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## G = C2×C23.Q8order 128 = 27

### Direct product of C2 and C23.Q8

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

Series: Derived Chief Lower central Upper central Jennings

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

Generators and relations for C2×C23.Q8
G = < a,b,c,d,e,f | a2=b2=c2=d2=e4=1, f2=ce2, ab=ba, ac=ca, ad=da, ae=ea, af=fa, fbf-1=bc=cb, ebe-1=bd=db, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e-1 >

Subgroups: 868 in 454 conjugacy classes, 164 normal (8 characteristic)
C1, C2, C2 [×14], C2 [×4], C4 [×18], C22, C22 [×34], C22 [×36], C2×C4 [×12], C2×C4 [×66], C23, C23 [×18], C23 [×52], C22⋊C4 [×24], C4⋊C4 [×24], C22×C4 [×24], C22×C4 [×30], C24, C24 [×6], C24 [×12], C2.C42 [×4], C2×C22⋊C4 [×12], C2×C22⋊C4 [×12], C2×C4⋊C4 [×12], C2×C4⋊C4 [×12], C23×C4 [×6], C25, C2×C2.C42, C23.Q8 [×8], C22×C22⋊C4 [×3], C22×C4⋊C4 [×3], C2×C23.Q8
Quotients: C1, C2 [×15], C22 [×35], D4 [×12], Q8 [×4], C23 [×15], C2×D4 [×18], C2×Q8 [×6], C4○D4 [×6], C24, C4⋊D4 [×12], C22⋊Q8 [×12], C422C2 [×4], C22×D4 [×3], C22×Q8, C2×C4○D4 [×3], C23.Q8 [×8], C2×C4⋊D4 [×3], C2×C22⋊Q8 [×3], C2×C422C2, C2×C23.Q8

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

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

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

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

44 conjugacy classes

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

44 irreducible representations

 dim 1 1 1 1 1 2 2 2 type + + + + + + - image C1 C2 C2 C2 C2 D4 Q8 C4○D4 kernel C2×C23.Q8 C2×C2.C42 C23.Q8 C22×C22⋊C4 C22×C4⋊C4 C22×C4 C24 C23 # reps 1 1 8 3 3 12 4 12

Matrix representation of C2×C23.Q8 in GL8(𝔽5)

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

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

C2×C23.Q8 in GAP, Magma, Sage, TeX

C_2\times C_2^3.Q_8
% in TeX

G:=Group("C2xC2^3.Q8");
// GroupNames label

G:=SmallGroup(128,1121);
// by ID

G=gap.SmallGroup(128,1121);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,336,253,120,758,723]);
// Polycyclic

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

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