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## G = C2×C23.4Q8order 128 = 27

### Direct product of C2 and C23.4Q8

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C23 — C2×C23.4Q8
 Chief series C1 — C2 — C22 — C23 — C24 — C25 — C22×C22⋊C4 — C2×C23.4Q8
 Lower central C1 — C23 — C2×C23.4Q8
 Upper central C1 — C24 — C2×C23.4Q8
 Jennings C1 — C23 — C2×C23.4Q8

Generators and relations for C2×C23.4Q8
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=ce-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, C23.4Q8 [×8], C22×C22⋊C4 [×3], C22×C4⋊C4 [×3], C2×C23.4Q8
Quotients: C1, C2 [×15], C22 [×35], D4 [×12], Q8 [×4], C23 [×15], C2×D4 [×18], C2×Q8 [×6], C4○D4 [×6], C24, C22⋊Q8 [×12], C22.D4 [×12], C41D4 [×4], C22×D4 [×3], C22×Q8, C2×C4○D4 [×3], C23.4Q8 [×8], C2×C22⋊Q8 [×3], C2×C22.D4 [×3], C2×C41D4, C2×C23.4Q8

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

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

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

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

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×C23.4Q8 C2×C2.C42 C23.4Q8 C22×C22⋊C4 C22×C4⋊C4 C22×C4 C24 C23 # reps 1 1 8 3 3 12 4 12

Matrix representation of C2×C23.4Q8 in GL8(𝔽5)

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

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

C2×C23.4Q8 in GAP, Magma, Sage, TeX

C_2\times C_2^3._4Q_8
% in TeX

G:=Group("C2xC2^3.4Q8");
// GroupNames label

G:=SmallGroup(128,1125);
// by ID

G=gap.SmallGroup(128,1125);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,112,253,344,758,723]);
// 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=c*e^-1>;
// generators/relations

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