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## G = C2×C42.7C22order 128 = 27

### Direct product of C2 and C42.7C22

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22 — C2×C42.7C22
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C2×C42 — C2×C42⋊C2 — C2×C42.7C22
 Lower central C1 — C22 — C2×C42.7C22
 Upper central C1 — C22×C4 — C2×C42.7C22
 Jennings C1 — C2 — C2 — C2×C4 — C2×C42.7C22

Generators and relations for C2×C42.7C22
G = < a,b,c,d,e | a2=b4=c4=e2=1, d2=c, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=b-1c2, ebe=bc2, cd=dc, ce=ec, ede=b2c2d >

Subgroups: 284 in 206 conjugacy classes, 140 normal (16 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, C23, C23, C23, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C22×C4, C22×C4, C24, C4×C8, C8⋊C4, C22⋊C8, C4⋊C8, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C22×C8, C23×C4, C2×C4×C8, C2×C8⋊C4, C2×C22⋊C8, C2×C4⋊C8, C42.7C22, C2×C42⋊C2, C2×C42.7C22
Quotients: C1, C2, C4, C22, C2×C4, C23, C22×C4, C4○D4, C24, C42⋊C2, C8○D4, C23×C4, C2×C4○D4, C42.7C22, C2×C42⋊C2, C2×C8○D4, C2×C42.7C22

Smallest permutation representation of C2×C42.7C22
On 64 points
Generators in S64
(1 51)(2 52)(3 53)(4 54)(5 55)(6 56)(7 49)(8 50)(9 63)(10 64)(11 57)(12 58)(13 59)(14 60)(15 61)(16 62)(17 29)(18 30)(19 31)(20 32)(21 25)(22 26)(23 27)(24 28)(33 41)(34 42)(35 43)(36 44)(37 45)(38 46)(39 47)(40 48)
(1 43 19 9)(2 14 20 48)(3 45 21 11)(4 16 22 42)(5 47 23 13)(6 10 24 44)(7 41 17 15)(8 12 18 46)(25 57 53 37)(26 34 54 62)(27 59 55 39)(28 36 56 64)(29 61 49 33)(30 38 50 58)(31 63 51 35)(32 40 52 60)
(1 3 5 7)(2 4 6 8)(9 11 13 15)(10 12 14 16)(17 19 21 23)(18 20 22 24)(25 27 29 31)(26 28 30 32)(33 35 37 39)(34 36 38 40)(41 43 45 47)(42 44 46 48)(49 51 53 55)(50 52 54 56)(57 59 61 63)(58 60 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 31)(2 56)(3 25)(4 50)(5 27)(6 52)(7 29)(8 54)(9 39)(10 64)(11 33)(12 58)(13 35)(14 60)(15 37)(16 62)(17 49)(18 26)(19 51)(20 28)(21 53)(22 30)(23 55)(24 32)(34 42)(36 44)(38 46)(40 48)(41 57)(43 59)(45 61)(47 63)

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

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

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

56 conjugacy classes

 class 1 2A ··· 2G 2H 2I 4A ··· 4H 4I ··· 4P 4Q ··· 4V 8A ··· 8P 8Q ··· 8X order 1 2 ··· 2 2 2 4 ··· 4 4 ··· 4 4 ··· 4 8 ··· 8 8 ··· 8 size 1 1 ··· 1 4 4 1 ··· 1 2 ··· 2 4 ··· 4 2 ··· 2 4 ··· 4

56 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 type + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C4 C4 C4 C4○D4 C8○D4 kernel C2×C42.7C22 C2×C4×C8 C2×C8⋊C4 C2×C22⋊C8 C2×C4⋊C8 C42.7C22 C2×C42⋊C2 C2×C22⋊C4 C2×C4⋊C4 C42⋊C2 C2×C4 C22 # reps 1 1 1 2 2 8 1 4 4 8 8 16

Matrix representation of C2×C42.7C22 in GL5(𝔽17)

 16 0 0 0 0 0 16 0 0 0 0 0 16 0 0 0 0 0 1 0 0 0 0 0 1
,
 16 0 0 0 0 0 13 0 0 0 0 0 4 0 0 0 0 0 0 16 0 0 0 1 0
,
 16 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 4 0 0 0 0 0 4
,
 13 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 15 0 0 0 0 0 15
,
 1 0 0 0 0 0 1 0 0 0 0 0 16 0 0 0 0 0 16 0 0 0 0 0 1

G:=sub<GL(5,GF(17))| [16,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,1,0,0,0,0,0,1],[16,0,0,0,0,0,13,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,16,0],[16,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,4,0,0,0,0,0,4],[13,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,15,0,0,0,0,0,15],[1,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,1] >;

C2×C42.7C22 in GAP, Magma, Sage, TeX

C_2\times C_4^2._7C_2^2
% in TeX

G:=Group("C2xC4^2.7C2^2");
// GroupNames label

G:=SmallGroup(128,1651);
// by ID

G=gap.SmallGroup(128,1651);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,723,100,124]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^4=c^4=e^2=1,d^2=c,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,d*b*d^-1=b^-1*c^2,e*b*e=b*c^2,c*d=d*c,c*e=e*c,e*d*e=b^2*c^2*d>;
// generators/relations

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