(C2xQ8).D14 - GroupNames

G = (C₂×Q₈).D₁₄ order 448 = 2⁶·7

2^nd non-split extension by C₂×Q₈ of D₁₄ acting via D₁₄/C₇=C₂²

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: (C₂×C₄).₄D₂₈, (C₂×C₂₈).₁₆D₄, C₄.₁₀D₄.D₇, (C₂×Q₈).₂D₁₄, (C₄×Dic₇).₂C₄, C₇⋊₁(C₄².₃C₄), (C₂×Dic₁₄).₃C₄, (Q₈×C₁₄).₂C₂², C₁₄.₁₄(C₂³⋊C₄), Dic₇⋊Q₈.₁C₂, C₂₈.₁₀D₄.₁C₂, C₂².₁₅(D₁₄⋊C₄), C₂.₁₅(C₂³.₁D₁₄), (C₂×C₄).₄(C₄×D₇), (C₂×C₂₈).₄(C₂×C₄), (C₂×C₄).₄(C₇⋊D₄), (C₂×C₁₄).₈(C₂²⋊C₄), (C₇×C₄.₁₀D₄).₁C₂, SmallGroup(448,35)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₂₈ — (C₂×Q₈).D₁₄

Chief series

C₁ — C₇ — C₁₄ — C₂×C₁₄ — C₂×C₂₈ — Q₈×C₁₄ — Dic₇⋊Q₈ — (C₂×Q₈).D₁₄

Lower central

C₇ — C₁₄ — C₂×C₁₄ — C₂×C₂₈ — (C₂×Q₈).D₁₄

Upper central

C₁ — C₂ — C₂² — C₂×Q₈ — C₄.₁₀D₄

Generators and relations for (C₂×Q₈).D₁₄
G = < a,b,c,d | a²=b²⁸=1, c⁴=d²=b¹⁴, ab=ba, cac^-1=ab¹⁴, ad=da, cbc^-1=dbd^-1=ab¹³, dcd^-1=b²¹c³ >

Subgroups: 300 in 60 conjugacy classes, 21 normal (all characteristic)
C₁, C₂, C₂, C₄, C₂², C₇, C₈, C₂×C₄, C₂×C₄, Q₈, C₁₄, C₁₄, C₄², C₄⋊C₄, M₄(2), C₂×Q₈, C₂×Q₈, Dic₇, C₂₈, C₂×C₁₄, C₄.₁₀D₄, C₄.₁₀D₄, C₄⋊Q₈, C₇⋊C₈, C₅₆, Dic₁₄, C₂×Dic₇, C₂×C₂₈, C₇×Q₈, C₄².₃C₄, C₄.Dic₇, C₄×Dic₇, Dic₇⋊C₄, C₇×M₄(2), C₂×Dic₁₄, Q₈×C₁₄, C₂₈.₁₀D₄, C₇×C₄.₁₀D₄, Dic₇⋊Q₈, (C₂×Q₈).D₁₄
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, D₄, D₇, C₂²⋊C₄, D₁₄, C₂³⋊C₄, C₄×D₇, D₂₈, C₇⋊D₄, C₄².₃C₄, D₁₄⋊C₄, C₂³.₁D₁₄, (C₂×Q₈).D₁₄

Smallest permutation representation of (C₂×Q₈).D₁₄
►On 112 points

Generators in S₁₁₂

(29 43)(30 44)(31 45)(32 46)(33 47)(34 48)(35 49)(36 50)(37 51)(38 52)(39 53)(40 54)(41 55)(42 56)(57 71)(58 72)(59 73)(60 74)(61 75)(62 76)(63 77)(64 78)(65 79)(66 80)(67 81)(68 82)(69 83)(70 84)

(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 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84)(85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112)

(1 55 107 76 15 41 93 62)(2 40 94 75 16 54 108 61)(3 53 109 74 17 39 95 60)(4 38 96 73 18 52 110 59)(5 51 111 72 19 37 97 58)(6 36 98 71 20 50 112 57)(7 49 85 70 21 35 99 84)(8 34 100 69 22 48 86 83)(9 47 87 68 23 33 101 82)(10 32 102 67 24 46 88 81)(11 45 89 66 25 31 103 80)(12 30 104 65 26 44 90 79)(13 43 91 64 27 29 105 78)(14 56 106 63 28 42 92 77)

(1 22 15 8)(2 7 16 21)(3 20 17 6)(4 5 18 19)(9 14 23 28)(10 27 24 13)(11 12 25 26)(29 60 43 74)(30 59 44 73)(31 58 45 72)(32 57 46 71)(33 84 47 70)(34 83 48 69)(35 82 49 68)(36 81 50 67)(37 80 51 66)(38 79 52 65)(39 78 53 64)(40 77 54 63)(41 76 55 62)(42 75 56 61)(85 94 99 108)(86 107 100 93)(87 92 101 106)(88 105 102 91)(89 90 103 104)(95 112 109 98)(96 97 110 111)

G:=sub<Sym(112)| (29,43)(30,44)(31,45)(32,46)(33,47)(34,48)(35,49)(36,50)(37,51)(38,52)(39,53)(40,54)(41,55)(42,56)(57,71)(58,72)(59,73)(60,74)(61,75)(62,76)(63,77)(64,78)(65,79)(66,80)(67,81)(68,82)(69,83)(70,84), (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)(57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112), (1,55,107,76,15,41,93,62)(2,40,94,75,16,54,108,61)(3,53,109,74,17,39,95,60)(4,38,96,73,18,52,110,59)(5,51,111,72,19,37,97,58)(6,36,98,71,20,50,112,57)(7,49,85,70,21,35,99,84)(8,34,100,69,22,48,86,83)(9,47,87,68,23,33,101,82)(10,32,102,67,24,46,88,81)(11,45,89,66,25,31,103,80)(12,30,104,65,26,44,90,79)(13,43,91,64,27,29,105,78)(14,56,106,63,28,42,92,77), (1,22,15,8)(2,7,16,21)(3,20,17,6)(4,5,18,19)(9,14,23,28)(10,27,24,13)(11,12,25,26)(29,60,43,74)(30,59,44,73)(31,58,45,72)(32,57,46,71)(33,84,47,70)(34,83,48,69)(35,82,49,68)(36,81,50,67)(37,80,51,66)(38,79,52,65)(39,78,53,64)(40,77,54,63)(41,76,55,62)(42,75,56,61)(85,94,99,108)(86,107,100,93)(87,92,101,106)(88,105,102,91)(89,90,103,104)(95,112,109,98)(96,97,110,111)>;

G:=Group( (29,43)(30,44)(31,45)(32,46)(33,47)(34,48)(35,49)(36,50)(37,51)(38,52)(39,53)(40,54)(41,55)(42,56)(57,71)(58,72)(59,73)(60,74)(61,75)(62,76)(63,77)(64,78)(65,79)(66,80)(67,81)(68,82)(69,83)(70,84), (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)(57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112), (1,55,107,76,15,41,93,62)(2,40,94,75,16,54,108,61)(3,53,109,74,17,39,95,60)(4,38,96,73,18,52,110,59)(5,51,111,72,19,37,97,58)(6,36,98,71,20,50,112,57)(7,49,85,70,21,35,99,84)(8,34,100,69,22,48,86,83)(9,47,87,68,23,33,101,82)(10,32,102,67,24,46,88,81)(11,45,89,66,25,31,103,80)(12,30,104,65,26,44,90,79)(13,43,91,64,27,29,105,78)(14,56,106,63,28,42,92,77), (1,22,15,8)(2,7,16,21)(3,20,17,6)(4,5,18,19)(9,14,23,28)(10,27,24,13)(11,12,25,26)(29,60,43,74)(30,59,44,73)(31,58,45,72)(32,57,46,71)(33,84,47,70)(34,83,48,69)(35,82,49,68)(36,81,50,67)(37,80,51,66)(38,79,52,65)(39,78,53,64)(40,77,54,63)(41,76,55,62)(42,75,56,61)(85,94,99,108)(86,107,100,93)(87,92,101,106)(88,105,102,91)(89,90,103,104)(95,112,109,98)(96,97,110,111) );

G=PermutationGroup([[(29,43),(30,44),(31,45),(32,46),(33,47),(34,48),(35,49),(36,50),(37,51),(38,52),(39,53),(40,54),(41,55),(42,56),(57,71),(58,72),(59,73),(60,74),(61,75),(62,76),(63,77),(64,78),(65,79),(66,80),(67,81),(68,82),(69,83),(70,84)], [(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),(57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84),(85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112)], [(1,55,107,76,15,41,93,62),(2,40,94,75,16,54,108,61),(3,53,109,74,17,39,95,60),(4,38,96,73,18,52,110,59),(5,51,111,72,19,37,97,58),(6,36,98,71,20,50,112,57),(7,49,85,70,21,35,99,84),(8,34,100,69,22,48,86,83),(9,47,87,68,23,33,101,82),(10,32,102,67,24,46,88,81),(11,45,89,66,25,31,103,80),(12,30,104,65,26,44,90,79),(13,43,91,64,27,29,105,78),(14,56,106,63,28,42,92,77)], [(1,22,15,8),(2,7,16,21),(3,20,17,6),(4,5,18,19),(9,14,23,28),(10,27,24,13),(11,12,25,26),(29,60,43,74),(30,59,44,73),(31,58,45,72),(32,57,46,71),(33,84,47,70),(34,83,48,69),(35,82,49,68),(36,81,50,67),(37,80,51,66),(38,79,52,65),(39,78,53,64),(40,77,54,63),(41,76,55,62),(42,75,56,61),(85,94,99,108),(86,107,100,93),(87,92,101,106),(88,105,102,91),(89,90,103,104),(95,112,109,98),(96,97,110,111)]])

46 conjugacy classes

class	1	2A	2B	4A	4B	4C	4D	4E	4F	7A	7B	7C	8A	8B	8C	8D	14A	14B	14C	14D	14E	14F	28A	···	28F	28G	···	28L	56A	···	56L
order	1	2	2	4	4	4	4	4	4	7	7	7	8	8	8	8	14	14	14	14	14	14	28	···	28	28	···	28	56	···	56
size	1	1	2	4	4	4	28	28	56	2	2	2	8	8	56	56	2	2	2	4	4	4	4	···	4	8	···	8	8	···	8

46 irreducible representations

dim	1	1	1	1	1	1	2	2	2	2	2	2	4	4	4	8
type	+	+	+	+			+	+	+		+		+	-		-
image	C₁	C₂	C₂	C₂	C₄	C₄	D₄	D₇	D₁₄	C₄×D₇	D₂₈	C₇⋊D₄	C₂³⋊C₄	C₄².₃C₄	C₂³.₁D₁₄	(C₂×Q₈).D₁₄
kernel	(C₂×Q₈).D₁₄	C₂₈.₁₀D₄	C₇×C₄.₁₀D₄	Dic₇⋊Q₈	C₄×Dic₇	C₂×Dic₁₄	C₂×C₂₈	C₄.₁₀D₄	C₂×Q₈	C₂×C₄	C₂×C₄	C₂×C₄	C₁₄	C₇	C₂	C₁
# reps	1	1	1	1	2	2	2	3	3	6	6	6	1	2	6	3

Matrix representation of (C₂×Q₈).D₁₄ ►in GL₆(𝔽₁₁₃)

1	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	112	0
0	0	0	0	0	112

1	89	0	0	0	0
24	103	0	0	0	0
0	0	10	99	0	0
0	0	96	103	0	0
0	0	0	0	10	99
0	0	0	0	96	103

17	8	0	0	0	0
77	96	0	0	0	0
0	0	0	0	89	79
0	0	0	0	7	24
0	0	112	0	0	0
0	0	0	112	0	0

112	0	0	0	0	0
89	1	0	0	0	0
0	0	10	99	0	0
0	0	96	103	0	0
0	0	0	0	89	79
0	0	0	0	7	24

G:=sub<GL(6,GF(113))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,112,0,0,0,0,0,0,112],[1,24,0,0,0,0,89,103,0,0,0,0,0,0,10,96,0,0,0,0,99,103,0,0,0,0,0,0,10,96,0,0,0,0,99,103],[17,77,0,0,0,0,8,96,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,89,7,0,0,0,0,79,24,0,0],[112,89,0,0,0,0,0,1,0,0,0,0,0,0,10,96,0,0,0,0,99,103,0,0,0,0,0,0,89,7,0,0,0,0,79,24] >;

(C₂×Q₈).D₁₄ in GAP, Magma, Sage, TeX

(C_2\times Q_8).D_{14}

% in TeX

G:=Group("(C2xQ8).D14");

// GroupNames label

G:=SmallGroup(448,35);

// by ID

G=gap.SmallGroup(448,35);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,224,141,36,422,184,1123,794,297,136,851,18822]);

// Polycyclic

G:=Group<a,b,c,d|a^2=b^28=1,c^4=d^2=b^14,a*b=b*a,c*a*c^-1=a*b^14,a*d=d*a,c*b*c^-1=d*b*d^-1=a*b^13,d*c*d^-1=b^21*c^3>;

// generators/relations

G = (C2×Q8).D14 order 448 = 26·7

2nd non-split extension by C2×Q8 of D14 acting via D14/C7=C22

G = (C₂×Q₈).D₁₄ order 448 = 2⁶·7

2^nd non-split extension by C₂×Q₈ of D₁₄ acting via D₁₄/C₇=C₂²