 [×31], C22 [×155], D4 [×8], C23 [×155], C2×D4 [×28], C24 [×31], C22×D4 [×14], 2+ 1+4 [×2], 2- 1+4 [×2], C25, C22.31C24 [×4], D4×C23, C2×2+ 1+4, C2×2- 1+4, C2×C22.31C24

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

44 irreducible representations

 dim 1 1 1 1 1 1 2 4 4 type + + + + + + + + - image C1 C2 C2 C2 C2 C2 D4 2+ 1+4 2- 1+4 kernel C2×C22.31C24 C22×C4⋊C4 C2×C4⋊D4 C2×C22⋊Q8 C22.31C24 C22×C4○D4 C22×C4 C22 C22 # reps 1 1 8 4 16 2 8 2 2

Matrix representation of C2×C22.31C24 in GL8(𝔽5)

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

G:=sub<GL(8,GF(5))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[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],[4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,1,0,1,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,2,1],[4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,4,0,3,0,0,0,0,4,3,4,3,0,0,0,0,2,2,1,4,0,0,0,0,0,1,3,1],[0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,2,0,4,0,0,0,0,3,0,3,4,0,0,0,0,0,0,2,2,0,0,0,0,0,0,1,3],[4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,4,0,1,1,0,0,0,0,0,0,4,4,0,0,0,0,0,0,2,1] >;

C2×C22.31C24 in GAP, Magma, Sage, TeX

C_2\times C_2^2._{31}C_2^4
% in TeX

G:=Group("C2xC2^2.31C2^4");
// GroupNames label

G:=SmallGroup(128,2180);
// by ID

G=gap.SmallGroup(128,2180);
# by ID

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

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