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## G = C4×C8.C22order 128 = 27

### Direct product of C4 and C8.C22

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C4 — C4×C8.C22
 Chief series C1 — C2 — C22 — C2×C4 — C22×C4 — C2×C42 — C2×C4×Q8 — C4×C8.C22
 Lower central C1 — C2 — C4 — C4×C8.C22
 Upper central C1 — C2×C4 — C2×C42 — C4×C8.C22
 Jennings C1 — C2 — C2 — C2×C4 — C4×C8.C22

Generators and relations for C4×C8.C22
G = < a,b,c,d | a4=b8=c2=d2=1, ab=ba, ac=ca, ad=da, cbc=b3, dbd=b5, dcd=b4c >

Subgroups: 372 in 242 conjugacy classes, 142 normal (44 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×4], C4 [×13], C22, C22 [×2], C22 [×6], C8 [×4], C8 [×2], C2×C4 [×6], C2×C4 [×4], C2×C4 [×24], D4 [×2], D4 [×5], Q8 [×6], Q8 [×7], C23, C23, C42 [×4], C42 [×7], C22⋊C4 [×3], C4⋊C4 [×2], C4⋊C4 [×2], C4⋊C4 [×7], C2×C8 [×4], M4(2) [×4], M4(2) [×2], SD16 [×8], Q16 [×8], C22×C4 [×3], C22×C4 [×5], C2×D4, C2×D4, C2×Q8, C2×Q8 [×6], C2×Q8 [×3], C4○D4 [×4], C4○D4 [×2], C4×C8 [×2], C8⋊C4 [×2], D4⋊C4 [×2], Q8⋊C4 [×6], C4.Q8 [×2], C2.D8 [×2], C2×C42, C2×C42 [×2], C2×C4⋊C4, C2×C4⋊C4, C42⋊C2, C42⋊C2, C4×D4 [×2], C4×D4 [×2], C4×Q8 [×2], C4×Q8 [×4], C4×Q8 [×2], C2×M4(2) [×2], C2×SD16 [×2], C2×Q16 [×2], C8.C22 [×8], C22×Q8, C2×C4○D4, C4×M4(2), C23.36D4, C23.38D4, M4(2)⋊C4, C4×SD16 [×2], C4×Q16 [×2], SD16⋊C4 [×2], Q16⋊C4 [×2], C2×C4×Q8, C4×C4○D4, C2×C8.C22, C4×C8.C22
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], D4 [×4], C23 [×15], C22×C4 [×14], C2×D4 [×6], C4○D4 [×2], C24, C4×D4 [×4], C8.C22 [×2], C23×C4, C22×D4, C2×C4○D4, C2×C4×D4, C2×C8.C22, D8⋊C22, C4×C8.C22

Smallest permutation representation of C4×C8.C22
On 64 points
Generators in S64
(1 61 53 48)(2 62 54 41)(3 63 55 42)(4 64 56 43)(5 57 49 44)(6 58 50 45)(7 59 51 46)(8 60 52 47)(9 21 39 32)(10 22 40 25)(11 23 33 26)(12 24 34 27)(13 17 35 28)(14 18 36 29)(15 19 37 30)(16 20 38 31)
(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 53)(2 56)(3 51)(4 54)(5 49)(6 52)(7 55)(8 50)(9 35)(10 38)(11 33)(12 36)(13 39)(14 34)(15 37)(16 40)(17 32)(18 27)(19 30)(20 25)(21 28)(22 31)(23 26)(24 29)(41 64)(42 59)(43 62)(44 57)(45 60)(46 63)(47 58)(48 61)
(1 28)(2 25)(3 30)(4 27)(5 32)(6 29)(7 26)(8 31)(9 57)(10 62)(11 59)(12 64)(13 61)(14 58)(15 63)(16 60)(17 53)(18 50)(19 55)(20 52)(21 49)(22 54)(23 51)(24 56)(33 46)(34 43)(35 48)(36 45)(37 42)(38 47)(39 44)(40 41)

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

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

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

44 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E ··· 4N 4O ··· 4AB 8A ··· 8H order 1 2 2 2 2 2 2 2 4 4 4 4 4 ··· 4 4 ··· 4 8 ··· 8 size 1 1 1 1 2 2 4 4 1 1 1 1 2 ··· 2 4 ··· 4 4 ··· 4

44 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 4 4 type + + + + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C4 D4 D4 C4○D4 C8.C22 D8⋊C22 kernel C4×C8.C22 C4×M4(2) C23.36D4 C23.38D4 M4(2)⋊C4 C4×SD16 C4×Q16 SD16⋊C4 Q16⋊C4 C2×C4×Q8 C4×C4○D4 C2×C8.C22 C8.C22 C42 C22×C4 C2×C4 C4 C2 # reps 1 1 1 1 1 2 2 2 2 1 1 1 16 2 2 4 2 2

Matrix representation of C4×C8.C22 in GL6(𝔽17)

 4 0 0 0 0 0 0 4 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 16 2 0 0 0 0 16 1 0 0 0 0 0 0 7 10 12 12 0 0 7 7 5 12 0 0 2 2 10 7 0 0 15 2 10 10
,
 16 0 0 0 0 0 16 1 0 0 0 0 0 0 16 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 16
,
 16 0 0 0 0 0 0 16 0 0 0 0 0 0 0 4 0 0 0 0 13 0 0 0 0 0 1 0 0 13 0 0 0 1 4 0

G:=sub<GL(6,GF(17))| [4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[16,16,0,0,0,0,2,1,0,0,0,0,0,0,7,7,2,15,0,0,10,7,2,2,0,0,12,5,10,10,0,0,12,12,7,10],[16,16,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,13,1,0,0,0,4,0,0,1,0,0,0,0,0,4,0,0,0,0,13,0] >;

C4×C8.C22 in GAP, Magma, Sage, TeX

C_4\times C_8.C_2^2
% in TeX

G:=Group("C4xC8.C2^2");
// GroupNames label

G:=SmallGroup(128,1677);
// by ID

G=gap.SmallGroup(128,1677);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,456,184,2019,2804,1411,172]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^8=c^2=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^3,d*b*d=b^5,d*c*d=b^4*c>;
// generators/relations

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