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G = C2×C42.78C22order 128 = 27

Direct product of C2 and C42.78C22

direct product, p-group, metabelian, nilpotent (class 3), monomial

Aliases: C2×C42.78C22, C42.354D4, C42.705C23, (C4×C8)⋊71C22, C4⋊C4.83C23, (C2×C8).490C23, (C2×C4).328C24, (C2×D4).97C23, C23.872(C2×D4), (C22×C4).565D4, (C2×Q8).85C23, Q8⋊C461C22, C4.20(C4.4D4), C22.97(C4○D8), C42.C233C22, (C22×C8).519C22, C22.588(C22×D4), D4⋊C4.143C22, (C22×C4).1550C23, (C2×C42).1123C22, C4.4D4.132C22, C22.82(C4.4D4), (C22×D4).365C22, (C22×Q8).298C22, (C2×C4×C8)⋊21C2, C2.28(C2×C4○D8), C4.37(C2×C4○D4), (C2×C4).693(C2×D4), (C2×Q8⋊C4)⋊19C2, (C2×C42.C2)⋊33C2, C2.39(C2×C4.4D4), (C2×D4⋊C4).19C2, (C2×C4).707(C4○D4), (C2×C4⋊C4).620C22, (C2×C4.4D4).38C2, SmallGroup(128,1862)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C2×C42.78C22
C1C2C4C2×C4C22×C4C22×C8C2×C4×C8 — C2×C42.78C22
C1C2C2×C4 — C2×C42.78C22
C1C23C2×C42 — C2×C42.78C22
C1C2C2C2×C4 — C2×C42.78C22

Generators and relations for C2×C42.78C22
 G = < a,b,c,d,e | a2=b4=c4=d2=1, e2=c, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd=b-1c2, be=eb, dcd=c-1, ce=ec, ede-1=b2cd >

Subgroups: 404 in 206 conjugacy classes, 100 normal (14 characteristic)
C1, C2, C2 [×6], C2 [×2], C4 [×4], C4 [×10], C22, C22 [×6], C22 [×10], C8 [×4], C2×C4 [×2], C2×C4 [×8], C2×C4 [×16], D4 [×6], Q8 [×6], C23, C23 [×8], C42 [×4], C22⋊C4 [×8], C4⋊C4 [×4], C4⋊C4 [×10], C2×C8 [×4], C2×C8 [×4], C22×C4, C22×C4 [×2], C22×C4 [×3], C2×D4 [×2], C2×D4 [×5], C2×Q8 [×2], C2×Q8 [×5], C24, C4×C8 [×4], D4⋊C4 [×8], Q8⋊C4 [×8], C2×C42, C2×C22⋊C4 [×2], C2×C4⋊C4 [×2], C2×C4⋊C4 [×2], C4.4D4 [×4], C4.4D4 [×2], C42.C2 [×4], C42.C2 [×2], C22×C8 [×2], C22×D4, C22×Q8, C2×C4×C8, C2×D4⋊C4 [×2], C2×Q8⋊C4 [×2], C42.78C22 [×8], C2×C4.4D4, C2×C42.C2, C2×C42.78C22
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×4], C24, C4.4D4 [×4], C4○D8 [×4], C22×D4, C2×C4○D4 [×2], C42.78C22 [×4], C2×C4.4D4, C2×C4○D8 [×2], C2×C42.78C22

Smallest permutation representation of C2×C42.78C22
On 64 points
Generators in S64
(1 53)(2 54)(3 55)(4 56)(5 49)(6 50)(7 51)(8 52)(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 46)(34 47)(35 48)(36 41)(37 42)(38 43)(39 44)(40 45)
(1 64 17 34)(2 57 18 35)(3 58 19 36)(4 59 20 37)(5 60 21 38)(6 61 22 39)(7 62 23 40)(8 63 24 33)(9 28 46 52)(10 29 47 53)(11 30 48 54)(12 31 41 55)(13 32 42 56)(14 25 43 49)(15 26 44 50)(16 27 45 51)
(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 25)(2 56)(3 31)(4 54)(5 29)(6 52)(7 27)(8 50)(9 35)(10 64)(11 33)(12 62)(13 39)(14 60)(15 37)(16 58)(17 49)(18 32)(19 55)(20 30)(21 53)(22 28)(23 51)(24 26)(34 47)(36 45)(38 43)(40 41)(42 61)(44 59)(46 57)(48 63)
(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)| (1,53)(2,54)(3,55)(4,56)(5,49)(6,50)(7,51)(8,52)(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)(33,46)(34,47)(35,48)(36,41)(37,42)(38,43)(39,44)(40,45), (1,64,17,34)(2,57,18,35)(3,58,19,36)(4,59,20,37)(5,60,21,38)(6,61,22,39)(7,62,23,40)(8,63,24,33)(9,28,46,52)(10,29,47,53)(11,30,48,54)(12,31,41,55)(13,32,42,56)(14,25,43,49)(15,26,44,50)(16,27,45,51), (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), (1,25)(2,56)(3,31)(4,54)(5,29)(6,52)(7,27)(8,50)(9,35)(10,64)(11,33)(12,62)(13,39)(14,60)(15,37)(16,58)(17,49)(18,32)(19,55)(20,30)(21,53)(22,28)(23,51)(24,26)(34,47)(36,45)(38,43)(40,41)(42,61)(44,59)(46,57)(48,63), (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( (1,53)(2,54)(3,55)(4,56)(5,49)(6,50)(7,51)(8,52)(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)(33,46)(34,47)(35,48)(36,41)(37,42)(38,43)(39,44)(40,45), (1,64,17,34)(2,57,18,35)(3,58,19,36)(4,59,20,37)(5,60,21,38)(6,61,22,39)(7,62,23,40)(8,63,24,33)(9,28,46,52)(10,29,47,53)(11,30,48,54)(12,31,41,55)(13,32,42,56)(14,25,43,49)(15,26,44,50)(16,27,45,51), (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), (1,25)(2,56)(3,31)(4,54)(5,29)(6,52)(7,27)(8,50)(9,35)(10,64)(11,33)(12,62)(13,39)(14,60)(15,37)(16,58)(17,49)(18,32)(19,55)(20,30)(21,53)(22,28)(23,51)(24,26)(34,47)(36,45)(38,43)(40,41)(42,61)(44,59)(46,57)(48,63), (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([(1,53),(2,54),(3,55),(4,56),(5,49),(6,50),(7,51),(8,52),(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),(33,46),(34,47),(35,48),(36,41),(37,42),(38,43),(39,44),(40,45)], [(1,64,17,34),(2,57,18,35),(3,58,19,36),(4,59,20,37),(5,60,21,38),(6,61,22,39),(7,62,23,40),(8,63,24,33),(9,28,46,52),(10,29,47,53),(11,30,48,54),(12,31,41,55),(13,32,42,56),(14,25,43,49),(15,26,44,50),(16,27,45,51)], [(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)], [(1,25),(2,56),(3,31),(4,54),(5,29),(6,52),(7,27),(8,50),(9,35),(10,64),(11,33),(12,62),(13,39),(14,60),(15,37),(16,58),(17,49),(18,32),(19,55),(20,30),(21,53),(22,28),(23,51),(24,26),(34,47),(36,45),(38,43),(40,41),(42,61),(44,59),(46,57),(48,63)], [(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···2G2H2I4A···4L4M···4R8A···8P
order12···2224···44···48···8
size11···1882···28···82···2

44 irreducible representations

dim11111112222
type+++++++++
imageC1C2C2C2C2C2C2D4D4C4○D4C4○D8
kernelC2×C42.78C22C2×C4×C8C2×D4⋊C4C2×Q8⋊C4C42.78C22C2×C4.4D4C2×C42.C2C42C22×C4C2×C4C22
# reps112281122816

Matrix representation of C2×C42.78C22 in GL5(𝔽17)

160000
01000
00100
000160
000016
,
160000
07800
0151000
000013
00040
,
10000
01000
00100
000016
00010
,
10000
016000
06100
000160
00001
,
10000
06200
081100
000314
00033

G:=sub<GL(5,GF(17))| [16,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,16],[16,0,0,0,0,0,7,15,0,0,0,8,10,0,0,0,0,0,0,4,0,0,0,13,0],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,16,0],[1,0,0,0,0,0,16,6,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,1],[1,0,0,0,0,0,6,8,0,0,0,2,11,0,0,0,0,0,3,3,0,0,0,14,3] >;

C2×C42.78C22 in GAP, Magma, Sage, TeX

C_2\times C_4^2._{78}C_2^2
% in TeX

G:=Group("C2xC4^2.78C2^2");
// GroupNames label

G:=SmallGroup(128,1862);
// by ID

G=gap.SmallGroup(128,1862);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,680,758,100,2804,172,4037,124]);
// Polycyclic

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

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