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G = C2×C23.8Q8order 128 = 27

Direct product of C2 and C23.8Q8

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22 — C2×C23.8Q8
 Chief series C1 — C2 — C22 — C23 — C24 — C25 — C24×C4 — C2×C23.8Q8
 Lower central C1 — C22 — C2×C23.8Q8
 Upper central C1 — C24 — C2×C23.8Q8
 Jennings C1 — C23 — C2×C23.8Q8

Generators and relations for C2×C23.8Q8
G = < a,b,c,d,e,f | a2=b2=c2=d2=e4=1, f2=e2, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, fbf-1=bd=db, be=eb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=ce-1 >

Subgroups: 1020 in 616 conjugacy classes, 236 normal (16 characteristic)
C1, C2 [×3], C2 [×12], C2 [×8], C4 [×20], C22 [×3], C22 [×40], C22 [×56], C2×C4 [×16], C2×C4 [×100], C23, C23 [×42], C23 [×56], C22⋊C4 [×16], C22⋊C4 [×8], C4⋊C4 [×16], C22×C4 [×28], C22×C4 [×84], C24, C24 [×14], C24 [×8], C2.C42 [×8], C2×C22⋊C4 [×16], C2×C22⋊C4 [×4], C2×C4⋊C4 [×8], C2×C4⋊C4 [×8], C23×C4 [×2], C23×C4 [×8], C23×C4 [×12], C25, C2×C2.C42 [×2], C23.8Q8 [×8], C22×C22⋊C4 [×2], C22×C4⋊C4 [×2], C24×C4, C2×C23.8Q8
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], D4 [×12], Q8 [×4], C23 [×15], C4⋊C4 [×16], C22×C4 [×14], C2×D4 [×18], C2×Q8 [×6], C4○D4 [×4], C24, C2×C4⋊C4 [×12], C4×D4 [×8], C22≀C2 [×4], C22⋊Q8 [×8], C22.D4 [×4], C23×C4, C22×D4 [×3], C22×Q8, C2×C4○D4 [×2], C23.8Q8 [×8], C22×C4⋊C4, C2×C4×D4 [×2], C2×C22≀C2, C2×C22⋊Q8 [×2], C2×C22.D4, C2×C23.8Q8

Smallest permutation representation of C2×C23.8Q8
On 64 points
Generators in S64
(1 5)(2 6)(3 7)(4 8)(9 23)(10 24)(11 21)(12 22)(13 32)(14 29)(15 30)(16 31)(17 45)(18 46)(19 47)(20 48)(25 38)(26 39)(27 40)(28 37)(33 42)(34 43)(35 44)(36 41)(49 60)(50 57)(51 58)(52 59)(53 64)(54 61)(55 62)(56 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 52)(26 49)(27 50)(28 51)(29 31)(30 32)(33 54)(34 55)(35 56)(36 53)(37 58)(38 59)(39 60)(40 57)(41 64)(42 61)(43 62)(44 63)(45 47)(46 48)
(1 9)(2 10)(3 11)(4 12)(5 23)(6 24)(7 21)(8 22)(13 18)(14 19)(15 20)(16 17)(25 61)(26 62)(27 63)(28 64)(29 47)(30 48)(31 45)(32 46)(33 59)(34 60)(35 57)(36 58)(37 53)(38 54)(39 55)(40 56)(41 51)(42 52)(43 49)(44 50)
(1 31)(2 32)(3 29)(4 30)(5 16)(6 13)(7 14)(8 15)(9 45)(10 46)(11 47)(12 48)(17 23)(18 24)(19 21)(20 22)(25 50)(26 51)(27 52)(28 49)(33 56)(34 53)(35 54)(36 55)(37 60)(38 57)(39 58)(40 59)(41 62)(42 63)(43 64)(44 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 56 3 54)(2 39 4 37)(5 63 7 61)(6 26 8 28)(9 40 11 38)(10 55 12 53)(13 51 15 49)(14 44 16 42)(17 52 19 50)(18 41 20 43)(21 25 23 27)(22 64 24 62)(29 35 31 33)(30 60 32 58)(34 46 36 48)(45 59 47 57)

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

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

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

56 conjugacy classes

 class 1 2A ··· 2O 2P ··· 2W 4A ··· 4P 4Q ··· 4AF order 1 2 ··· 2 2 ··· 2 4 ··· 4 4 ··· 4 size 1 1 ··· 1 2 ··· 2 2 ··· 2 4 ··· 4

56 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 type + + + + + + + + - image C1 C2 C2 C2 C2 C2 C4 D4 D4 Q8 C4○D4 kernel C2×C23.8Q8 C2×C2.C42 C23.8Q8 C22×C22⋊C4 C22×C4⋊C4 C24×C4 C2×C22⋊C4 C22×C4 C24 C24 C23 # reps 1 2 8 2 2 1 16 8 4 4 8

Matrix representation of C2×C23.8Q8 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 4 0 0 0 0 0 0 4 0 0 0 0 0 0 1
,
 4 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 0 1 0 0 0 0 0 0 1
,
 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 4 0 0 0 0 0 0 4
,
 2 0 0 0 0 0 0 3 0 0 0 0 0 0 2 2 0 0 0 0 0 3 0 0 0 0 0 0 2 0 0 0 0 0 0 3
,
 3 0 0 0 0 0 0 2 0 0 0 0 0 0 3 0 0 0 0 0 4 2 0 0 0 0 0 0 0 1 0 0 0 0 4 0

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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,1],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[2,0,0,0,0,0,0,3,0,0,0,0,0,0,2,0,0,0,0,0,2,3,0,0,0,0,0,0,2,0,0,0,0,0,0,3],[3,0,0,0,0,0,0,2,0,0,0,0,0,0,3,4,0,0,0,0,0,2,0,0,0,0,0,0,0,4,0,0,0,0,1,0] >;

C2×C23.8Q8 in GAP, Magma, Sage, TeX

C_2\times C_2^3._8Q_8
% in TeX

G:=Group("C2xC2^3.8Q8");
// GroupNames label

G:=SmallGroup(128,1018);
// by ID

G=gap.SmallGroup(128,1018);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,456,758]);
// Polycyclic

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

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