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## G = C2×C24.C22order 128 = 27

### Direct product of C2 and C24.C22

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

Series: Derived Chief Lower central Upper central Jennings

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

Generators and relations for C2×C24.C22
G = < a,b,c,d,e,f,g | a2=b2=c2=d2=e2=1, f2=e, g2=c, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, fbf-1=bc=cb, gbg-1=bd=db, be=eb, cd=dc, ce=ec, cf=fc, cg=gc, de=ed, gfg-1=df=fd, dg=gd, ef=fe, eg=ge >

Subgroups: 844 in 464 conjugacy classes, 196 normal (32 characteristic)
C1, C2 [×7], C2 [×8], C2 [×4], C4 [×20], C22 [×7], C22 [×28], C22 [×36], C2×C4 [×16], C2×C4 [×68], C23, C23 [×18], C23 [×52], C42 [×8], C22⋊C4 [×16], C22⋊C4 [×16], C4⋊C4 [×8], C22×C4 [×28], C22×C4 [×28], C24, C24 [×6], C24 [×12], C2.C42 [×8], C2×C42 [×4], C2×C42 [×4], C2×C22⋊C4 [×20], C2×C22⋊C4 [×8], C2×C4⋊C4 [×4], C2×C4⋊C4 [×4], C23×C4 [×6], C25, C2×C2.C42 [×2], C24.C22 [×8], C22×C42, C22×C22⋊C4 [×3], C22×C4⋊C4, C2×C24.C22
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], D4 [×8], C23 [×15], C22×C4 [×14], C2×D4 [×12], C4○D4 [×8], C24, C42⋊C2 [×4], C4×D4 [×8], C4⋊D4 [×4], C22.D4 [×4], C4.4D4 [×4], C422C2 [×4], C23×C4, C22×D4 [×2], C2×C4○D4 [×4], C24.C22 [×8], C2×C42⋊C2, C2×C4×D4 [×2], C2×C4⋊D4, C2×C22.D4, C2×C4.4D4, C2×C422C2, C2×C24.C22

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

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

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

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

56 conjugacy classes

 class 1 2A ··· 2O 2P 2Q 2R 2S 4A ··· 4X 4Y ··· 4AJ order 1 2 ··· 2 2 2 2 2 4 ··· 4 4 ··· 4 size 1 1 ··· 1 4 4 4 4 2 ··· 2 4 ··· 4

56 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 type + + + + + + + image C1 C2 C2 C2 C2 C2 C4 D4 C4○D4 kernel C2×C24.C22 C2×C2.C42 C24.C22 C22×C42 C22×C22⋊C4 C22×C4⋊C4 C2×C22⋊C4 C22×C4 C23 # reps 1 2 8 1 3 1 16 8 16

Matrix representation of C2×C24.C22 in GL6(𝔽5)

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

G:=sub<GL(6,GF(5))| [1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,0,4],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[2,0,0,0,0,0,0,1,0,0,0,0,0,0,0,4,0,0,0,0,4,0,0,0,0,0,0,0,2,2,0,0,0,0,1,3],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,1,0,0,0,0,4,0,0,0,0,0,0,0,3,0,0,0,0,0,0,3] >;

C2×C24.C22 in GAP, Magma, Sage, TeX

C_2\times C_2^4.C_2^2
% in TeX

G:=Group("C2xC2^4.C2^2");
// GroupNames label

G:=SmallGroup(128,1021);
// by ID

G=gap.SmallGroup(128,1021);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,568,758,100]);
// Polycyclic

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

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