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G = C2xC8:6D4order 128 = 27

Direct product of C2 and C8:6D4

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

Aliases: C2xC8:6D4, C42.680C23, (C2xC8):33D4, C8:12(C2xD4), C4o(C8:6D4), C4:C8:86C22, (C4xC8):80C22, (C4xD4).27C4, C4.184(C4xD4), C4:1(C2xM4(2)), (C2xC4):8M4(2), C24.79(C2xC4), C22:C8:75C22, (C2xC8).477C23, (C2xC4).647C24, C42.282(C2xC4), (C22xD4).39C4, C22.114(C4xD4), C4.193(C22xD4), (C4xD4).285C22, C22.42(C8oD4), (C2xM4(2)):75C22, (C22xM4(2)):24C2, (C22xC4).915C23, (C22xC8).509C22, (C23xC4).525C22, C23.104(C22xC4), C22.174(C23xC4), C2.11(C22xM4(2)), C22.63(C2xM4(2)), (C2xC42).1109C22, (C2xC4xC8):42C2, (C2xC4:C8):48C2, C2.45(C2xC4xD4), (C2xC4xD4).71C2, (C2xC4:C4).71C4, (C2xC4)o(C8:6D4), C2.15(C2xC8oD4), C4:C4.222(C2xC4), (C2xC22:C8):43C2, C4.298(C2xC4oD4), (C2xD4).231(C2xC4), (C2xC4).1572(C2xD4), (C2xC22:C4).48C4, C22:C4.72(C2xC4), (C2xC4).957(C4oD4), (C2xC4).463(C22xC4), (C22xC4).339(C2xC4), SmallGroup(128,1660)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C2xC8:6D4
C1C2C4C2xC4C22xC4C22xC8C22xM4(2) — C2xC8:6D4
C1C22 — C2xC8:6D4
C1C22xC4 — C2xC8:6D4
C1C2C2C2xC4 — C2xC8:6D4

Generators and relations for C2xC8:6D4
 G = < a,b,c,d | a2=b8=c4=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b5, dcd=c-1 >

Subgroups: 420 in 276 conjugacy classes, 156 normal (22 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C8, C8, C2xC4, C2xC4, C2xC4, D4, C23, C23, C23, C42, C22:C4, C4:C4, C2xC8, C2xC8, M4(2), C22xC4, C22xC4, C22xC4, C2xD4, C2xD4, C24, C4xC8, C22:C8, C4:C8, C2xC42, C2xC22:C4, C2xC4:C4, C4xD4, C22xC8, C22xC8, C2xM4(2), C2xM4(2), C23xC4, C22xD4, C2xC4xC8, C2xC22:C8, C2xC4:C8, C8:6D4, C2xC4xD4, C22xM4(2), C2xC8:6D4
Quotients: C1, C2, C4, C22, C2xC4, D4, C23, M4(2), C22xC4, C2xD4, C4oD4, C24, C4xD4, C2xM4(2), C8oD4, C23xC4, C22xD4, C2xC4oD4, C8:6D4, C2xC4xD4, C22xM4(2), C2xC8oD4, C2xC8:6D4

Smallest permutation representation of C2xC8:6D4
On 64 points
Generators in S64
(1 19)(2 20)(3 21)(4 22)(5 23)(6 24)(7 17)(8 18)(9 41)(10 42)(11 43)(12 44)(13 45)(14 46)(15 47)(16 48)(25 39)(26 40)(27 33)(28 34)(29 35)(30 36)(31 37)(32 38)(49 60)(50 61)(51 62)(52 63)(53 64)(54 57)(55 58)(56 59)
(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 42 36)(2 59 43 37)(3 60 44 38)(4 61 45 39)(5 62 46 40)(6 63 47 33)(7 64 48 34)(8 57 41 35)(9 29 18 54)(10 30 19 55)(11 31 20 56)(12 32 21 49)(13 25 22 50)(14 26 23 51)(15 27 24 52)(16 28 17 53)
(1 36)(2 33)(3 38)(4 35)(5 40)(6 37)(7 34)(8 39)(9 50)(10 55)(11 52)(12 49)(13 54)(14 51)(15 56)(16 53)(17 28)(18 25)(19 30)(20 27)(21 32)(22 29)(23 26)(24 31)(41 61)(42 58)(43 63)(44 60)(45 57)(46 62)(47 59)(48 64)

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

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

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

56 conjugacy classes

class 1 2A···2G2H2I2J2K4A···4H4I···4P4Q4R4S4T8A···8P8Q···8X
order12···222224···44···444448···88···8
size11···144441···12···244442···24···4

56 irreducible representations

dim111111111112222
type++++++++
imageC1C2C2C2C2C2C2C4C4C4C4D4M4(2)C4oD4C8oD4
kernelC2xC8:6D4C2xC4xC8C2xC22:C8C2xC4:C8C8:6D4C2xC4xD4C22xM4(2)C2xC22:C4C2xC4:C4C4xD4C22xD4C2xC8C2xC4C2xC4C22
# reps112181242824848

Matrix representation of C2xC8:6D4 in GL5(F17)

160000
01000
00100
000160
000016
,
160000
04700
081300
000130
000013
,
10000
081400
016900
000013
000130
,
160000
09500
01800
000013
00040

G:=sub<GL(5,GF(17))| [16,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,16],[16,0,0,0,0,0,4,8,0,0,0,7,13,0,0,0,0,0,13,0,0,0,0,0,13],[1,0,0,0,0,0,8,16,0,0,0,14,9,0,0,0,0,0,0,13,0,0,0,13,0],[16,0,0,0,0,0,9,1,0,0,0,5,8,0,0,0,0,0,0,4,0,0,0,13,0] >;

C2xC8:6D4 in GAP, Magma, Sage, TeX

C_2\times C_8\rtimes_6D_4
% in TeX

G:=Group("C2xC8:6D4");
// GroupNames label

G:=SmallGroup(128,1660);
// by ID

G=gap.SmallGroup(128,1660);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,1430,723,184,124]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^8=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=b^5,d*c*d=c^-1>;
// generators/relations

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