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G = C4○D32order 128 = 27

Central product of C4 and D32

p-group, metabelian, nilpotent (class 5), monomial

Aliases: C4D32, C4Q64, C4SD64, D323C2, Q643C2, C8.22D8, SD643C2, C4.20D16, C16.13D4, C32.6C22, C16.9C23, C22.1D16, D16.2C22, Q32.2C22, (C2×C32)⋊6C2, C4○D161C2, C4.16(C2×D8), (C2×C4).91D8, C8.48(C2×D4), C2.15(C2×D16), (C2×C8).271D4, (C2×C16).96C22, SmallGroup(128,994)

Series: Derived Chief Lower central Upper central Jennings

C1C16 — C4○D32
C1C2C4C8C16C2×C16C4○D16 — C4○D32
C1C2C4C8C16 — C4○D32
C1C4C2×C4C2×C8C2×C16 — C4○D32
C1C2C2C2C2C2C2C2C2C4C4C4C4C8C8C16 — C4○D32

Generators and relations for C4○D32
 G = < a,b,c | a4=c2=1, b16=a2, ab=ba, ac=ca, cbc=a2b15 >

2C2
16C2
16C2
8C4
8C22
8C4
8C22
4D4
4D4
4Q8
4Q8
8C2×C4
8D4
8D4
8C2×C4
2Q16
2D8
2Q16
2D8
4C4○D4
4SD16
4C4○D4
4SD16
2SD32
2C4○D8
2C4○D8
2SD32

Smallest permutation representation of C4○D32
On 64 points
Generators in S64
(1 60 17 44)(2 61 18 45)(3 62 19 46)(4 63 20 47)(5 64 21 48)(6 33 22 49)(7 34 23 50)(8 35 24 51)(9 36 25 52)(10 37 26 53)(11 38 27 54)(12 39 28 55)(13 40 29 56)(14 41 30 57)(15 42 31 58)(16 43 32 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 32)(2 31)(3 30)(4 29)(5 28)(6 27)(7 26)(8 25)(9 24)(10 23)(11 22)(12 21)(13 20)(14 19)(15 18)(16 17)(33 54)(34 53)(35 52)(36 51)(37 50)(38 49)(39 48)(40 47)(41 46)(42 45)(43 44)(55 64)(56 63)(57 62)(58 61)(59 60)

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

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

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

38 conjugacy classes

class 1 2A2B2C2D4A4B4C4D4E8A8B8C8D16A···16H32A···32P
order1222244444888816···1632···32
size1121616112161622222···22···2

38 irreducible representations

dim1111112222222
type++++++++++++
imageC1C2C2C2C2C2D4D4D8D8D16D16C4○D32
kernelC4○D32C2×C32D32SD64Q64C4○D16C16C2×C8C8C2×C4C4C22C1
# reps11121211224416

Matrix representation of C4○D32 in GL2(𝔽97) generated by

220
022
,
5770
2757
,
5770
7040
G:=sub<GL(2,GF(97))| [22,0,0,22],[57,27,70,57],[57,70,70,40] >;

C4○D32 in GAP, Magma, Sage, TeX

C_4\circ D_{32}
% in TeX

G:=Group("C4oD32");
// GroupNames label

G:=SmallGroup(128,994);
// by ID

G=gap.SmallGroup(128,994);
# by ID

G:=PCGroup([7,-2,2,2,-2,-2,-2,-2,141,352,675,346,192,1684,851,242,4037,2028,124]);
// Polycyclic

G:=Group<a,b,c|a^4=c^2=1,b^16=a^2,a*b=b*a,a*c=c*a,c*b*c=a^2*b^15>;
// generators/relations

Export

Subgroup lattice of C4○D32 in TeX

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