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## G = Q8×C23order 64 = 26

### Direct product of C23 and Q8

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

Aliases: Q8×C23, C2.2C25, C4.11C24, C22.14C24, C23.74C23, C24.37C22, (C23×C4).12C2, (C2×C4).140C23, (C22×C4).133C22, SmallGroup(64,262)

Series: Derived Chief Lower central Upper central Jennings

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

Generators and relations for Q8×C23
G = < a,b,c,d,e | a2=b2=c2=d4=1, e2=d2, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >

Subgroups: 425, all normal (4 characteristic)
C1, C2, C2 [×14], C4 [×24], C22 [×35], C2×C4 [×84], Q8 [×64], C23 [×15], C22×C4 [×42], C2×Q8 [×112], C24, C23×C4 [×3], C22×Q8 [×28], Q8×C23
Quotients: C1, C2 [×31], C22 [×155], Q8 [×8], C23 [×155], C2×Q8 [×28], C24 [×31], C22×Q8 [×14], C25, Q8×C23

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

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

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

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

Q8×C23 is a maximal subgroup of
C24.636C23  C4.C22≀C2  C24.155D4  C23.192C24  C23.309C24  C23.334C24  C24.565C23  C23.514C24  C24.178D4  C22.75C25
Q8×C23 is a maximal quotient of
C22.47C25  C22.90C25  C22.91C25  C22.92C25  C22.93C25

40 conjugacy classes

 class 1 2A ··· 2O 4A ··· 4X order 1 2 ··· 2 4 ··· 4 size 1 1 ··· 1 2 ··· 2

40 irreducible representations

 dim 1 1 1 2 type + + + - image C1 C2 C2 Q8 kernel Q8×C23 C23×C4 C22×Q8 C23 # reps 1 3 28 8

Matrix representation of Q8×C23 in GL5(𝔽5)

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

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

Q8×C23 in GAP, Magma, Sage, TeX

Q_8\times C_2^3
% in TeX

G:=Group("Q8xC2^3");
// GroupNames label

G:=SmallGroup(64,262);
// by ID

G=gap.SmallGroup(64,262);
# by ID

G:=PCGroup([6,-2,2,2,2,2,-2,192,409,199]);
// Polycyclic

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

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