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G = C23xD8order 128 = 27

Direct product of C23 and D8

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

Aliases: C23xD8, C8:2C24, D4:1C24, C4.1C25, C24.195D4, (C23xC8):9C2, (C2xC8):14C23, (D4xC23):17C2, (C2xD4):20C23, C4.27(C22xD4), C2.36(D4xC23), (C2xC4).607C24, (C22xC8):66C22, (C22xC4).627D4, C23.893(C2xD4), (C22xD4):64C22, (C23xC4).711C22, C22.164(C22xD4), (C22xC4).1589C23, (C2xC4).880(C2xD4), SmallGroup(128,2306)

Series: Derived Chief Lower central Upper central Jennings

C1C4 — C23xD8
C1C2C4C2xC4C22xC4C23xC4D4xC23 — C23xD8
C1C2C4 — C23xD8
C1C24C23xC4 — C23xD8
C1C2C2C4 — C23xD8

Generators and relations for C23xD8
 G = < a,b,c,d,e | a2=b2=c2=d8=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 2012 in 988 conjugacy classes, 476 normal (7 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C8, C2xC4, D4, D4, C23, C23, C2xC8, D8, C22xC4, C2xD4, C2xD4, C24, C24, C22xC8, C2xD8, C23xC4, C22xD4, C22xD4, C25, C23xC8, C22xD8, D4xC23, C23xD8
Quotients: C1, C2, C22, D4, C23, D8, C2xD4, C24, C2xD8, C22xD4, C25, C22xD8, D4xC23, C23xD8

Smallest permutation representation of C23xD8
On 64 points
Generators in S64
(1 24)(2 17)(3 18)(4 19)(5 20)(6 21)(7 22)(8 23)(9 26)(10 27)(11 28)(12 29)(13 30)(14 31)(15 32)(16 25)(33 56)(34 49)(35 50)(36 51)(37 52)(38 53)(39 54)(40 55)(41 62)(42 63)(43 64)(44 57)(45 58)(46 59)(47 60)(48 61)
(1 53)(2 54)(3 55)(4 56)(5 49)(6 50)(7 51)(8 52)(9 47)(10 48)(11 41)(12 42)(13 43)(14 44)(15 45)(16 46)(17 39)(18 40)(19 33)(20 34)(21 35)(22 36)(23 37)(24 38)(25 59)(26 60)(27 61)(28 62)(29 63)(30 64)(31 57)(32 58)
(1 62)(2 63)(3 64)(4 57)(5 58)(6 59)(7 60)(8 61)(9 36)(10 37)(11 38)(12 39)(13 40)(14 33)(15 34)(16 35)(17 42)(18 43)(19 44)(20 45)(21 46)(22 47)(23 48)(24 41)(25 50)(26 51)(27 52)(28 53)(29 54)(30 55)(31 56)(32 49)
(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 28)(2 27)(3 26)(4 25)(5 32)(6 31)(7 30)(8 29)(9 18)(10 17)(11 24)(12 23)(13 22)(14 21)(15 20)(16 19)(33 46)(34 45)(35 44)(36 43)(37 42)(38 41)(39 48)(40 47)(49 58)(50 57)(51 64)(52 63)(53 62)(54 61)(55 60)(56 59)

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

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

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

56 conjugacy classes

class 1 2A···2O2P···2AE4A···4H8A···8P
order12···22···24···48···8
size11···14···42···22···2

56 irreducible representations

dim1111222
type+++++++
imageC1C2C2C2D4D4D8
kernelC23xD8C23xC8C22xD8D4xC23C22xC4C24C23
# reps112827116

Matrix representation of C23xD8 in GL5(F17)

10000
016000
00100
000160
000016
,
160000
01000
001600
000160
000016
,
10000
016000
001600
00010
00001
,
160000
01000
00100
000011
000311
,
10000
01000
001600
000160
000161

G:=sub<GL(5,GF(17))| [1,0,0,0,0,0,16,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,16],[16,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,16],[1,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,1,0,0,0,0,0,1],[16,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,3,0,0,0,11,11],[1,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,16,16,0,0,0,0,1] >;

C23xD8 in GAP, Magma, Sage, TeX

C_2^3\times D_8
% in TeX

G:=Group("C2^3xD8");
// GroupNames label

G:=SmallGroup(128,2306);
// by ID

G=gap.SmallGroup(128,2306);
# by ID

G:=PCGroup([7,-2,2,2,2,2,-2,-2,477,4037,2028,124]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^8=e^2=1,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=d^-1>;
// generators/relations

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